Tetrahedron - Knowledge

819: 14026: 12732: 2861: 2942: 14441: 14412: 12958: 12951: 12965: 12944: 5104: 13932: 187: 13946: 2561: 12767: 12753: 14130: 8469: 10045: 13772: 12760: 3110: 12562: 12548: 5576: 12739: 4869: 12937: 12555: 5570: 170: 13100: 12534: 12746: 12978: 13599: 12527: 5253: 13093: 12910: 12541: 5393: 3040: 12328: 1636: 3150: 3031: 13086: 13072: 13058: 13800: 13779: 13592: 13079: 13065: 13051: 12903: 12917: 14421: 12896: 6709: 2935: 5303: 13793: 13786: 13585: 13578: 13571: 12889: 12777: 12701: 5494: 5002: 3175: 13807: 12708: 9662: 13564: 13425: 12515: 5500: 13918: 7791: 12501: 158: 14662: 13432: 13411: 13397: 12480: 2968: 2829: 14118: 12687: 13828: 13620: 12508: 12680: 13849: 13842: 13835: 13641: 13634: 13627: 1175: 5177: 12694: 2217: 13613: 13606: 14503: 2836: 13439: 12494: 317: 11912: 13821: 13814: 13418: 13390: 12487: 13404: 14336: 12715: 6528: 814: 10040:{\displaystyle {\begin{vmatrix}-1&\cos {(\alpha _{12})}&\cos {(\alpha _{13})}&\cos {(\alpha _{14})}\\\cos {(\alpha _{12})}&-1&\cos {(\alpha _{23})}&\cos {(\alpha _{24})}\\\cos {(\alpha _{13})}&\cos {(\alpha _{23})}&-1&\cos {(\alpha _{34})}\\\cos {(\alpha _{14})}&\cos {(\alpha _{24})}&\cos {(\alpha _{34})}&-1\\\end{vmatrix}}=0\,} 1103: 14432: 7469: 1996: 9576:: the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of 7454: 6056: 12323:{\displaystyle {\begin{aligned}C&=A^{-1}B&{\text{where}}&\ &A=\left({\begin{matrix}\left^{T}\\\left^{T}\\\left^{T}\end{matrix}}\right)&\ &{\text{and}}&\ &B={\frac {1}{2}}\left({\begin{matrix}\|x_{1}\|^{2}-\|x_{0}\|^{2}\\\|x_{2}\|^{2}-\|x_{0}\|^{2}\\\|x_{3}\|^{2}-\|x_{0}\|^{2}\end{matrix}}\right)\\\end{aligned}}} 10886: 1631:{\displaystyle {\begin{aligned}{\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+{\frac {16R^{4}}{9}}&=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+{\frac {2R^{2}}{3}}\right)^{2},\\4\left(a^{4}+d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}\right)&=\left(a^{2}+d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}\right)^{2}.\end{aligned}}} 4833:) copies of itself to tile space. The cube can be dissected into six 3-orthoschemes, three left-handed and three right-handed (one of each at each cube face), and cubes can fill space, so the characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense. (The characteristic orthoscheme of the cube is one of the 1835: 2896: 6863: 641: 9347: 6704:{\displaystyle 36\cdot V^{2}={\begin{vmatrix}\mathbf {a^{2}} &\mathbf {a} \cdot \mathbf {b} &\mathbf {a} \cdot \mathbf {c} \\\mathbf {a} \cdot \mathbf {b} &\mathbf {b^{2}} &\mathbf {b} \cdot \mathbf {c} \\\mathbf {a} \cdot \mathbf {c} &\mathbf {b} \cdot \mathbf {c} &\mathbf {c^{2}} \end{vmatrix}}} 939: 14109:, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is to serve as a reference to pilots indicating wind direction. 7786:{\displaystyle 288\cdot V^{2}={\begin{vmatrix}0&1&1&1&1\\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\\1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\\1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2}&0\end{vmatrix}}} 3213:

in which all edges are mutually perpendicular. In a 3-dimensional orthoscheme, the tree consists of three perpendicular edges connecting all four vertices in a linear path that makes two right-angled turns. The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, so another

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yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides,

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The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, five is the minimum number of tetrahedra required to compose a cube. To see this, starting from a base tetrahedron with 4 vertices, each added tetrahedra adds at most 1 new vertex, so at least 4 more must be

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The iterative LEB of the regular tetrahedron has been shown to produce only 8 similarity classes. Furthermore, in the case of nearly equilateral tetrahedra where their two longest edges are not connected to each other, and the ratio between their longest and their shortest edge is less than or equal

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Tetrahedra subdivision is a process used in computational geometry and 3D modeling to divide a tetrahedron into several smaller tetrahedra. This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and

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Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such

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is the set of tetrahedra with the same geometric shape, regardless of their specific position, orientation, and scale. So, any two tetrahedra belonging to the same similarity class may be transformed to each other by an affine transformation. The outcome of having a limited number of similarity

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If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of

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to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the

7155: 5796: 10670: 6513: 8464: 7229: 2212:{\displaystyle {\begin{aligned}\left({\sqrt {\frac {8}{9}}},0,-{\frac {1}{3}}\right),&\quad \left(-{\sqrt {\frac {2}{9}}},{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right),\\\left(-{\sqrt {\frac {2}{9}}},-{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right),&\quad (0,0,1)\end{aligned}}} 10419:

is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.

5646:, which identifies the longest edge of the tetrahedron and bisects it at its midpoint, generating two new, smaller tetrahedra. When this process is repeated multiple times, bisecting all the tetrahedra generated in each previous iteration, the process is called iterative LEB. 1701: 10398: 8998: 11383: 5789:

is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of these faces. Another way is by dissecting a triangular prism into three pieces.

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It has 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group

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classes in iterative subdivision methods is significant for computational modeling and simulation. It reduces the variability in the shapes and sizes of generated tetrahedra, preventing the formation of highly irregular elements that could compromise simulation results.

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is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid. Other names for the same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron.

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Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups. For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is

2888:. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become 6140: 2417: 9650:

twelve-point center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.

809:{\displaystyle {\begin{aligned}\arccos \left({\frac {1}{3}}\right)&=\arctan \left(2{\sqrt {2}}\right)\approx 70.529^{\circ },\\\arccos \left(-{\frac {1}{3}}\right)&=2\arctan \left({\sqrt {2}}\right)\approx 109.471^{\circ }.\end{aligned}}} 9634:, one third of the way from the Monge point toward each of the four vertices. Finally it passes through the four base points of orthogonal lines dropped from each Euler point to the face not containing the vertex that generated the Euler point. 11647: 6234: 463: 500: 8875: 7834:

is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. A negative value of the determinant means that a tetrahedron cannot be constructed with the given distances. This formula, sometimes called

6308: 4527:, first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 14235:, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds. 8134: 14480:

computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand.

1098:{\displaystyle {\begin{aligned}R={\frac {\sqrt {6}}{4}}a,&\qquad r={\frac {1}{3}}R={\frac {a}{\sqrt {24}}},\\r_{\mathrm {M} }={\sqrt {rR}}={\frac {a}{\sqrt {8}}},&\qquad r_{\mathrm {E} }={\frac {a}{\sqrt {6}}}.\end{aligned}}} 10148:. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space, for an example in 8319: 11863:

The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron. With this definition, the circumcenter

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to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see

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we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustrated

10084:, and at which the angles subtended by opposite edges are equal. A solid angle of π sr is one quarter of that subtended by all of space. When all the solid angles at the vertices of a tetrahedron are smaller than π sr, 8880: 7449:{\displaystyle 6\cdot V=\left|\det \left({\begin{matrix}a_{1}&b_{1}&c_{1}&d_{1}\\a_{2}&b_{2}&c_{2}&d_{2}\\a_{3}&b_{3}&c_{3}&d_{3}\\1&1&1&1\end{matrix}}\right)\right|\,.} 7006: 9552: 6319: 1864: 11524:

This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. Since the four subtetrahedra fill the volume, we have

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The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters,

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The tetrahedron with regular faces is a solution to an old puzzle asking to form four equilateral triangles using six unbroken matchsticks. The solution places the matchsticks along the edges of a tetrahedron.

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An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.

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This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is

4923:. The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be 10259: 6051:{\displaystyle {\begin{aligned}\mathbf {a} &=(a_{1},a_{2},a_{3}),\\\mathbf {b} &=(b_{1},b_{2},b_{3}),\\\mathbf {c} &=(c_{1},c_{2},c_{3}),\\\mathbf {d} &=(d_{1},d_{2},d_{3}).\end{aligned}}} 5120:

It gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group

6998: 6954: 6910: 10881:{\displaystyle \Delta _{i}^{2}=\Delta _{j}^{2}+\Delta _{k}^{2}+\Delta _{l}^{2}-2(\Delta _{j}\Delta _{k}\cos \theta _{il}+\Delta _{j}\Delta _{l}\cos \theta _{ik}+\Delta _{k}\Delta _{l}\cos \theta _{ij})} 12333:

In contrast to the centroid, the circumcenter may not always lay on the inside of a tetrahedron. Analogously to an obtuse triangle, the circumcenter is outside of the object for an obtuse tetrahedron.

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of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid not mapped to itself by

11518: 4947:, ) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges. 4139: 4049: 3959: 3783: 3558: 2538: 11917: 9008: 8885: 8144: 5801: 2274: 2269: 2001: 1706: 1180: 944: 646: 10953: 1830:{\displaystyle {\begin{aligned}\arccos \left({\frac {23}{27}}\right)&={\frac {\pi }{2}}-3\arcsin \left({\frac {1}{3}}\right)\\&=3\arccos \left({\frac {1}{3}}\right)-\pi \end{aligned}}} 4232: 11214: 2564:

The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) in the symmetry group of the regular tetrahedron

6858:{\displaystyle {\begin{cases}\mathbf {a} \cdot \mathbf {b} =ab\cos {\gamma },\\\mathbf {b} \cdot \mathbf {c} =bc\cos {\alpha },\\\mathbf {a} \cdot \mathbf {c} =ac\cos {\beta }.\end{cases}}} 3828: 3701: 9342:{\displaystyle {\begin{aligned}X=(w-U+v)(U+v+w),&\quad x=(U-v+w)(v-w+U),\\Y=(u-V+w)(V+w+u),&\quad y=(V-w+u)(w-u+V),\\Z=(v-W+u)(W+u+v),&\quad z=(W-u+v)\,(u-v+W).\end{aligned}}} 368: 2770:

reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion (

2604: 9449: 4814: 4781: 4748: 4715: 4682: 4649: 4616: 4583: 4524: 4491: 4432: 4399: 4366: 4329: 4274: 8719: 2253: 7993: 6064: 495: 5743: 2527: 3865: 3738: 3628: 3501: 3462: 934: 902: 3230:, which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length 5688: 15591:, Dover Publications, 2003 (orig. ed. 1962), p. 107. Note however that Sierpiński repeats an erroneous calculation of the volume of the Heronian tetrahedron example above. 13333: 13323: 13313: 13304: 13294: 13284: 13275: 13255: 13246: 13217: 13207: 13178: 13149: 13139: 13110: 11528: 10624: 6175: 5454: 5444: 5434: 5338: 5328: 5039: 3591: 3326: 3248: 2664: 2465: 2444: 497:. The volume of a regular tetrahedron can be ascertained similarly as the other pyramids, one-third of the base and its height. Because the base is an equilateral, it is: 10567: 4186: 14190:), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called 13265: 13236: 13226: 13197: 13188: 13168: 13159: 13130: 13120: 10661: 9374:, after Jun Murakami and Masakazu Yano. However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist. 5348: 5059: 5049: 4899:. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point. (The 3374: 3364: 3354: 3346: 3336: 3311:

instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once). The characteristic tetrahedron of the cube is an example of a

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as a rectified tetrahedron. The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.

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This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group

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This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group

5054: 5044: 4099: 4009: 3919: 3369: 3359: 3341: 3331: 3291: 3281: 3263: 3253: 2480: 2470: 6245: 10594: 10540: 10513: 10486: 10459: 2694: 2638: 1170: 14771:. This angle (in radians) is also the length of the circular arc on the unit sphere resulting from centrally projecting one edge of the tetrahedron to the sphere. 14327:

Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).

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to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-

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Five tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. These points are then attached to each other and

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of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle.

261:, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., 4848:. Regular tetrahedra, however, cannot fill space by themselves (moreover, it is not scissors-congruent to any other polyhedra which can fill the space, see 624:{\displaystyle V={\frac {1}{3}}\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)\cdot {\frac {\sqrt {6}}{3}}a={\frac {a^{3}}{6{\sqrt {2}}}}\approx 0.118a^{3}.} 15314: 818: 13533: 11695: 10667:

for a tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation:

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of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies one third of the way from the Monge point

638:—the angle between two planar—and its angle between lines from the center of a regular tetrahedron between two vertices is respectively: 12406:

A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48.

7150:{\displaystyle V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}},\,} 2884:. When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a 6508:{\displaystyle {\begin{cases}\mathbf {a} =(a_{1},a_{2},a_{3}),\\\mathbf {b} =(b_{1},b_{2},b_{3}),\\\mathbf {c} =(c_{1},c_{2},c_{3}),\end{cases}}} 8459:{\displaystyle V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}}} 3113:

Kepler's drawing of a regular tetrahedron inscribed in a cube, and one of the four trirectangular tetrahedra that surround it, filling the cube.

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representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a single

4888:. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as 2262:

in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the

1688: 14012:(itself is the dual of Szilassi polyhedron) and the tetrahedron are the only two known polyhedra in which every diagonal lies on the sides. 12616: 4935:

The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a

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Lindelof, L. (1867). "Sur les maxima et minima d'une fonction des rayons vecteurs menés d'un point mobile à plusieurs centres fixes".

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constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.

10393:{\displaystyle \sin \angle OAB\cdot \sin \angle OBC\cdot \sin \angle OCA=\sin \angle OAC\cdot \sin \angle OCB\cdot \sin \angle OBA.\,} 8993:{\displaystyle {\begin{aligned}p={\sqrt {xYZ}},&\quad q={\sqrt {yZX}},\\r={\sqrt {zXY}},&\quad s={\sqrt {xyz}},\end{aligned}}} 16673: 3378:

by its planes of symmetry. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.

11378:{\displaystyle {\frac {1}{r_{1}^{2}}}+{\frac {1}{r_{2}^{2}}}+{\frac {1}{r_{3}^{2}}}+{\frac {1}{r_{4}^{2}}}\leq {\frac {2}{r^{2}}},} 6959: 6915: 6871: 17658: 14029:

An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.

13994:. A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull. 13526: 2710:

rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together

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can be grouped into two groups of four, each forming a regular tetrahedron, showing one of the two tetrahedra in the cube. The

1957:{\displaystyle \left(\pm 1,0,-{\frac {1}{\sqrt {2}}}\right)\quad {\mbox{and}}\quad \left(0,\pm 1,{\frac {1}{\sqrt {2}}}\right)} 14008:

and the tetrahedron are the only two known polyhedra in which each face shares an edge with each other face. Furthermore, the

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There are molecules with the shape based on four nearby atoms whose bonds form the sides of a tetrahedral structure, such as

11437: 8309:{\displaystyle {\begin{aligned}X&=b^{2}+c^{2}-x^{2},\\Y&=a^{2}+c^{2}-y^{2},\\Z&=a^{2}+b^{2}-z^{2}.\end{aligned}}} 3190:. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a 10403:

One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.

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A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as

13028: 11065:{\displaystyle PA\cdot F_{\mathrm {a} }+PB\cdot F_{\mathrm {b} }+PC\cdot F_{\mathrm {c} }+PD\cdot F_{\mathrm {d} }\geq 9V.} 3796: 3669: 3077: 4872:

For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra. They can be seen as points on and within a cube.

4107: 4017: 3927: 3751: 3526: 3322:. There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron 17093: 16287: 15951: 15056: 9599:

of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all

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The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume:

14025: 2927:, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. 13732: 13519: 4531:

which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges

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define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a

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The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.

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The absolute value of the scalar triple product can be represented as the following absolute values of determinants:

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together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.

15704: 15329: 13683: 10198: 10183: 9611:). The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the 4936: 4853: 4845: 2701: 234:, who associated those four solids with nature. The regular tetrahedron was considered as the classical element of 13931: 9566:

and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes.

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are the lengths of the tetrahedron's edges as in the following image. Here, the first three form a triangle, with

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Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess.

2795:, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A 2578: 15284:

Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54

14928: 13945: 211:. In other words, all of its faces are the same size and shape (congruent) and all edges are the same length. A 15996: 15982: 15012: 14986: 14866: 14768: 14141: 14058: 9424: 1861:, defining the four vertices of a tetrahedron with edge length 2, centered at the origin, and two-level edges: 12346: 6135:{\textstyle {\frac {1}{6}}\det(\mathbf {a} -\mathbf {d} ,\mathbf {b} -\mathbf {d} ,\mathbf {c} -\mathbf {d} )} 3475: 3436: 288: 118:(any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". 17695: 16585: 15240: 14882: 13725: 10429: 10228: 9626:

of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the

7460: 4788: 4755: 4722: 4689: 4656: 4623: 4590: 4557: 4498: 4465: 4406: 4373: 4340: 4303: 4248: 114:

base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a

12375:. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces are 9630:

and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute

3018:

by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.

2412:{\displaystyle {\begin{aligned}(1,1,1),&\quad (1,-1,-1),\\(-1,1,-1),&\quad (-1,-1,1).\end{aligned}}} 17700: 17690: 15663: 15157: 14627: 14464: 14062: 6143: 5406:

It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the

4924: 4244:

If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths

2222: 16879: 16820: 15303:. Dept of Mathematics, Chulalongkorn University, Bangkok. Archived from the original on 27 February 2009. 13959: 13937: 3571: 3191: 2495: 468: 9370:

of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the

4169: 3841: 3714: 3604: 17685: 16909: 16869: 16475: 16021: 15772:

Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography

15558: 14739: 14678: 13987: 13951: 13891: 12602: 10189:

On otherhand, several irregular tetrahedra are known, of which copies can tile space, for instance the

9452: 4849: 3118: 3104: 2988: 2972: 296: 10178:

claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a

3091:

has concurrent cevians that join the vertices to the points of contact of the opposite faces with the

910: 878: 16904: 16899: 14599: 14246: 13678: 13673: 11642:{\displaystyle V={\frac {1}{3}}A_{1}r+{\frac {1}{3}}A_{2}r+{\frac {1}{3}}A_{3}r+{\frac {1}{3}}A_{4}r} 9653:

The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron.

9371: 6229:{\displaystyle 6\cdot V={\begin{Vmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{Vmatrix}}} 2573: 458:{\displaystyle A=4\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)=a^{2}{\sqrt {3}}\approx 1.732a^{2}.} 6729: 6328: 17680: 17116: 16469: 16000: 15986: 14603: 14459: 14253: 13703:

The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3

12861: 12856: 12371:

There exist tetrahedra having integer-valued edge lengths, face areas and volume. These are called

9577: 5661: 3058: 2954: 2920: 2700:(in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the 262: 235: 15602: 14309:

of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how

10599: 3135:

discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.

2647: 2422: 17086: 17010: 17005: 16884: 16790: 16597: 16525: 16445: 16280: 15902: 15838: 14845: 14540: 14530: 14473: 14302: 14050: 14009: 12874: 12866: 10545: 9608: 9607:

of the tetrahedron. In addition the four medians are divided in a 3:1 ratio by the centroid (see

2949: 2806: 2550: 15109:"Finite number of similarity classes in Longest Edge Bisection of nearly equilateral tetrahedra" 14683: 12731: 10629: 9381:, and the distance between the edges is defined as the distance between the two skew lines. Let 3198: 300: 16874: 16815: 16805: 16750: 16531: 15814: 15766: 15582: 14595: 14395: 14391: 14385: 13966:

of five tetrahedra has been known for hundreds of years. It comes up regularly in the world of

13917: 12851: 10076:

found that, corresponding to any given tetrahedron is a point now known as an isogonic center,

8870:{\displaystyle V={\frac {\sqrt {\,(-p+q+r+s)\,(p-q+r+s)\,(p+q-r+s)\,(p+q+r-s)}}{192\,u\,v\,w}}} 7836: 7459:

Given the distances between the vertices of a tetrahedron the volume can be computed using the

6303:{\displaystyle 6\cdot V={\begin{Vmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{Vmatrix}}} 4900: 2995: 2455: 280:, a regular tetrahedron with four triangular pyramids attached to each of its faces. i.e., its 15886: 14876: 4975: 4895:

For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a

17630: 17623: 17616: 16894: 16810: 16765: 16253: 16248: 15882: 15587: 15479: 15167: 15151: 14938: 14510: 14192: 14151: 14070: 13356: 13346: 12668: 8129:{\displaystyle V={\frac {\sqrt {4a^{2}b^{2}c^{2}-a^{2}X^{2}-b^{2}Y^{2}-c^{2}Z^{2}+XYZ}}{12}}} 7840: 5296: 4067: 3977: 3887: 2904: 2792: 2263: 1858: 273: 269: 30: 17155: 17133: 17121: 15842: 15232: 17287: 17234: 16854: 16780: 16728: 16173: 15776: 15637: 15370: 14733: 14713: 14550: 14078: 13471: 13371: 13039: 12784: 12660: 12650: 12372: 12366: 10572: 10518: 10491: 10464: 10437: 10171: 4915: 4889: 4844:

A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in the

3312: 2860: 2672: 2616: 1148: 905: 250: 208: 10215: 9351:

Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron

4439:

of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is

8: 17705: 17675: 17642: 17541: 17291: 17020: 16889: 16864: 16849: 16785: 16733: 16190: 15460: 15046: 15044: 14922: 14728: 14520: 14306: 14005: 13963: 13489:

This polyhedron is topologically related as a part of sequence of regular polyhedra with

13476: 13461: 13451: 12655: 10164: 10080:, at which the solid angles subtended by the faces are equal, having a common value of π 9656:

There is a relation among the angles made by the faces of a general tetrahedron given by

4885: 3210: 1846: 277: 246: 15780: 15641: 14892: 1973: 17511: 17461: 17411: 17368: 17338: 17298: 17261: 17079: 17035: 17000: 16859: 16754: 16703: 16642: 16519: 16513: 16273: 16182: 16064: 16056: 16006: 15807: 15387: 15308: 15267: 15249: 14723: 14560: 14082: 14054: 14034: 12985: 12589: 12446: 12415: 12376: 10209: 9458: 9404: 9384: 8699: 8679: 8659: 8639: 8619: 8599: 8579: 8559: 8539: 8519: 8499: 8479: 7973: 7953: 7933: 7913: 7893: 7873: 7853: 7844: 5772: 5748: 5487: 5386: 4877: 4838: 4535: 4443: 4281: 3641: 3408: 2999: 2941: 1128: 1108: 855: 832: 348: 328: 292: 107: 93: 15920: 15179: 15072: 15041: 13490: 10073: 5064: 3187: 3007: 3003: 2490: 17650: 17015: 16825: 16800: 16744: 16632: 16556: 16508: 16481: 16451: 16258:

that also includes a description of a "rotating ring of tetrahedra", also known as a

16231: 16208: 16161: 16083: 16068: 16025: 15955: 15917: 15818: 15784: 15741: 15608: 15422: 15365: 15130: 15033: 15016: 14820: 14764: 14583: 14440: 14411: 14074: 13998: 13881: 13668: 13376: 13015: 12957: 12950: 12451: 12357:

The sum of the areas of any three faces is greater than the area of the fourth face.

9623: 9363: 5103: 4985: 4528: 3011: 2667: 1667: 89: 78: 12964: 10626:

be the dihedral angle between the two faces of the tetrahedron adjacent to the edge

5266:. An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ). 17654: 17219: 17208: 17197: 17186: 17177: 17168: 17107: 17103: 16954: 16637: 16617: 16439: 16151: 16115: 16048: 15898: 15645: 15541: 15414: 15405:

Rassat, André; Fowler, Patrick W. (2004). "Is There a "Most Chiral Tetrahedron"?".

15379: 15259: 15120: 15028: 14718: 14570: 14398:

to explain the formation of the Earth, was popular through the early 20th century.

14346: 13991: 13923: 13886: 13876: 13498: 12943: 12929: 12645: 11848:{\displaystyle R={\frac {\sqrt {(aA+bB+cC)(aA+bB-cC)(aA-bB+cC)(-aA+bB+cC)}}{24V}}.} 10149: 10097: 10069: 9600: 9359: 8476:

The volume of a tetrahedron can be ascertained by using the Heron formula. Suppose

5521: 5407: 4980: 3092: 3081: 2916: 2889: 304: 227: 176: 126: 122: 16098: 15466: 8468: 186: 17244: 17229: 16591: 16503: 16498: 16463: 16418: 16408: 16398: 16393: 16234: 16169: 16111: 16077: 15770: 14823: 14591: 14455: 14365: 14361: 12766: 12752: 12424: 12342: 10153: 5766: 5575: 5129: 5033: 5022: 4834: 3132: 3109: 2697: 2611: 2554: 2459: 2447: 2259: 1663: 242: 230:. Known since antiquity, the Platonic solid is named after the Greek philosopher 82: 74: 70: 16015: 15843:"William Lowthian Green and his Theory of the Evolution of the Earth's Features" 14364:, with the number rolled appearing around the bottom or on the top vertex. Some 14129: 13771: 12759: 10415:

triangles, there are four such constraints on sums of angles, and the number of

6166:

of the volume of any parallelepiped that shares three converging edges with it.

5569: 5302: 4911:

which is multiplied by mirror reflections into the vertices of the polyhedron.)

4868: 2895: 2560: 2537: 17594: 16775: 16698: 16647: 16550: 16413: 16403: 16156: 16139: 16100:

What has the Volume of a Tetrahedron to do with Computer Programming Languages?

14791:𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the 14651: 14647: 14631: 13754: 12738: 12561: 12547: 12441: 10664: 9563: 9367: 6147: 4990: 3227: 3154: 2569: 635: 299:, which is a tessellation. Some tetrahedra that are not regular, including the 238:, because of his interpretation of its sharpest corner being most penetrating. 223: 193: 15125: 15108: 14242:

of mixtures of chemical substances are represented graphically as tetrahedra.

14001:

is another polyhedron with four faces, but it does not have triangular faces.

12936: 12554: 10104:, of the vertices. In the event that the solid angle at one of the vertices, 5493: 5252: 5001: 219:. There are eight convex deltahedra, one of which is the regular tetrahedron. 17669: 17611: 17499: 17492: 17485: 17449: 17442: 17435: 17399: 17392: 16980: 16836: 16770: 16568: 16562: 16457: 16388: 16378: 16165: 15860: 15802: 15134: 14693: 14587: 14469: 14298: 14239: 14197: 14147: 13704: 13366: 13099: 13092: 12836: 12819: 12816:

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual

12745: 12533: 9569: 6142:, or any other combination of pairs of vertices that form a simply connected 5499: 5392: 5263: 4881: 3054: 2924: 2799: 2451: 1842: 169: 15545: 13598: 13085: 13071: 13057: 12977: 12526: 3149: 3039: 3030: 125:, a tetrahedron can be folded from a single sheet of paper. It has two such 17551: 16383: 16259: 15788: 15426: 15418: 14792: 14607: 14339: 14261: 14066: 14042: 13983: 13979: 13971: 13871: 13799: 13778: 13591: 13481: 13078: 13064: 13050: 12970: 12909: 12902: 12540: 10234: 10179: 9595:

and a line segment joining the midpoints of two opposite edges is called a

5601: 5197: 4896: 4830: 827: 133: 20: 12916: 2744:

rotation by an angle of 180° such that an edge maps to the opposite edge:

157: 17560: 17521: 17471: 17421: 17378: 17348: 16423: 16357: 16347: 16337: 14744: 14707: 14639: 14373: 14357: 14353: 14322: 14277: 14273: 14134: 14038: 14037:, complicated three-dimensional shapes are commonly broken down into, or 13975: 13663: 6059: 4837:, a family of space-filling tetrahedra. All space-filling tetrahedra are 3206: 3126: 3061:. When only one pair of opposite edges are perpendicular, it is called a 2607: 1967: 1695: 216: 14420: 13792: 13785: 13584: 13577: 13570: 12895: 12888: 12573:

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform

9547:{\displaystyle V={\frac {d|(\mathbf {a} \times \mathbf {(b-c)} )|}{6}}.} 7843:

in the 15th century, as a three-dimensional analogue of the 1st century

2934: 2778:): the rotations correspond to those of the cube about face-to-face axes 17546: 17530: 17480: 17430: 17387: 17357: 17271: 17045: 16933: 16723: 16690: 16608: 16362: 16352: 16342: 16327: 16317: 16296: 16060: 15628:

Brittin, W. E. (1945). "Valence angle of the tetrahedral carbon atom".

15391: 15271: 14623: 14094: 14046: 13856: 13658: 13351: 13019: 12881: 12700: 12638: 12631: 10216:

A law of sines for tetrahedra and the space of all shapes of tetrahedra

9616: 9378: 5642:

computer graphics. One of the commonly used subdivision methods is the

5288: 4857: 3162: 3144: 631:

Its volume can also be obtained by dissecting a cube into three parts.

258: 212: 66: 24: 15649: 15107:

Trujillo-Pino, Agustín; Suárez, Jose Pablo; Padrón, Miguel A. (2024).

13424: 12707: 12514: 10167:

gives two more regular compounds, containing five and ten tetrahedra.

215:

polyhedron in which all of its faces are equilateral triangles is the

17602: 17516: 17466: 17416: 17373: 17343: 17312: 17040: 17030: 16975: 16959: 16795: 16622: 16239: 15925: 14828: 14155: 13431: 13410: 13396: 13014:

A truncation process applied to the tetrahedron produces a series of

12776: 12686: 12596: 12576: 12500: 12479: 10175: 10081: 9352: 3153:

A space-filling tetrahedral disphenoid inside a cube. Two edges have

2881: 2696:—the identity and 11 proper rotations—with the following 1857:

One way to construct a regular tetrahedron is by using the following

1838: 873: 16052: 15383: 15263: 14117: 14061:. These methods have wide applications in practical applications in 13827: 13806: 13619: 13563: 12679: 12507: 3174: 2967: 2782: 17576: 17331: 17327: 17254: 16926: 16658: 16322: 15506: 14477: 14369: 14335: 14283: 14175: 14122: 14106: 13848: 13841: 13834: 13648: 13640: 13633: 13626: 12693: 12431: 11229: 10088:

lies inside the tetrahedron, and because the sum of distances from

9588: 6146:. Comparing this formula with that used to compute the volume of a 5176: 4856:

fills space with alternating regular tetrahedron cells and regular

3318:

Every regular polytope, including the regular tetrahedron, has its

3015: 2976: 2899:

A tetragonal disphenoid viewed orthogonally to the two green edges.

1675: 850: 281: 137: 115: 42: 15254: 15093:, p. 63, §4.3 Rotation groups in two dimensions; notion of a 14661: 14349:, dating from 2600 BC, was played with a set of tetrahedral dice. 13612: 9572:

found a center that exists in every tetrahedron, now known as the

3350:

is subdivided into 24 instances of its characteristic tetrahedron

2828: 17585: 17555: 17322: 17317: 17308: 17249: 17050: 17025: 16627: 15969:

Bottema, O. (1969). "A Theorem of Bobillier on the Tetrahedron".

14697: 14688: 14310: 14294: 14159: 13967: 12493: 10145: 5262:

Its only isometry is the identity, and the symmetry group is the

3202: 141: 111: 96: 15948:

A Mathematical Space Odyssey: Solid Geometry in the 21st Century

13605: 9401:

be the distance between the skew lines formed by opposite edges

5690:, the iterated LEB produces no more than 37 similarity classes. 4332:

around its exterior right-triangle face (the edges opposite the

3244:, so all its edges are edges or diagonals of the cube. The cube 1172:

from an arbitrary point in 3-space to its four vertices, it is:

295:

in the ratio of two tetrahedra to one octahedron, they form the

163:

Regular tetrahedron, described as the classical element of fire.

17525: 17475: 17425: 17382: 17352: 17303: 17239: 15600: 14905: 14903: 14901: 14787:) uses the greek letter 𝝓 (phi) to represent one of the three 14626:

of the tetrahedron (comprising the vertices and edges) forms a

14502: 14287: 13438: 3073: 2984: 2885: 2869: 2835: 15849:. Vol. XXV. Geological Publishing Company. pp. 1–10. 13820: 13813: 13417: 13389: 12714: 12486: 9556: 7970:

be those of the opposite edges. The volume of the tetrahedron

5530:. A digonal disphenoid has Schläfli symbol { }∨{ }. 2446:, centered at the origin. For the other tetrahedron (which is 316: 16125:

Lee, Jung Rye (1997). "The Law of Cosines in a Tetrahedron".

14219: 14200:

between any two vertices of a perfect tetrahedron is arccos(−

13403: 12581: 11673:

be the lengths of the three edges that meet at a vertex, and

10174:

by themselves, although this result seems likely enough that

6058:

The volume of a tetrahedron can be ascertained in terms of a

1641: 231: 16265: 16205:

The Routledge International Handbook of Innovation Education

14898: 14630:, with 4 vertices, and 6 edges. It is a special case of the 13990:, in which the ten tetrahedra are arranged as five pairs of 6993:{\displaystyle c={\begin{Vmatrix}\mathbf {c} \end{Vmatrix}}} 6949:{\displaystyle b={\begin{Vmatrix}\mathbf {b} \end{Vmatrix}}} 6905:{\displaystyle a={\begin{Vmatrix}\mathbf {a} \end{Vmatrix}}} 17275: 16718: 16249:

Free paper models of a tetrahedron and many other polyhedra

15508:

Spherical Trigonometry: For the Use of Colleges and Schools

13866: 13456: 11115:

to the faces, and suppose the faces have equal areas, then

9645:

towards the circumcenter. Also, an orthogonal line through

7180:, is the angle between the two edges connecting the vertex 6851: 6501: 6150:, we conclude that the volume of a tetrahedron is equal to 2546: 15915: 15233:"Altitudes of a tetrahedron and traceless quadratic forms" 14158:) at the four corners of a tetrahedron. For instance in a 9587:

A line segment joining a vertex of a tetrahedron with the

4914:

Among the Goursat tetrahedra which generate 3-dimensional

3053:

If all three pairs of opposite edges of a tetrahedron are

1970:, centroid at the origin, with lower face parallel to the 136:) on which all four vertices lie, and another sphere (the 16140:"On the volume of a hyperbolic and spherical tetrahedron" 15330:"Déterminants sphérique et hyperbolique de Cayley-Menger" 15106: 14779: 14777: 14260:-butyltetrahedrane, known derivative of the hypothetical 11388:

with equality if and only if the tetrahedron is regular.

5139:. A triangular pyramid has Schläfli symbol {3}∨( ). 268:

The tetrahedron is yet related to another two solids: By

15731:"Radial and Pruned Tetrahedral Interpolation Techniques" 15212: 14431: 14249:

are represented graphically on a two-dimensional plane.

12823:, containing 6 vertices, in two sets of colinear edges. 12341:

The tetrahedron's center of mass can be computed as the

10221: 7910:

be the lengths of three edges that meet at a point, and

222:

The regular tetrahedron is also one of the five regular

15861:"Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron" 15607:, U. S. Government Printing Office, p. 13-10, 15532:

centroidal Voronoi tessellation and its applications",

11513:{\displaystyle r={\frac {3V}{A_{1}+A_{2}+A_{3}+A_{4}}}} 9455:. Then another formula for the volume of a tetrahedron 3300:

four different ways, with all six surrounding the same

2855: 2568:

The regular tetrahedron has 24 isometries, forming the

226:, a set of polyhedrons in which all of their faces are 15190: 15188: 14774: 14736:– constructed by joining two tetrahedra along one face 12150: 11971: 9671: 7497: 7258: 6974: 6930: 6886: 6556: 6266: 6196: 6067: 5793:

Given the vertices of a tetrahedron in the following:

4794: 4761: 4728: 4695: 4662: 4629: 4596: 4563: 4504: 4471: 4412: 4379: 4346: 4309: 4254: 4206: 4113: 4023: 3933: 3846: 3801: 3757: 3719: 3674: 3609: 3532: 3307:

cube diagonal. The cube can also be dissected into 48

3178:

A cube dissected into six characteristic orthoschemes.

2225: 1910: 471: 196:

is left, where the five edge angles do not quite meet.

132:

For any tetrahedron there exists a sphere (called the

81:. The tetrahedron is the simplest of all the ordinary 14993: 11915: 11698: 11531: 11440: 11254: 11124: 10956: 10673: 10632: 10602: 10575: 10548: 10521: 10494: 10467: 10440: 10262: 9665: 9481: 9461: 9427: 9407: 9387: 9006: 8883: 8722: 8702: 8682: 8662: 8642: 8622: 8602: 8582: 8562: 8542: 8522: 8502: 8482: 8322: 8142: 7996: 7976: 7956: 7936: 7916: 7896: 7876: 7856: 7472: 7232: 7009: 6962: 6918: 6874: 6723: 6531: 6322: 6248: 6178: 5799: 5775: 5751: 5709: 5664: 5636: 5482:

Generalized disphenoids (2 pairs of equal triangles)

4930: 4791: 4758: 4725: 4692: 4659: 4626: 4593: 4560: 4538: 4501: 4468: 4446: 4409: 4376: 4343: 4306: 4284: 4251: 4204: 4172: 4110: 4070: 4020: 3980: 3930: 3890: 3844: 3799: 3754: 3717: 3672: 3644: 3607: 3574: 3529: 3478: 3439: 3411: 3044:

Tetrahedral symmetries shown in tetrahedral diagrams

2675: 2650: 2619: 2581: 2498: 2425: 2272: 1999: 1976: 1867: 1704: 1178: 1151: 1131: 1111: 942: 913: 881: 858: 835: 822:

Regular tetrahedron ABCD and its circumscribed sphere

644: 503: 371: 351: 331: 16229: 14818: 10919:, and for which the areas of the opposite faces are 4925:

dissected into characteristic tetrahedra of the cube

4134:{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408} 4044:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707} 3954:{\displaystyle {\sqrt {\tfrac {3}{2}}}\approx 1.225} 3778:{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577} 3553:{\displaystyle {\sqrt {\tfrac {4}{3}}}\approx 1.155} 2983:

Regular tetrahedra can be stacked face-to-face in a

325:

Given that the regular tetrahedron with edge length

15378:(5). Mathematical Association of America: 227–243. 15200: 15185: 15180:"Simplex Volumes and the Cayley-Menger Determinant" 13970:. Joining the twenty vertices would form a regular 9377:Any two opposite edges of a tetrahedron lie on two 3222:tetrahedron because it contains four right angles. 3157:

of 90°, and four edges have dihedral angles of 60°.

365:is four times the area of an equilateral triangle: 16255:An Amazing, Space Filling, Non-regular Tetrahedron 15806: 15480:"Einige Bemerkungen über die dreiseitige Pyramide" 13958:An interesting polyhedron can be constructed from 12414:A regular tetrahedron can be seen as a triangular 12409: 12322: 11847: 11641: 11512: 11377: 11208: 11064: 10880: 10655: 10618: 10588: 10561: 10534: 10507: 10480: 10453: 10392: 10039: 9546: 9467: 9443: 9413: 9393: 9341: 8992: 8869: 8708: 8688: 8668: 8648: 8628: 8608: 8588: 8568: 8548: 8528: 8508: 8488: 8458: 8308: 8128: 7982: 7962: 7942: 7922: 7902: 7882: 7862: 7785: 7448: 7149: 6992: 6948: 6904: 6857: 6703: 6507: 6302: 6228: 6134: 6050: 5781: 5757: 5737: 5682: 4829:packs with directly congruent or enantiomorphous ( 4808: 4775: 4742: 4709: 4676: 4643: 4610: 4577: 4544: 4518: 4485: 4452: 4426: 4393: 4360: 4323: 4290: 4268: 4226: 4180: 4133: 4093: 4043: 4003: 3953: 3913: 3859: 3822: 3777: 3732: 3695: 3650: 3622: 3585: 3552: 3495: 3456: 3417: 2767:reflections in a plane perpendicular to an edge: 6 2688: 2658: 2632: 2598: 2521: 2438: 2411: 2247: 2211: 1985: 1956: 1829: 1630: 1164: 1137: 1117: 1097: 928: 896: 864: 841: 808: 623: 489: 457: 357: 337: 249:figure comprising two such dual tetrahedra form a 241:The regular tetrahedron is self-dual, meaning its 16127:J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math 15484:Sammlung mathematischer Aufsätze u. Bemerkungen 1 15081:, pp. 33–34, §3.1 Congruent transformations. 15053:, pp. 71–72, §4.7 Characteristic tetrahedra. 10899:be any interior point of a tetrahedron of volume 9523: 9511: 2783:Orthogonal projections of the regular tetrahedron 17667: 15313:: CS1 maint: bot: original URL status unknown ( 15065:, pp. 292–293, Table I(i); "Tetrahedron, 𝛼 14967: 14854: 13986:of each other. Superimposing both forms gives a 10423: 10406:Putting any of the four vertices in the role of 7250: 6078: 4227:{\displaystyle {\tfrac {{\text{arc sec }}3}{2}}} 15841:(January 1900). Winchell, Newton Horace (ed.). 15433: 14844:Ford, Walter Burton; Ammerman, Charles (1913), 14049:in the process of setting up the equations for 13018:. Truncating edges down to points produces the 11209:{\displaystyle PA+PB+PC+PD\geq 3(PJ+PK+PL+PM).} 5324:. A tetragonal disphenoid has Coxeter diagram 2876:The two skew perpendicular opposite edges of a 2258:A regular tetrahedron can be embedded inside a 14267: 12386:and the other two are isosceles with areas of 3823:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}} 3696:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}} 3272:can be dissected into six such 3-orthoschemes 3194:with right triangle or obtuse triangle faces. 17087: 16674: 16281: 15887:"The tetrahedral principle in kite structure" 15486:(in German). Berlin: Maurer. pp. 105–132 14313:of silicon form and what shapes they assume. 13733: 13527: 12610: 10201:. The complete list remains an open problem. 7208:, is defined by the position of the vertices 3098: 2915:The tetrahedron can also be represented as a 207:is a tetrahedron in which all four faces are 16137: 15945: 15903:10.1038/scientificamerican06131903-22947supp 15621: 15404: 15230: 15218: 15147: 14934: 14843: 14767:at a vertex. In chemistry, it is called the 13511:32 symmetry mutation of regular tilings: {3, 12299: 12285: 12273: 12259: 12246: 12232: 12220: 12206: 12193: 12179: 12167: 12153: 11906:can be formulated as matrix-vector product: 11657:Denote the circumradius of a tetrahedron as 11427:denote the area of each faces, the value of 10160:added to make a cube, which has 8 vertices. 9615:of the tetrahedron that is analogous to the 5430:. A rhombic disphenoid has Coxeter diagram 2907:when applied to the two special edge pairs. 2516: 2504: 345:. The surface area of a regular tetrahedron 16117:Harmonices Mundi (The Harmony of the World) 14368:-like puzzles are tetrahedral, such as the 14146:The tetrahedron shape is seen in nature in 10120:. If however, a tetrahedron has a vertex, 9557:Properties analogous to those of a triangle 4903:of the generated polyhedron contains three 4820: 3384:Characteristics of the regular tetrahedron 2599:{\displaystyle \mathrm {T} _{\mathrm {d} }} 2419:This yields a tetrahedron with edge-length 1966:Expressed symmetrically as 4 points on the 1682:to a vertex of the base is twice that from 1105:For a regular tetrahedron with side length 17094: 17080: 16681: 16667: 16288: 16274: 15604:Pilot's Handbook of Aeronautical Knowledge 15576: 13740: 13726: 13717:32 symmetry mutation of regular tilings: { 13534: 13520: 12617: 12603: 11224:Denoting the inradius of a tetrahedron as 3186:is a tetrahedron where all four faces are 3025: 16202: 16155: 15837: 15775:. Vol. Part I. London: E. Stanford. 15504: 15297:The Various Kinds of Centres of Simplices 15253: 15124: 15032: 14872: 14088: 10389: 10163:Inscribing tetrahedra inside the regular 10036: 9444:{\displaystyle \mathbf {b} -\mathbf {c} } 9310: 8860: 8856: 8852: 8819: 8791: 8763: 8732: 7442: 7172:are the plane angles occurring in vertex 7146: 1125:, the radius of its circumscribed sphere 16713: 16013: 15601:Federal Aviation Administration (2009), 15368:(1981). "Which tetrahedra fill space?". 15364: 15345: 14888: 14334: 14133:Calculation of the central angle with a 14128: 14116: 14024: 11689:be the volume of the tetrahedron. Then 10569:be the area of the face opposite vertex 8467: 4867: 3396: 3393: 3173: 3148: 3108: 3035:Tetrahedral symmetry subgroup relations 2966: 2894: 2859: 2559: 2536: 1852: 817: 314: 17659:List of regular polytopes and compounds 16144:Communications in Analysis and Geometry 16039:Cundy, H. Martyn (1952). "Deltahedra". 15995: 15981: 15968: 15728: 15627: 15522: 15454: 15452: 15450: 15448: 15206: 15090: 15078: 15062: 15050: 15011: 14959: 14953: 14909: 14784: 13029:Family of uniform tetrahedral polyhedra 11685:the length of the opposite edges. Let 10237:is that in a tetrahedron with vertices 6518:are expressed as row or column vectors. 4809:{\displaystyle {\sqrt {\tfrac {1}{6}}}} 4776:{\displaystyle {\sqrt {\tfrac {3}{2}}}} 4743:{\displaystyle {\sqrt {\tfrac {4}{3}}}} 4710:{\displaystyle {\sqrt {\tfrac {1}{6}}}} 4677:{\displaystyle {\sqrt {\tfrac {1}{2}}}} 4644:{\displaystyle {\sqrt {\tfrac {1}{3}}}} 4611:{\displaystyle {\sqrt {\tfrac {1}{2}}}} 4578:{\displaystyle {\sqrt {\tfrac {3}{2}}}} 4519:{\displaystyle {\sqrt {\tfrac {1}{6}}}} 4486:{\displaystyle {\sqrt {\tfrac {1}{3}}}} 4427:{\displaystyle {\sqrt {\tfrac {1}{6}}}} 4394:{\displaystyle {\sqrt {\tfrac {1}{2}}}} 4361:{\displaystyle {\sqrt {\tfrac {3}{2}}}} 4324:{\displaystyle {\sqrt {\tfrac {1}{3}}}} 4269:{\displaystyle {\sqrt {\tfrac {4}{3}}}} 3021: 3014:) can be constructed as tilings of the 2644:It has rotational tetrahedral symmetry 465:The height of a regular tetrahedron is 17668: 16138:Murakami, Jun; Yano, Masakazu (2005). 16110: 16075: 15801: 15477: 15293: 15163: 14999: 14472:, a cognitive scientist and expert on 14245:However, quaternary phase diagrams in 10191:characteristic orthoscheme of the cube 10139: 10124:, with solid angle greater than π sr, 5580: 5504: 5397: 5307: 5257: 5181: 5108: 5006: 4876:An irregular tetrahedron which is the 4863: 2640:. They can be categorized as follows: 2248:{\textstyle {\frac {2{\sqrt {6}}}{3}}} 1666:along an edge is twice that along the 1662:distance covered from the base to the 1658:), corresponding to the fact that the 147: 92:case of the more general concept of a 16662: 16269: 16230: 16096: 16038: 15916: 15765: 15687: 15327: 15231:Havlicek, Hans; Weiß, Gunter (2003). 15194: 14860: 14819: 14020: 12360: 5693: 2919:, and projected onto the plane via a 291:(fill space), but if alternated with 16688: 16180: 15881: 15445: 15348:Acta Societatis Scientiarum Fennicae 14973: 14814: 14812: 14488: 10542:be the points of a tetrahedron. Let 10208:The tetrahedron is unique among the 7839:, is essentially due to the painter 4890:Wythoff's kaleidoscopic construction 2856:Cross section of regular tetrahedron 490:{\textstyle {\frac {\sqrt {6}}{3}}a} 245:is another regular tetrahedron. The 110:, which is a polyhedron with a flat 16124: 15952:Mathematical Association of America 15439: 15113:Applied Mathematics and Computation 10411:the result is the fourth identity. 10205:tetrahedron have the same volume.) 5738:{\displaystyle V={\frac {1}{3}}Ah.} 2962: 2910: 2522:{\displaystyle \mathrm {h} \{4,3\}} 1698:at a vertex subtended by a face is 1640:With respect to the base plane the 50: 13: 16305:Listed by number of faces and type 15946:Alsina, C.; Nelsen, R. B. (2015). 14468:to be a tetrahedron, according to 14401: 14154:atoms are surrounded by atoms (or 14083:naval architecture and engineering 12580:, where base polygons are reduced 11044: 11020: 10996: 10972: 10847: 10837: 10805: 10795: 10763: 10753: 10729: 10711: 10693: 10675: 10550: 10374: 10353: 10332: 10311: 10290: 10269: 5637:Subdivision and similarity classes 4939:is formed. Two other isometries (C 4931:Isometries of irregular tetrahedra 4718:, and a right triangle with edges 3860:{\displaystyle {\tfrac {\pi }{3}}} 3733:{\displaystyle {\tfrac {\pi }{3}}} 3623:{\displaystyle {\tfrac {\pi }{3}}} 2979:, seen in stereographic projection 2652: 2590: 2584: 2500: 1067: 1021: 920: 888: 14: 17717: 16223: 15740:. HPL-98-95: 1–32. Archived from 14809: 10890: 10212:in possessing no parallel faces. 10190: 9591:of the opposite face is called a 5187:triangles with a common base edge 3496:{\displaystyle \pi -2{\text{𝟁}}} 3457:{\displaystyle \pi -2{\text{𝜿}}} 2666:. This symmetry is isomorphic to 321:3D model of a regular tetrahedron 15523:Lévy, Bruno; Liu, Yang (2010), " 15294:Outudee, Somluck; New, Stephen. 14925:, Mathematische Basteleien, 2001 14660: 14501: 14439: 14430: 14419: 14410: 13944: 13930: 13916: 13847: 13840: 13833: 13826: 13819: 13812: 13805: 13798: 13791: 13784: 13777: 13770: 13639: 13632: 13625: 13618: 13611: 13604: 13597: 13590: 13583: 13576: 13569: 13562: 13437: 13430: 13423: 13416: 13409: 13402: 13395: 13388: 13331: 13326: 13321: 13316: 13311: 13302: 13297: 13292: 13287: 13282: 13273: 13268: 13263: 13258: 13253: 13244: 13239: 13234: 13229: 13224: 13215: 13210: 13205: 13200: 13195: 13186: 13181: 13176: 13171: 13166: 13157: 13152: 13147: 13142: 13137: 13128: 13123: 13118: 13113: 13108: 13098: 13091: 13084: 13077: 13070: 13063: 13056: 13049: 12976: 12963: 12956: 12949: 12942: 12935: 12915: 12908: 12901: 12894: 12887: 12775: 12765: 12758: 12751: 12744: 12737: 12730: 12713: 12706: 12699: 12692: 12685: 12678: 12560: 12553: 12546: 12539: 12532: 12525: 12513: 12506: 12499: 12492: 12485: 12478: 10220: 10199:disphenoid tetrahedral honeycomb 10184:tetrahedral-octahedral honeycomb 9520: 9517: 9514: 9503: 9437: 9429: 6978: 6934: 6890: 6821: 6813: 6781: 6773: 6741: 6733: 6687: 6683: 6675: 6667: 6660: 6652: 6643: 6635: 6626: 6622: 6614: 6606: 6597: 6589: 6582: 6574: 6565: 6561: 6446: 6389: 6332: 6288: 6279: 6270: 6214: 6207: 6200: 6125: 6117: 6109: 6101: 6093: 6085: 5988: 5927: 5866: 5805: 5574: 5568: 5498: 5492: 5452: 5447: 5442: 5437: 5432: 5391: 5346: 5341: 5336: 5331: 5326: 5301: 5251: 5175: 5102: 5057: 5052: 5047: 5042: 5037: 5000: 4854:tetrahedral-octahedral honeycomb 4846:disphenoid tetrahedral honeycomb 3372: 3367: 3362: 3357: 3352: 3344: 3339: 3334: 3329: 3324: 3294: 3289: 3284: 3279: 3274: 3266: 3261: 3256: 3251: 3246: 3038: 3029: 2940: 2933: 2834: 2827: 2483: 2478: 2473: 2468: 2463: 929:{\displaystyle r_{\mathrm {E} }} 897:{\displaystyle r_{\mathrm {M} }} 287:Regular tetrahedra alone do not 276:. The dual of this solid is the 185: 168: 156: 99:, and may thus also be called a 15938: 15909: 15875: 15853: 15831: 15795: 15759: 15729:Vondran, Gary L. (April 1998). 15722: 15690:"Resistance-Distance Sum Rules" 15681: 15656: 15594: 15551: 15516: 15498: 15471: 15398: 15358: 15339: 15321: 15287: 15278: 15224: 15173: 15100: 15005: 14979: 14944: 14015: 12410:Related polyhedra and compounds 11870:of a tetrahedron with vertices 11858: 11652: 10194: 10057:is the angle between the faces 9355:the volume of the tetrahedron. 9282: 9173: 9064: 8963: 8912: 8472:Six edge-lengths of Tetrahedron 4920: 3169: 2903:This property also applies for 2374: 2305: 2183: 2053: 1916: 1908: 1678:of the base, the distance from 1060: 973: 315: 106:The tetrahedron is one kind of 15809:Principles of physical geology 14915: 14837: 14757: 14615:Table of graphs and parameters 14316: 14142:Tetrahedral molecular geometry 14059:partial differential equations 11828: 11798: 11795: 11768: 11765: 11738: 11735: 11708: 11200: 11164: 10875: 10749: 10136:lies outside the tetrahedron. 10108:, measures exactly π sr, then 10092:to the vertices is a minimum, 10010: 9997: 9984: 9971: 9958: 9945: 9930: 9917: 9896: 9883: 9870: 9857: 9842: 9829: 9816: 9803: 9782: 9769: 9754: 9741: 9728: 9715: 9702: 9689: 9531: 9527: 9499: 9495: 9329: 9311: 9307: 9289: 9274: 9256: 9253: 9235: 9219: 9201: 9198: 9180: 9165: 9147: 9144: 9126: 9110: 9092: 9089: 9071: 9056: 9038: 9035: 9017: 8844: 8820: 8816: 8792: 8788: 8764: 8760: 8733: 6985: 6971: 6941: 6927: 6897: 6883: 6492: 6453: 6435: 6396: 6378: 6339: 6295: 6263: 6221: 6193: 6129: 6081: 6038: 5999: 5977: 5938: 5916: 5877: 5855: 5816: 5114:triangle base and three equal 4619:, a right triangle with edges 3129:, as at the corner of a cube. 3076:that join the vertices to the 2702:unit quaternion representation 2399: 2375: 2364: 2340: 2330: 2306: 2295: 2277: 2202: 2184: 1837:This is approximately 0.55129 1670:of a face. In other words, if 310: 1: 16516:(two infinite groups and 75) 16295: 16203:Shavinina, Larisa V. (2013). 15630:Journal of Chemical Education 15407:Chemistry: A European Journal 15241:American Mathematical Monthly 14850:, Macmillan, pp. 294–295 14802: 14763:It is also the angle between 14216:), or approximately 109.47°. 10430:Trigonometry of a tetrahedron 10424:Law of cosines for tetrahedra 10229:Trigonometry of a tetrahedron 5683:{\displaystyle {\sqrt {3/2}}} 5458:and Schläfli symbol sr{2,2}. 5423:, present as the point group 3138: 3063:semi-orthocentric tetrahedron 2971:A single 30-tetrahedron ring 2864:A central cross section of a 17061:Degenerate polyhedra are in 16534:(two infinite groups and 50) 16183:"Regular polytope distances" 15703:(2): 633–649. Archived from 15534:ACM Transactions on Graphics 15034:10.1016/0898-1221(89)90148-X 14658: 14112: 14063:computational fluid dynamics 13960:five intersecting tetrahedra 12983: 12927: 12879: 12834: 10619:{\displaystyle \theta _{ij}} 7847:for the area of a triangle. 5644:Longest Edge Bisection (LEB) 5352:and Schläfli symbol s{2,4}. 5032:. A regular tetrahedron has 5014:It forms the symmetry group 4238: 4236: 4197: 4192: 4190: 4149: 4147: 4145: 4143: 4103: 4059: 4057: 4055: 4053: 4013: 3969: 3967: 3965: 3963: 3923: 3837: 3832: 3792: 3787: 3747: 3710: 3705: 3665: 3660: 3637: 3600: 3595: 3586:{\displaystyle 2{\text{𝜿}}} 3567: 3562: 3522: 3471: 3466: 3432: 3427: 3404: 3390: 3002:with tetrahedral cells (the 2931: 2841: 2822: 2659:{\displaystyle \mathrm {T} } 2439:{\displaystyle 2{\sqrt {2}}} 1651:) is twice that of an edge ( 194:a thin volume of empty space 144:to the tetrahedron's faces. 69:composed of four triangular 7: 16880:pentagonal icositetrahedron 16821:truncated icosidodecahedron 16014:Cromwell, Peter R. (1997). 15017:"Trisecting an Orthoscheme" 14987:"Sections of a Tetrahedron" 14672: 14476:who advised Kubrick on the 14356:, this solid is known as a 14268:Electricity and electronics 14100: 13938:Compound of five tetrahedra 13383:Duals to uniform polyhedra 12336: 11232:of its triangular faces as 11219: 11111:of the perpendiculars from 10903:for which the vertices are 10562:{\displaystyle \Delta _{i}} 10182:that can tile space as the 7196:, does so for the vertices 4181:{\displaystyle {\text{𝜿}}} 4165: 4063: 3973: 3883: 3744: 3634: 3519: 3401: 3388: 2987:aperiodic chain called the 2532: 10: 17722: 17648: 17075: 16910:pentagonal hexecontahedron 16870:deltoidal icositetrahedron 16157:10.4310/cag.2005.v13.n2.a5 16022:Cambridge University Press 16005:(3rd ed.). New York: 15688:Klein, Douglas J. (2002). 15459:Inequalities proposed in “ 15328:Audet, Daniel (May 2011). 14740:Trirectangular tetrahedron 14394:, originally published by 14383: 14379: 14320: 14271: 14139: 13988:compound of ten tetrahedra 13952:Compound of ten tetrahedra 13712: 13506: 12364: 12345:of its four vertices, see 10656:{\displaystyle P_{i}P_{j}} 10427: 10226: 10170:Regular tetrahedra cannot 7219:If we do not require that 3320:characteristic orthoscheme 3228:characteristic of the cube 3142: 3119:trirectangular tetrahedron 3105:Trirectangular tetrahedron 3102: 3099:Trirectangular tetrahedron 3080:of the opposite faces are 2454:, a polyhedron that is by 297:alternated cubic honeycomb 272:the tetrahedron becomes a 18: 16:Polyhedron with four faces 17059: 16993: 16968: 16950: 16943: 16918: 16905:disdyakis triacontahedron 16900:deltoidal hexecontahedron 16834: 16742: 16697: 16607: 16586:Kepler–Poinsot polyhedron 16578: 16543: 16491: 16432: 16371: 16310: 16303: 16106:(Thesis). pp. 16–17. 15897:(1432supp): s2294–22950. 15668:American Chemical Society 15126:10.1016/j.amc.2024.128631 14650:, each a skeleton of its 14613: 14579: 14569: 14559: 14549: 14539: 14529: 14519: 14509: 14500: 14495: 14360:, one of the more common 14247:communication engineering 13764: 13758: 13750: 13556: 13550: 13544: 13382: 13034: 13027: 12875:Apeirogonal trapezohedron 12472: 12469: 12423: 10233:A corollary of the usual 7461:Cayley–Menger determinant 5698: 5567: 5564: 5491: 5486: 5481: 5390: 5385: 5300: 5295: 5287: 5250: 5245: 5196:, also isomorphic to the 5174: 5171: 5101: 5098: 4999: 4996: 4967: 4962: 4955: 4952: 4937:3-dimensional point group 4860:cells in a ratio of 2:1. 4827:space-filling tetrahedron 3383: 3216:birectangular tetrahedron 3121:the three face angles at 2953: 2606:. This symmetry group is 2574:full tetrahedral symmetry 1993:plane, the vertices are: 16041:The Mathematical Gazette 15839:Hitchcock, Charles Henry 15219:Murakami & Yano 2005 15148:Alsina & Nelsen 2015 14935:Alsina & Nelsen 2015 14847:Plane and Solid Geometry 14750: 14710:– 4-dimensional analogue 14458:originally intended the 14330: 13924:Two tetrahedra in a cube 12862:Pentagonal trapezohedron 12857:Tetragonal trapezohedron 12352: 9578:orthocentric tetrahedron 9362:or in three-dimensional 4821:Space-filling tetrahedra 3059:orthocentric tetrahedron 2955:Stereographic projection 2921:stereographic projection 2541:The cube and tetrahedron 2219:with the edge length of 19:Not to be confused with 17011:gyroelongated bipyramid 16885:rhombic triacontahedron 16791:truncated cuboctahedron 16598:Uniform star polyhedron 16526:quasiregular polyhedron 16181:Park, Poo-Sung (2016). 15971:Elemente der Mathematik 15767:Green, William Lowthian 15546:10.1145/1778765.1778856 14474:artificial intelligence 14303:solid-state electronics 14051:finite element analysis 13912:Compounds of tetrahedra 13497:}, continuing into the 12867:Hexagonal trapezohedron 12727:Spherical tiling image 7807:represent the vertices 5291:(Four equal triangles) 4850:Hilbert's third problem 4094:{\displaystyle _{2}R/l} 4004:{\displaystyle _{1}R/l} 3914:{\displaystyle _{0}R/l} 3057:, then it is called an 2950:Orthographic projection 2807:Orthographic projection 88:The tetrahedron is the 17006:truncated trapezohedra 16875:disdyakis dodecahedron 16841:(duals of Archimedean) 16816:rhombicosidodecahedron 16806:truncated dodecahedron 16532:semiregular polyhedron 16076:Fekete, A. E. (1985). 15883:Bell, Alexander Graham 15847:The American Geologist 15572:(5): 162–166, May 1985 15505:Todhunter, I. (1886), 15478:Crelle, A. L. (1821). 15419:10.1002/chem.200400869 15021:Computers Math. Applic 14769:tetrahedral bond angle 14704:-dimensional analogues 14679:Boerdijk–Coxeter helix 14396:William Lowthian Green 14392:tetrahedral hypothesis 14386:tetrahedral hypothesis 14342: 14137: 14126: 14089:Structural engineering 14085:, and related fields. 14071:electromagnetic fields 14030: 13765:Noncompact hyperbolic 13557:Noncompact hyperbolic 12852:Trigonal trapezohedron 12393:, while the volume is 12324: 11849: 11643: 11514: 11379: 11245:= 1, 2, 3, 4, we have 11210: 11066: 10882: 10657: 10620: 10590: 10563: 10536: 10509: 10482: 10455: 10394: 10165:compound of five cubes 10041: 9603:at a point called the 9548: 9469: 9445: 9415: 9395: 9343: 8994: 8871: 8710: 8690: 8670: 8650: 8630: 8610: 8590: 8570: 8550: 8530: 8510: 8490: 8473: 8460: 8310: 8130: 7984: 7964: 7944: 7924: 7904: 7884: 7864: 7787: 7450: 7151: 6994: 6950: 6906: 6859: 6705: 6509: 6304: 6230: 6136: 6052: 5783: 5759: 5739: 5684: 5259:Four unequal triangles 4901:Coxeter-Dynkin diagram 4873: 4810: 4777: 4744: 4711: 4678: 4645: 4612: 4579: 4546: 4520: 4487: 4454: 4428: 4395: 4362: 4325: 4292: 4270: 4228: 4182: 4135: 4095: 4045: 4005: 3955: 3915: 3861: 3824: 3779: 3734: 3697: 3652: 3624: 3587: 3554: 3497: 3458: 3419: 3218:. It is also called a 3179: 3158: 3114: 3070:isodynamic tetrahedron 2989:Boerdijk–Coxeter helix 2980: 2973:Boerdijk–Coxeter helix 2905:tetragonal disphenoids 2900: 2873: 2793:orthogonal projections 2707:identity (identity; 1) 2690: 2660: 2634: 2600: 2565: 2542: 2523: 2458:a cube. This form has 2440: 2413: 2249: 2213: 1987: 1958: 1831: 1632: 1166: 1139: 1119: 1099: 930: 898: 866: 843: 823: 810: 625: 491: 459: 359: 339: 322: 16895:pentakis dodecahedron 16811:truncated icosahedron 16766:truncated tetrahedron 16579:non-convex polyhedron 16097:Kahan, W. M. (2012). 16082:. Marcel Dekker Inc. 15697:Croatica Chemica Acta 15588:Pythagorean Triangles 14789:characteristic angles 14465:2001: A Space Odyssey 14338: 14132: 14120: 14028: 12842:Digonal trapezohedron 12669:Apeirogonal antiprism 12325: 11850: 11644: 11515: 11380: 11211: 11067: 10883: 10658: 10621: 10591: 10589:{\displaystyle P_{i}} 10564: 10537: 10535:{\displaystyle P_{4}} 10510: 10508:{\displaystyle P_{3}} 10483: 10481:{\displaystyle P_{2}} 10456: 10454:{\displaystyle P_{1}} 10395: 10144:A tetrahedron is a 3- 10128:still corresponds to 10042: 9549: 9470: 9446: 9416: 9396: 9372:Murakami–Yano formula 9344: 8995: 8872: 8711: 8691: 8671: 8651: 8631: 8611: 8591: 8571: 8551: 8531: 8511: 8491: 8471: 8461: 8311: 8131: 7985: 7965: 7945: 7925: 7905: 7885: 7865: 7841:Piero della Francesca 7796:where the subscripts 7788: 7451: 7152: 6995: 6951: 6907: 6860: 6706: 6510: 6305: 6231: 6137: 6053: 5784: 5760: 5740: 5685: 5297:Tetragonal disphenoid 5246:Irregular tetrahedron 4871: 4811: 4778: 4745: 4712: 4679: 4646: 4613: 4580: 4547: 4521: 4488: 4455: 4429: 4396: 4363: 4334:characteristic angles 4326: 4293: 4271: 4229: 4183: 4136: 4096: 4046: 4006: 3956: 3916: 3862: 3825: 3780: 3735: 3698: 3653: 3625: 3588: 3555: 3498: 3459: 3420: 3177: 3152: 3112: 2970: 2923:. This projection is 2898: 2863: 2691: 2689:{\displaystyle A_{4}} 2661: 2635: 2633:{\displaystyle S_{4}} 2601: 2563: 2540: 2524: 2441: 2414: 2264:Cartesian coordinates 2250: 2214: 1988: 1959: 1859:Cartesian coordinates 1853:Cartesian coordinates 1832: 1633: 1167: 1165:{\displaystyle d_{i}} 1140: 1120: 1100: 931: 899: 867: 844: 821: 811: 626: 492: 460: 360: 340: 320: 274:truncated tetrahedron 257:. Its interior is an 209:equilateral triangles 17696:Prismatoid polyhedra 16855:rhombic dodecahedron 16781:truncated octahedron 15371:Mathematics Magazine 14734:Triangular dipyramid 14714:Synergetics (Fuller) 14684:Möbius configuration 14305:, and silicon has a 14256:allotrope and tetra- 14079:chemical engineering 12831:-gonal trapezohedra 12661:Heptagonal antiprism 12651:Pentagonal antiprism 12639:Triangular antiprism 12367:Heronian tetrahedron 11913: 11696: 11529: 11438: 11252: 11122: 10954: 10671: 10630: 10600: 10573: 10546: 10519: 10492: 10465: 10438: 10260: 9663: 9609:Commandino's theorem 9479: 9459: 9425: 9405: 9385: 9004: 8881: 8720: 8700: 8680: 8660: 8640: 8620: 8600: 8580: 8560: 8540: 8520: 8500: 8480: 8320: 8140: 7994: 7974: 7954: 7934: 7914: 7894: 7874: 7854: 7470: 7230: 7007: 6960: 6916: 6872: 6721: 6529: 6320: 6246: 6176: 6065: 5797: 5773: 5749: 5707: 5662: 5520:, isomorphic to the 5128:, isomorphic to the 5021:, isomorphic to the 4997:Regular tetrahedron 4789: 4756: 4723: 4690: 4657: 4624: 4591: 4558: 4536: 4499: 4466: 4444: 4437:characteristic radii 4435:(edges that are the 4407: 4374: 4341: 4304: 4282: 4249: 4202: 4170: 4108: 4068: 4018: 3978: 3928: 3888: 3842: 3797: 3752: 3715: 3670: 3642: 3605: 3572: 3527: 3476: 3437: 3409: 3313:Heronian tetrahedron 3095:of the tetrahedron. 3089:isogonic tetrahedron 3072:is one in which the 3022:Irregular tetrahedra 2673: 2648: 2617: 2579: 2496: 2423: 2270: 2266:of the vertices are 2223: 1997: 1974: 1865: 1702: 1176: 1149: 1129: 1109: 940: 911: 879: 856: 833: 642: 501: 469: 369: 349: 329: 301:Schläfli orthoscheme 251:stellated octahedron 17701:Pyramids (geometry) 17691:Self-dual polyhedra 17643:pentagonal polytope 17542:Uniform 10-polytope 17102:Fundamental convex 16890:triakis icosahedron 16865:tetrakis hexahedron 16850:triakis tetrahedron 16786:rhombicuboctahedron 16191:Forum Geometricorum 16079:Real Linear Algebra 15921:"Tetrahedral graph" 15891:Scientific American 15781:1875vmge.book.....G 15738:HP Technical Report 15642:1945JChEd..22..145B 15566:Crux Mathematicorum 15540:(4): 119:1–119:11, 15461:Crux Mathematicorum 14729:Tetrahedron packing 14600:distance-transitive 14449:Tetrahedral objects 14297:is the most common 14156:lone electron pairs 14006:Szilassi polyhedron 12832: 12772:Plane tiling image 12656:Hexagonal antiprism 12624: 12377:isosceles triangles 12373:Heronian tetrahedra 11349: 11324: 11299: 11274: 10742: 10724: 10706: 10688: 10140:Geometric relations 10096:coincides with the 9628:twelve-point sphere 7837:Tartaglia's formula 7769: 7752: 7735: 7711: 7689: 7672: 7648: 7631: 7609: 7585: 7568: 7551: 5582:Two pairs of equal 5565:Phyllic disphenoid 5506:Two pairs of equal 5099:Triangular pyramid 4886:Goursat tetrahedron 4884:is an example of a 4864:Fundamental domains 3000:regular 4-polytopes 2878:regular tetrahedron 2866:regular tetrahedron 2809: 1609: 1591: 1573: 1555: 1509: 1491: 1473: 1455: 1370: 1352: 1334: 1316: 1254: 1236: 1218: 1200: 278:triakis tetrahedron 265:the tetrahedron). 205:regular tetrahedron 148:Regular tetrahedron 61:), also known as a 17512:Uniform 9-polytope 17462:Uniform 8-polytope 17412:Uniform 7-polytope 17369:Uniform 6-polytope 17339:Uniform 5-polytope 17299:Uniform polychoron 17262:Uniform polyhedron 17110:in dimensions 2–10 16860:triakis octahedron 16745:Archimedean solids 16520:regular polyhedron 16514:uniform polyhedron 16476:Hectotriadiohedron 16232:Weisstein, Eric W. 16007:Dover Publications 15918:Weisstein, Eric W. 15813:. Nelson. p. 15664:"White phosphorus" 15366:Senechal, Marjorie 15354:(Part 1): 189–203. 15095:fundamental region 14821:Weisstein, Eric W. 14724:Tetrahedral number 14604:3-vertex-connected 14343: 14138: 14127: 14125:ion is tetrahedral 14055:numerical solution 14053:especially in the 14035:numerical analysis 14031: 14021:Numerical analysis 14010:Császár polyhedron 13992:stellae octangulae 12986:Face configuration 12826: 12587: 12361:Integer tetrahedra 12320: 12318: 12310: 12104: 11845: 11639: 11510: 11375: 11335: 11310: 11285: 11260: 11206: 11062: 10878: 10728: 10710: 10692: 10674: 10653: 10616: 10586: 10559: 10532: 10505: 10478: 10451: 10417:degrees of freedom 10390: 10037: 10024: 9544: 9465: 9441: 9411: 9391: 9358:For tetrahedra in 9339: 9337: 8990: 8988: 8867: 8706: 8686: 8666: 8646: 8626: 8606: 8586: 8566: 8546: 8526: 8506: 8486: 8474: 8456: 8306: 8304: 8126: 7980: 7960: 7940: 7920: 7900: 7880: 7860: 7783: 7777: 7755: 7738: 7721: 7697: 7675: 7658: 7634: 7617: 7595: 7571: 7554: 7537: 7446: 7431: 7147: 6990: 6984: 6946: 6940: 6902: 6896: 6855: 6850: 6701: 6695: 6505: 6500: 6300: 6294: 6226: 6220: 6132: 6048: 6046: 5779: 5755: 5735: 5694:General properties 5680: 5600:isomorphic to the 5488:Digonal disphenoid 5387:Rhombic disphenoid 5172:Mirrored sphenoid 4878:fundamental domain 4874: 4839:scissors-congruent 4806: 4803: 4773: 4770: 4740: 4737: 4707: 4704: 4674: 4671: 4641: 4638: 4608: 4605: 4575: 4572: 4542: 4516: 4513: 4483: 4480: 4450: 4424: 4421: 4391: 4388: 4358: 4355: 4336:𝟀, 𝝉, 𝟁), plus 4321: 4318: 4288: 4266: 4263: 4224: 4222: 4178: 4131: 4122: 4091: 4041: 4032: 4001: 3951: 3942: 3911: 3857: 3855: 3820: 3810: 3775: 3766: 3730: 3728: 3693: 3683: 3648: 3620: 3618: 3583: 3550: 3541: 3493: 3454: 3415: 3237:and one of length 3180: 3159: 3115: 2981: 2901: 2874: 2805: 2686: 2656: 2630: 2596: 2566: 2545:The vertices of a 2543: 2519: 2436: 2409: 2407: 2245: 2209: 2207: 1986:{\displaystyle xy} 1983: 1954: 1914: 1827: 1825: 1628: 1626: 1595: 1577: 1559: 1541: 1495: 1477: 1459: 1441: 1356: 1338: 1320: 1302: 1240: 1222: 1204: 1186: 1162: 1135: 1115: 1095: 1093: 926: 894: 862: 839: 824: 806: 804: 621: 487: 455: 355: 335: 323: 307:, can tessellate. 63:triangular pyramid 17686:Individual graphs 17664: 17663: 17651:Polytope families 17108:uniform polytopes 17070: 17069: 16989: 16988: 16826:snub dodecahedron 16801:icosidodecahedron 16656: 16655: 16557:Archimedean solid 16544:convex polyhedron 16452:Icosidodecahedron 16214:978-0-203-38714-6 16089:978-0-8247-7238-3 16031:978-0-521-55432-9 16002:Regular Polytopes 15991:. Methuen and Co. 15988:Regular Polytopes 15983:Coxeter, H. S. M. 15961:978-1-61444-216-5 15650:10.1021/ed022p145 15583:Wacław Sierpiński 15413:(24): 6575–6580. 14670: 14669: 14646:. It is one of 5 14620: 14619: 14496:Tetrahedral graph 14489:Tetrahedral graph 14148:covalently bonded 14075:civil engineering 13999:square hosohedron 13982:forms, which are 13974:. There are both 13909: 13908: 13701: 13700: 13487: 13486: 13016:uniform polyhedra 13012: 13011: 12814: 12813: 12675:Polyhedron image 12632:Digonal antiprism 12571: 12570: 12143: 12126: 12120: 12114: 11957: 11951: 11840: 11831: 11624: 11598: 11572: 11546: 11508: 11370: 11350: 11325: 11300: 11275: 11091:, interior point 10210:uniform polyhedra 9624:nine-point circle 9539: 9468:{\displaystyle V} 9414:{\displaystyle a} 9394:{\displaystyle d} 9364:elliptic geometry 8981: 8956: 8930: 8905: 8865: 8847: 8709:{\displaystyle W} 8689:{\displaystyle w} 8669:{\displaystyle V} 8649:{\displaystyle v} 8629:{\displaystyle U} 8609:{\displaystyle u} 8589:{\displaystyle w} 8569:{\displaystyle v} 8549:{\displaystyle u} 8529:{\displaystyle W} 8509:{\displaystyle V} 8489:{\displaystyle U} 8454: 8345: 8124: 8120: 7983:{\displaystyle V} 7963:{\displaystyle z} 7943:{\displaystyle y} 7923:{\displaystyle x} 7903:{\displaystyle c} 7883:{\displaystyle b} 7863:{\displaystyle a} 7141: 7032: 6076: 5782:{\displaystyle h} 5758:{\displaystyle A} 5724: 5678: 5634: 5633: 4953:Tetrahedron name 4816: 4804: 4802: 4783: 4771: 4769: 4750: 4738: 4736: 4717: 4705: 4703: 4684: 4672: 4670: 4651: 4639: 4637: 4618: 4606: 4604: 4585: 4573: 4571: 4552: 4545:{\displaystyle 1} 4529:60-90-30 triangle 4526: 4514: 4512: 4493: 4481: 4479: 4460: 4453:{\displaystyle 1} 4434: 4422: 4420: 4401: 4389: 4387: 4368: 4356: 4354: 4331: 4319: 4317: 4298: 4291:{\displaystyle 1} 4276: 4264: 4262: 4242: 4241: 4234: 4221: 4212: 4195: 4188: 4176: 4141: 4123: 4121: 4101: 4051: 4033: 4031: 4011: 3961: 3943: 3941: 3921: 3867: 3854: 3835: 3830: 3818: 3809: 3790: 3785: 3767: 3765: 3740: 3727: 3708: 3703: 3691: 3682: 3663: 3658: 3651:{\displaystyle 1} 3630: 3617: 3598: 3593: 3581: 3565: 3560: 3542: 3540: 3503: 3491: 3469: 3464: 3452: 3430: 3425: 3418:{\displaystyle 2} 3220:quadrirectangular 3048: 3047: 2998:, all the convex 2960: 2959: 2853: 2852: 2698:conjugacy classes 2668:alternating group 2434: 2243: 2237: 2169: 2153: 2152: 2135: 2134: 2103: 2087: 2086: 2072: 2071: 2041: 2019: 2018: 1947: 1946: 1913: 1901: 1900: 1811: 1774: 1748: 1727: 1400: 1375: 1284: 1259: 1138:{\displaystyle R} 1118:{\displaystyle a} 1086: 1085: 1053: 1052: 1038: 1006: 1005: 988: 963: 959: 865:{\displaystyle r} 842:{\displaystyle R} 826:The radii of its 780: 748: 699: 667: 600: 597: 568: 564: 538: 534: 518: 482: 478: 434: 399: 395: 358:{\displaystyle A} 338:{\displaystyle a} 293:regular octahedra 90:three-dimensional 17713: 17655:Regular polytope 17216: 17205: 17194: 17153: 17096: 17089: 17082: 17073: 17072: 16948: 16947: 16944:Dihedral uniform 16919:Dihedral regular 16842: 16758: 16707: 16683: 16676: 16669: 16660: 16659: 16492:elemental things 16470:Enneacontahedron 16440:Icositetrahedron 16290: 16283: 16276: 16267: 16266: 16245: 16244: 16218: 16199: 16187: 16177: 16159: 16134: 16121: 16120:. Johann Planck. 16112:Kepler, Johannes 16107: 16105: 16093: 16072: 16047:(318): 263–266. 16035: 16010: 15992: 15978: 15965: 15932: 15931: 15930: 15913: 15907: 15906: 15879: 15873: 15872: 15870: 15868: 15863:. Web of Stories 15857: 15851: 15850: 15835: 15829: 15828: 15812: 15799: 15793: 15792: 15763: 15757: 15756: 15754: 15752: 15746: 15735: 15726: 15720: 15719: 15717: 15715: 15709: 15694: 15685: 15679: 15678: 15676: 15674: 15660: 15654: 15653: 15625: 15619: 15617: 15598: 15592: 15580: 15574: 15573: 15563: 15555: 15549: 15548: 15531: 15520: 15514: 15512: 15502: 15496: 15495: 15493: 15491: 15475: 15469: 15456: 15443: 15437: 15431: 15430: 15402: 15396: 15395: 15362: 15356: 15355: 15343: 15337: 15336: 15334: 15325: 15319: 15318: 15312: 15304: 15302: 15291: 15285: 15282: 15276: 15275: 15257: 15237: 15228: 15222: 15216: 15210: 15204: 15198: 15192: 15183: 15177: 15171: 15161: 15155: 15145: 15139: 15138: 15128: 15104: 15098: 15088: 15082: 15076: 15070: 15060: 15054: 15048: 15039: 15038: 15036: 15009: 15003: 14997: 14991: 14990: 14983: 14977: 14971: 14965: 14948: 14942: 14932: 14926: 14921:Köller, Jürgen, 14919: 14913: 14907: 14896: 14886: 14880: 14870: 14864: 14858: 14852: 14851: 14841: 14835: 14834: 14833: 14816: 14796: 14781: 14772: 14761: 14719:Tetrahedral kite 14666:3-fold symmetry 14664: 14657: 14656: 14596:distance-regular 14571:Chromatic number 14505: 14493: 14492: 14443: 14434: 14423: 14414: 14347:Royal Game of Ur 14254:white phosphorus 14234: 14232: 14231: 14215: 14213: 14212: 14209: 14206: 14189: 14188: 14187: 14173: 14172: 14171: 13948: 13934: 13920: 13851: 13844: 13837: 13830: 13823: 13816: 13809: 13802: 13795: 13788: 13781: 13774: 13759:Compact hyperb. 13742: 13735: 13728: 13710: 13709: 13643: 13636: 13629: 13622: 13615: 13608: 13601: 13594: 13587: 13580: 13573: 13566: 13536: 13529: 13522: 13504: 13503: 13499:hyperbolic plane 13491:Schläfli symbols 13441: 13434: 13427: 13420: 13413: 13406: 13399: 13392: 13336: 13335: 13334: 13330: 13329: 13325: 13324: 13320: 13319: 13315: 13314: 13307: 13306: 13305: 13301: 13300: 13296: 13295: 13291: 13290: 13286: 13285: 13278: 13277: 13276: 13272: 13271: 13267: 13266: 13262: 13261: 13257: 13256: 13249: 13248: 13247: 13243: 13242: 13238: 13237: 13233: 13232: 13228: 13227: 13220: 13219: 13218: 13214: 13213: 13209: 13208: 13204: 13203: 13199: 13198: 13191: 13190: 13189: 13185: 13184: 13180: 13179: 13175: 13174: 13170: 13169: 13162: 13161: 13160: 13156: 13155: 13151: 13150: 13146: 13145: 13141: 13140: 13133: 13132: 13131: 13127: 13126: 13122: 13121: 13117: 13116: 13112: 13111: 13102: 13095: 13088: 13081: 13074: 13067: 13060: 13053: 13025: 13024: 12980: 12967: 12960: 12953: 12946: 12939: 12930:Spherical tiling 12919: 12912: 12905: 12898: 12891: 12833: 12825: 12779: 12769: 12762: 12755: 12748: 12741: 12734: 12717: 12710: 12703: 12696: 12689: 12682: 12646:Square antiprism 12625: 12619: 12612: 12605: 12586: 12564: 12557: 12550: 12543: 12536: 12529: 12517: 12510: 12503: 12496: 12489: 12482: 12425:Regular pyramids 12421: 12420: 12402: 12401: 12398: 12392: 12391: 12385: 12384: 12329: 12327: 12326: 12321: 12319: 12315: 12311: 12307: 12306: 12297: 12296: 12281: 12280: 12271: 12270: 12254: 12253: 12244: 12243: 12228: 12227: 12218: 12217: 12201: 12200: 12191: 12190: 12175: 12174: 12165: 12164: 12144: 12136: 12124: 12121: 12118: 12112: 12109: 12105: 12101: 12100: 12095: 12091: 12090: 12089: 12077: 12076: 12057: 12056: 12051: 12047: 12046: 12045: 12033: 12032: 12013: 12012: 12007: 12003: 12002: 12001: 11989: 11988: 11955: 11952: 11949: 11942: 11941: 11905: 11896: 11887: 11878: 11869: 11854: 11852: 11851: 11846: 11841: 11839: 11707: 11706: 11648: 11646: 11645: 11640: 11635: 11634: 11625: 11617: 11609: 11608: 11599: 11591: 11583: 11582: 11573: 11565: 11557: 11556: 11547: 11539: 11519: 11517: 11516: 11511: 11509: 11507: 11506: 11505: 11493: 11492: 11480: 11479: 11467: 11466: 11456: 11448: 11384: 11382: 11381: 11376: 11371: 11369: 11368: 11356: 11351: 11348: 11343: 11331: 11326: 11323: 11318: 11306: 11301: 11298: 11293: 11281: 11276: 11273: 11268: 11256: 11215: 11213: 11212: 11207: 11071: 11069: 11068: 11063: 11049: 11048: 11047: 11025: 11024: 11023: 11001: 11000: 10999: 10977: 10976: 10975: 10887: 10885: 10884: 10879: 10874: 10873: 10855: 10854: 10845: 10844: 10832: 10831: 10813: 10812: 10803: 10802: 10790: 10789: 10771: 10770: 10761: 10760: 10741: 10736: 10723: 10718: 10705: 10700: 10687: 10682: 10662: 10660: 10659: 10654: 10652: 10651: 10642: 10641: 10625: 10623: 10622: 10617: 10615: 10614: 10595: 10593: 10592: 10587: 10585: 10584: 10568: 10566: 10565: 10560: 10558: 10557: 10541: 10539: 10538: 10533: 10531: 10530: 10514: 10512: 10511: 10506: 10504: 10503: 10487: 10485: 10484: 10479: 10477: 10476: 10460: 10458: 10457: 10452: 10450: 10449: 10399: 10397: 10396: 10391: 10224: 10172:tessellate space 10150:electromagnetism 10098:geometric median 10070:geometric median 10046: 10044: 10043: 10038: 10029: 10028: 10013: 10009: 10008: 9987: 9983: 9982: 9961: 9957: 9956: 9933: 9929: 9928: 9899: 9895: 9894: 9873: 9869: 9868: 9845: 9841: 9840: 9819: 9815: 9814: 9785: 9781: 9780: 9757: 9753: 9752: 9731: 9727: 9726: 9705: 9701: 9700: 9553: 9551: 9550: 9545: 9540: 9535: 9534: 9526: 9506: 9498: 9489: 9474: 9472: 9471: 9466: 9450: 9448: 9447: 9442: 9440: 9432: 9420: 9418: 9417: 9412: 9400: 9398: 9397: 9392: 9360:hyperbolic space 9348: 9346: 9345: 9340: 9338: 8999: 8997: 8996: 8991: 8989: 8982: 8971: 8957: 8946: 8931: 8920: 8906: 8895: 8876: 8874: 8873: 8868: 8866: 8864: 8731: 8730: 8715: 8713: 8712: 8707: 8695: 8693: 8692: 8687: 8675: 8673: 8672: 8667: 8655: 8653: 8652: 8647: 8635: 8633: 8632: 8627: 8615: 8613: 8612: 8607: 8595: 8593: 8592: 8587: 8575: 8573: 8572: 8567: 8555: 8553: 8552: 8547: 8535: 8533: 8532: 8527: 8515: 8513: 8512: 8507: 8495: 8493: 8492: 8487: 8465: 8463: 8462: 8457: 8455: 8453: 8445: 8444: 8432: 8424: 8423: 8411: 8403: 8402: 8390: 8379: 8368: 8348: 8346: 8341: 8330: 8315: 8313: 8312: 8307: 8305: 8298: 8297: 8285: 8284: 8272: 8271: 8245: 8244: 8232: 8231: 8219: 8218: 8192: 8191: 8179: 8178: 8166: 8165: 8135: 8133: 8132: 8127: 8125: 8107: 8106: 8097: 8096: 8084: 8083: 8074: 8073: 8061: 8060: 8051: 8050: 8038: 8037: 8028: 8027: 8018: 8017: 8005: 8004: 7989: 7987: 7986: 7981: 7969: 7967: 7966: 7961: 7949: 7947: 7946: 7941: 7929: 7927: 7926: 7921: 7909: 7907: 7906: 7901: 7889: 7887: 7886: 7881: 7869: 7867: 7866: 7861: 7826: 7806: 7792: 7790: 7789: 7784: 7782: 7781: 7768: 7763: 7751: 7746: 7734: 7729: 7710: 7705: 7688: 7683: 7671: 7666: 7647: 7642: 7630: 7625: 7608: 7603: 7584: 7579: 7567: 7562: 7550: 7545: 7488: 7487: 7455: 7453: 7452: 7447: 7441: 7437: 7436: 7432: 7406: 7405: 7394: 7393: 7382: 7381: 7370: 7369: 7356: 7355: 7344: 7343: 7332: 7331: 7320: 7319: 7306: 7305: 7294: 7293: 7282: 7281: 7270: 7269: 7184:to the vertices 7156: 7154: 7153: 7148: 7142: 7140: 7132: 7131: 7119: 7111: 7110: 7098: 7090: 7089: 7077: 7066: 7055: 7035: 7033: 7028: 7017: 6999: 6997: 6996: 6991: 6989: 6988: 6981: 6955: 6953: 6952: 6947: 6945: 6944: 6937: 6911: 6909: 6908: 6903: 6901: 6900: 6893: 6864: 6862: 6861: 6856: 6854: 6853: 6844: 6824: 6816: 6804: 6784: 6776: 6764: 6744: 6736: 6717: 6713: 6710: 6708: 6707: 6702: 6700: 6699: 6692: 6691: 6690: 6678: 6670: 6663: 6655: 6646: 6638: 6631: 6630: 6629: 6617: 6609: 6600: 6592: 6585: 6577: 6570: 6569: 6568: 6547: 6546: 6517: 6514: 6512: 6511: 6506: 6504: 6503: 6491: 6490: 6478: 6477: 6465: 6464: 6449: 6434: 6433: 6421: 6420: 6408: 6407: 6392: 6377: 6376: 6364: 6363: 6351: 6350: 6335: 6316: 6312: 6309: 6307: 6306: 6301: 6299: 6298: 6291: 6282: 6273: 6242: 6238: 6235: 6233: 6232: 6227: 6225: 6224: 6217: 6210: 6203: 6165: 6163: 6162: 6159: 6156: 6141: 6139: 6138: 6133: 6128: 6120: 6112: 6104: 6096: 6088: 6077: 6069: 6057: 6055: 6054: 6049: 6047: 6037: 6036: 6024: 6023: 6011: 6010: 5991: 5976: 5975: 5963: 5962: 5950: 5949: 5930: 5915: 5914: 5902: 5901: 5889: 5888: 5869: 5854: 5853: 5841: 5840: 5828: 5827: 5808: 5788: 5786: 5785: 5780: 5764: 5762: 5761: 5756: 5744: 5742: 5741: 5736: 5725: 5717: 5689: 5687: 5686: 5681: 5679: 5674: 5666: 5651:similarity class 5578: 5572: 5522:Klein four-group 5502: 5496: 5457: 5456: 5455: 5451: 5450: 5446: 5445: 5441: 5440: 5436: 5435: 5408:Klein four-group 5395: 5351: 5350: 5349: 5345: 5344: 5340: 5339: 5335: 5334: 5330: 5329: 5305: 5255: 5179: 5106: 5062: 5061: 5060: 5056: 5055: 5051: 5050: 5046: 5045: 5041: 5040: 5004: 4950: 4949: 4909:generating point 4815: 4813: 4812: 4807: 4805: 4795: 4793: 4785: 4782: 4780: 4779: 4774: 4772: 4762: 4760: 4752: 4749: 4747: 4746: 4741: 4739: 4729: 4727: 4719: 4716: 4714: 4713: 4708: 4706: 4696: 4694: 4686: 4683: 4681: 4680: 4675: 4673: 4663: 4661: 4653: 4650: 4648: 4647: 4642: 4640: 4630: 4628: 4620: 4617: 4615: 4614: 4609: 4607: 4597: 4595: 4587: 4584: 4582: 4581: 4576: 4574: 4564: 4562: 4554: 4551: 4549: 4548: 4543: 4532: 4525: 4523: 4522: 4517: 4515: 4505: 4503: 4495: 4492: 4490: 4489: 4484: 4482: 4472: 4470: 4462: 4459: 4457: 4456: 4451: 4440: 4433: 4431: 4430: 4425: 4423: 4413: 4411: 4403: 4400: 4398: 4397: 4392: 4390: 4380: 4378: 4370: 4367: 4365: 4364: 4359: 4357: 4347: 4345: 4337: 4330: 4328: 4327: 4322: 4320: 4310: 4308: 4300: 4297: 4295: 4294: 4289: 4278: 4275: 4273: 4272: 4267: 4265: 4255: 4253: 4245: 4233: 4231: 4230: 4225: 4223: 4217: 4213: 4210: 4207: 4198: 4193: 4187: 4185: 4184: 4179: 4177: 4174: 4166: 4140: 4138: 4137: 4132: 4124: 4114: 4112: 4104: 4100: 4098: 4097: 4092: 4087: 4079: 4078: 4064: 4050: 4048: 4047: 4042: 4034: 4024: 4022: 4014: 4010: 4008: 4007: 4002: 3997: 3989: 3988: 3974: 3960: 3958: 3957: 3952: 3944: 3934: 3932: 3924: 3920: 3918: 3917: 3912: 3907: 3899: 3898: 3884: 3866: 3864: 3863: 3858: 3856: 3847: 3838: 3833: 3829: 3827: 3826: 3821: 3819: 3816: 3811: 3802: 3793: 3788: 3784: 3782: 3781: 3776: 3768: 3758: 3756: 3748: 3739: 3737: 3736: 3731: 3729: 3720: 3711: 3706: 3702: 3700: 3699: 3694: 3692: 3689: 3684: 3675: 3666: 3661: 3657: 3655: 3654: 3649: 3638: 3629: 3627: 3626: 3621: 3619: 3610: 3601: 3596: 3592: 3590: 3589: 3584: 3582: 3579: 3568: 3563: 3559: 3557: 3556: 3551: 3543: 3533: 3531: 3523: 3502: 3500: 3499: 3494: 3492: 3489: 3472: 3467: 3463: 3461: 3460: 3455: 3453: 3450: 3433: 3428: 3424: 3422: 3421: 3416: 3405: 3381: 3380: 3377: 3376: 3375: 3371: 3370: 3366: 3365: 3361: 3360: 3356: 3355: 3349: 3348: 3347: 3343: 3342: 3338: 3337: 3333: 3332: 3328: 3327: 3306: 3305: 3299: 3298: 3297: 3293: 3292: 3288: 3287: 3283: 3282: 3278: 3277: 3271: 3270: 3269: 3265: 3264: 3260: 3259: 3255: 3254: 3250: 3249: 3243: 3242: 3236: 3235: 3201:is an irregular 3093:inscribed sphere 3042: 3033: 3026: 2963:Helical stacking 2944: 2937: 2930: 2929: 2917:spherical tiling 2911:Spherical tiling 2838: 2831: 2810: 2804: 2791:has two special 2761: 2747: 2740: 2738: 2737: 2734: 2731: 2713: 2695: 2693: 2692: 2687: 2685: 2684: 2665: 2663: 2662: 2657: 2655: 2639: 2637: 2636: 2631: 2629: 2628: 2605: 2603: 2602: 2597: 2595: 2594: 2593: 2587: 2528: 2526: 2525: 2520: 2503: 2488: 2487: 2486: 2482: 2481: 2477: 2476: 2472: 2471: 2467: 2466: 2445: 2443: 2442: 2437: 2435: 2430: 2418: 2416: 2415: 2410: 2408: 2254: 2252: 2251: 2246: 2244: 2239: 2238: 2233: 2227: 2218: 2216: 2215: 2210: 2208: 2175: 2171: 2170: 2162: 2154: 2145: 2144: 2136: 2127: 2126: 2109: 2105: 2104: 2096: 2088: 2079: 2078: 2073: 2064: 2063: 2047: 2043: 2042: 2034: 2020: 2011: 2010: 1992: 1990: 1989: 1984: 1963: 1961: 1960: 1955: 1953: 1949: 1948: 1942: 1938: 1915: 1911: 1907: 1903: 1902: 1896: 1892: 1836: 1834: 1833: 1828: 1826: 1816: 1812: 1804: 1783: 1779: 1775: 1767: 1749: 1741: 1732: 1728: 1720: 1657: 1656: 1650: 1649: 1637: 1635: 1634: 1629: 1627: 1620: 1619: 1614: 1610: 1608: 1603: 1590: 1585: 1572: 1567: 1554: 1549: 1537: 1536: 1514: 1510: 1508: 1503: 1490: 1485: 1472: 1467: 1454: 1449: 1437: 1436: 1412: 1411: 1406: 1402: 1401: 1396: 1395: 1394: 1381: 1376: 1371: 1369: 1364: 1351: 1346: 1333: 1328: 1315: 1310: 1300: 1285: 1280: 1279: 1278: 1265: 1260: 1255: 1253: 1248: 1235: 1230: 1217: 1212: 1199: 1194: 1184: 1171: 1169: 1168: 1163: 1161: 1160: 1145:, and distances 1144: 1142: 1141: 1136: 1124: 1122: 1121: 1116: 1104: 1102: 1101: 1096: 1094: 1087: 1081: 1077: 1072: 1071: 1070: 1054: 1048: 1044: 1039: 1031: 1026: 1025: 1024: 1007: 1001: 997: 989: 981: 964: 955: 954: 935: 933: 932: 927: 925: 924: 923: 903: 901: 900: 895: 893: 892: 891: 871: 869: 868: 863: 848: 846: 845: 840: 815: 813: 812: 807: 805: 798: 797: 785: 781: 776: 754: 750: 749: 741: 718: 717: 705: 701: 700: 695: 672: 668: 660: 630: 628: 627: 622: 617: 616: 601: 599: 598: 593: 587: 586: 577: 569: 560: 559: 554: 550: 549: 548: 539: 530: 529: 519: 511: 496: 494: 493: 488: 483: 474: 473: 464: 462: 461: 456: 451: 450: 435: 430: 428: 427: 415: 411: 410: 409: 400: 391: 390: 364: 362: 361: 356: 344: 342: 341: 336: 319: 305:Hill tetrahedron 255:stella octangula 228:regular polygons 189: 177:stella octangula 172: 160: 123:convex polyhedra 83:convex polyhedra 52: 17721: 17720: 17716: 17715: 17714: 17712: 17711: 17710: 17681:Platonic solids 17666: 17665: 17634: 17627: 17620: 17503: 17496: 17489: 17453: 17446: 17439: 17403: 17396: 17230:Regular polygon 17223: 17214: 17207: 17203: 17196: 17192: 17183: 17174: 17167: 17163: 17151: 17145: 17141: 17129: 17111: 17100: 17071: 17066: 17055: 16994:Dihedral others 16985: 16964: 16939: 16914: 16843: 16840: 16839: 16830: 16759: 16748: 16747: 16738: 16701: 16699:Platonic solids 16693: 16687: 16657: 16652: 16603: 16592:Star polyhedron 16574: 16539: 16487: 16464:Hexecontahedron 16446:Triacontahedron 16428: 16419:Enneadecahedron 16409:Heptadecahedron 16399:Pentadecahedron 16394:Tetradecahedron 16367: 16306: 16299: 16294: 16226: 16221: 16215: 16185: 16103: 16090: 16053:10.2307/3608204 16032: 15997:Coxeter, H.S.M. 15962: 15941: 15936: 15935: 15914: 15910: 15880: 15876: 15866: 15864: 15859: 15858: 15854: 15836: 15832: 15825: 15800: 15796: 15764: 15760: 15750: 15748: 15744: 15733: 15727: 15723: 15713: 15711: 15710:on 10 June 2007 15707: 15692: 15686: 15682: 15672: 15670: 15662: 15661: 15657: 15626: 15622: 15615: 15599: 15595: 15581: 15577: 15561: 15557: 15556: 15552: 15530: 15524: 15521: 15517: 15503: 15499: 15489: 15487: 15476: 15472: 15457: 15446: 15438: 15434: 15403: 15399: 15384:10.2307/2689983 15363: 15359: 15344: 15340: 15335:. Bulletin AMQ. 15332: 15326: 15322: 15306: 15305: 15300: 15292: 15288: 15283: 15279: 15264:10.2307/3647851 15235: 15229: 15225: 15217: 15213: 15205: 15201: 15193: 15186: 15182:, MathPages.com 15178: 15174: 15162: 15158: 15146: 15142: 15105: 15101: 15089: 15085: 15077: 15073: 15068: 15061: 15057: 15049: 15042: 15013:Coxeter, H.S.M. 15010: 15006: 14998: 14994: 14985: 14984: 14980: 14972: 14968: 14964: 14949: 14945: 14933: 14929: 14920: 14916: 14908: 14899: 14887: 14883: 14871: 14867: 14859: 14855: 14842: 14838: 14817: 14810: 14805: 14800: 14799: 14782: 14775: 14765:Plateau borders 14762: 14758: 14753: 14675: 14665: 14648:Platonic graphs 14645: 14637: 14491: 14456:Stanley Kubrick 14453: 14452: 14451: 14450: 14446: 14445: 14444: 14436: 14435: 14426: 14425: 14424: 14416: 14415: 14404: 14402:Popular culture 14388: 14382: 14362:polyhedral dice 14333: 14325: 14319: 14280: 14272:Main articles: 14270: 14230: 14227: 14226: 14225: 14223: 14210: 14207: 14204: 14203: 14201: 14186: 14183: 14182: 14181: 14179: 14170: 14167: 14166: 14165: 14163: 14150:molecules. All 14144: 14115: 14103: 14091: 14023: 14018: 13954: 13949: 13940: 13935: 13926: 13921: 13746: 13551:Compact hyper. 13540: 13332: 13327: 13322: 13317: 13312: 13310: 13303: 13298: 13293: 13288: 13283: 13281: 13274: 13269: 13264: 13259: 13254: 13252: 13245: 13240: 13235: 13230: 13225: 13223: 13216: 13211: 13206: 13201: 13196: 13194: 13187: 13182: 13177: 13172: 13167: 13165: 13158: 13153: 13148: 13143: 13138: 13136: 13129: 13124: 13119: 13114: 13109: 13107: 12843: 12644: 12637: 12628:Antiprism name 12623: 12412: 12399: 12396: 12394: 12389: 12387: 12382: 12380: 12369: 12363: 12355: 12343:arithmetic mean 12339: 12317: 12316: 12309: 12308: 12302: 12298: 12292: 12288: 12276: 12272: 12266: 12262: 12256: 12255: 12249: 12245: 12239: 12235: 12223: 12219: 12213: 12209: 12203: 12202: 12196: 12192: 12186: 12182: 12170: 12166: 12160: 12156: 12149: 12145: 12135: 12127: 12122: 12117: 12115: 12110: 12103: 12102: 12096: 12085: 12081: 12072: 12068: 12067: 12063: 12062: 12059: 12058: 12052: 12041: 12037: 12028: 12024: 12023: 12019: 12018: 12015: 12014: 12008: 11997: 11993: 11984: 11980: 11979: 11975: 11974: 11970: 11966: 11958: 11953: 11948: 11946: 11934: 11930: 11923: 11916: 11914: 11911: 11910: 11904: 11898: 11895: 11889: 11886: 11880: 11877: 11871: 11865: 11861: 11832: 11705: 11697: 11694: 11693: 11655: 11630: 11626: 11616: 11604: 11600: 11590: 11578: 11574: 11564: 11552: 11548: 11538: 11530: 11527: 11526: 11501: 11497: 11488: 11484: 11475: 11471: 11462: 11458: 11457: 11449: 11447: 11439: 11436: 11435: 11426: 11417: 11408: 11399: 11364: 11360: 11355: 11344: 11339: 11330: 11319: 11314: 11305: 11294: 11289: 11280: 11269: 11264: 11255: 11253: 11250: 11249: 11240: 11222: 11123: 11120: 11119: 11043: 11042: 11038: 11019: 11018: 11014: 10995: 10994: 10990: 10971: 10970: 10966: 10955: 10952: 10951: 10946: 10939: 10932: 10925: 10893: 10866: 10862: 10850: 10846: 10840: 10836: 10824: 10820: 10808: 10804: 10798: 10794: 10782: 10778: 10766: 10762: 10756: 10752: 10737: 10732: 10719: 10714: 10701: 10696: 10683: 10678: 10672: 10669: 10668: 10647: 10643: 10637: 10633: 10631: 10628: 10627: 10607: 10603: 10601: 10598: 10597: 10580: 10576: 10574: 10571: 10570: 10553: 10549: 10547: 10544: 10543: 10526: 10522: 10520: 10517: 10516: 10499: 10495: 10493: 10490: 10489: 10472: 10468: 10466: 10463: 10462: 10445: 10441: 10439: 10436: 10435: 10432: 10426: 10261: 10258: 10257: 10231: 10218: 10154:Thomson problem 10142: 10074:Lorenz Lindelöf 10055: 10023: 10022: 10014: 10004: 10000: 9996: 9988: 9978: 9974: 9970: 9962: 9952: 9948: 9944: 9935: 9934: 9924: 9920: 9916: 9908: 9900: 9890: 9886: 9882: 9874: 9864: 9860: 9856: 9847: 9846: 9836: 9832: 9828: 9820: 9810: 9806: 9802: 9794: 9786: 9776: 9772: 9768: 9759: 9758: 9748: 9744: 9740: 9732: 9722: 9718: 9714: 9706: 9696: 9692: 9688: 9680: 9667: 9666: 9664: 9661: 9660: 9619:of a triangle. 9559: 9530: 9510: 9502: 9494: 9490: 9488: 9480: 9477: 9476: 9460: 9457: 9456: 9436: 9428: 9426: 9423: 9422: 9406: 9403: 9402: 9386: 9383: 9382: 9368:dihedral angles 9336: 9335: 9280: 9226: 9225: 9171: 9117: 9116: 9062: 9007: 9005: 9002: 9001: 8987: 8986: 8970: 8961: 8945: 8936: 8935: 8919: 8910: 8894: 8884: 8882: 8879: 8878: 8848: 8729: 8721: 8718: 8717: 8701: 8698: 8697: 8681: 8678: 8677: 8661: 8658: 8657: 8641: 8638: 8637: 8621: 8618: 8617: 8601: 8598: 8597: 8581: 8578: 8577: 8561: 8558: 8557: 8541: 8538: 8537: 8521: 8518: 8517: 8501: 8498: 8497: 8481: 8478: 8477: 8449: 8440: 8436: 8428: 8419: 8415: 8407: 8398: 8394: 8386: 8375: 8364: 8347: 8331: 8329: 8321: 8318: 8317: 8303: 8302: 8293: 8289: 8280: 8276: 8267: 8263: 8256: 8250: 8249: 8240: 8236: 8227: 8223: 8214: 8210: 8203: 8197: 8196: 8187: 8183: 8174: 8170: 8161: 8157: 8150: 8143: 8141: 8138: 8137: 8102: 8098: 8092: 8088: 8079: 8075: 8069: 8065: 8056: 8052: 8046: 8042: 8033: 8029: 8023: 8019: 8013: 8009: 8003: 7995: 7992: 7991: 7975: 7972: 7971: 7955: 7952: 7951: 7935: 7932: 7931: 7915: 7912: 7911: 7895: 7892: 7891: 7875: 7872: 7871: 7855: 7852: 7851: 7845:Heron's formula 7832: 7808: 7797: 7776: 7775: 7770: 7764: 7759: 7753: 7747: 7742: 7736: 7730: 7725: 7719: 7713: 7712: 7706: 7701: 7695: 7690: 7684: 7679: 7673: 7667: 7662: 7656: 7650: 7649: 7643: 7638: 7632: 7626: 7621: 7615: 7610: 7604: 7599: 7593: 7587: 7586: 7580: 7575: 7569: 7563: 7558: 7552: 7546: 7541: 7535: 7530: 7524: 7523: 7518: 7513: 7508: 7503: 7493: 7492: 7483: 7479: 7471: 7468: 7467: 7430: 7429: 7424: 7419: 7414: 7408: 7407: 7401: 7397: 7395: 7389: 7385: 7383: 7377: 7373: 7371: 7365: 7361: 7358: 7357: 7351: 7347: 7345: 7339: 7335: 7333: 7327: 7323: 7321: 7315: 7311: 7308: 7307: 7301: 7297: 7295: 7289: 7285: 7283: 7277: 7273: 7271: 7265: 7261: 7257: 7253: 7249: 7245: 7231: 7228: 7227: 7136: 7127: 7123: 7115: 7106: 7102: 7094: 7085: 7081: 7073: 7062: 7051: 7034: 7018: 7016: 7008: 7005: 7004: 6983: 6982: 6977: 6970: 6969: 6961: 6958: 6957: 6939: 6938: 6933: 6926: 6925: 6917: 6914: 6913: 6895: 6894: 6889: 6882: 6881: 6873: 6870: 6869: 6849: 6848: 6840: 6820: 6812: 6809: 6808: 6800: 6780: 6772: 6769: 6768: 6760: 6740: 6732: 6725: 6724: 6722: 6719: 6718: 6715: 6711: 6694: 6693: 6686: 6682: 6681: 6679: 6674: 6666: 6664: 6659: 6651: 6648: 6647: 6642: 6634: 6632: 6625: 6621: 6620: 6618: 6613: 6605: 6602: 6601: 6596: 6588: 6586: 6581: 6573: 6571: 6564: 6560: 6559: 6552: 6551: 6542: 6538: 6530: 6527: 6526: 6515: 6499: 6498: 6486: 6482: 6473: 6469: 6460: 6456: 6445: 6442: 6441: 6429: 6425: 6416: 6412: 6403: 6399: 6388: 6385: 6384: 6372: 6368: 6359: 6355: 6346: 6342: 6331: 6324: 6323: 6321: 6318: 6317: 6314: 6310: 6293: 6292: 6287: 6284: 6283: 6278: 6275: 6274: 6269: 6262: 6261: 6247: 6244: 6243: 6240: 6236: 6219: 6218: 6213: 6211: 6206: 6204: 6199: 6192: 6191: 6177: 6174: 6173: 6160: 6157: 6154: 6153: 6151: 6124: 6116: 6108: 6100: 6092: 6084: 6068: 6066: 6063: 6062: 6045: 6044: 6032: 6028: 6019: 6015: 6006: 6002: 5992: 5987: 5984: 5983: 5971: 5967: 5958: 5954: 5945: 5941: 5931: 5926: 5923: 5922: 5910: 5906: 5897: 5893: 5884: 5880: 5870: 5865: 5862: 5861: 5849: 5845: 5836: 5832: 5823: 5819: 5809: 5804: 5800: 5798: 5795: 5794: 5774: 5771: 5770: 5750: 5747: 5746: 5716: 5708: 5705: 5704: 5701: 5696: 5670: 5665: 5663: 5660: 5659: 5639: 5622: 5610: 5599: 5591: 5573: 5559: 5554: 5550: 5547: 5541: 5540: 5529: 5519: 5511: 5497: 5469: 5453: 5448: 5443: 5438: 5433: 5431: 5429: 5422: 5415: 5404: 5380: 5375: 5371: 5368: 5364: 5363: 5347: 5342: 5337: 5332: 5327: 5325: 5323: 5314: 5275: 5260: 5247: 5233: 5226: 5225: 5218: 5217: 5206: 5195: 5188: 5166: 5161: 5157: 5154: 5150: 5149: 5138: 5130:symmetric group 5127: 5119: 5093: 5088: 5084: 5078: 5077: 5065:Schläfli symbol 5058: 5053: 5048: 5043: 5038: 5036: 5034:Coxeter diagram 5031: 5023:symmetric group 5020: 5013: 4959: 4957: 4946: 4942: 4933: 4866: 4835:Hill tetrahedra 4823: 4792: 4790: 4787: 4786: 4759: 4757: 4754: 4753: 4726: 4724: 4721: 4720: 4693: 4691: 4688: 4687: 4660: 4658: 4655: 4654: 4627: 4625: 4622: 4621: 4594: 4592: 4589: 4588: 4561: 4559: 4556: 4555: 4537: 4534: 4533: 4502: 4500: 4497: 4496: 4469: 4467: 4464: 4463: 4445: 4442: 4441: 4410: 4408: 4405: 4404: 4377: 4375: 4372: 4371: 4344: 4342: 4339: 4338: 4307: 4305: 4302: 4301: 4283: 4280: 4279: 4252: 4250: 4247: 4246: 4209: 4208: 4205: 4203: 4200: 4199: 4173: 4171: 4168: 4167: 4111: 4109: 4106: 4105: 4083: 4074: 4071: 4069: 4066: 4065: 4021: 4019: 4016: 4015: 3993: 3984: 3981: 3979: 3976: 3975: 3931: 3929: 3926: 3925: 3903: 3894: 3891: 3889: 3886: 3885: 3845: 3843: 3840: 3839: 3815: 3800: 3798: 3795: 3794: 3755: 3753: 3750: 3749: 3718: 3716: 3713: 3712: 3688: 3673: 3671: 3668: 3667: 3643: 3640: 3639: 3608: 3606: 3603: 3602: 3578: 3573: 3570: 3569: 3530: 3528: 3525: 3524: 3488: 3477: 3474: 3473: 3449: 3438: 3435: 3434: 3410: 3407: 3406: 3373: 3368: 3363: 3358: 3353: 3351: 3345: 3340: 3335: 3330: 3325: 3323: 3303: 3301: 3295: 3290: 3285: 3280: 3275: 3273: 3267: 3262: 3257: 3252: 3247: 3245: 3240: 3238: 3233: 3231: 3214:name for it is 3188:right triangles 3172: 3155:dihedral angles 3147: 3141: 3107: 3101: 3043: 3034: 3024: 2996:four dimensions 2965: 2913: 2858: 2844: 2798: 2785: 2749: 2745: 2735: 2732: 2718: 2717: 2715: 2711: 2680: 2676: 2674: 2671: 2670: 2651: 2649: 2646: 2645: 2624: 2620: 2618: 2615: 2614: 2612:symmetric group 2589: 2588: 2583: 2582: 2580: 2577: 2576: 2555:point inversion 2535: 2499: 2497: 2494: 2493: 2491:Schläfli symbol 2484: 2479: 2474: 2469: 2464: 2462: 2460:Coxeter diagram 2429: 2424: 2421: 2420: 2406: 2405: 2370: 2337: 2336: 2301: 2273: 2271: 2268: 2267: 2232: 2228: 2226: 2224: 2221: 2220: 2206: 2205: 2179: 2161: 2143: 2125: 2121: 2117: 2114: 2113: 2095: 2077: 2062: 2058: 2054: 2051: 2033: 2009: 2008: 2004: 2000: 1998: 1995: 1994: 1975: 1972: 1971: 1937: 1921: 1917: 1909: 1891: 1872: 1868: 1866: 1863: 1862: 1855: 1824: 1823: 1803: 1799: 1781: 1780: 1766: 1762: 1740: 1733: 1719: 1715: 1705: 1703: 1700: 1699: 1654: 1652: 1647: 1645: 1625: 1624: 1615: 1604: 1599: 1586: 1581: 1568: 1563: 1550: 1545: 1532: 1528: 1527: 1523: 1522: 1515: 1504: 1499: 1486: 1481: 1468: 1463: 1450: 1445: 1432: 1428: 1427: 1423: 1417: 1416: 1407: 1390: 1386: 1382: 1380: 1365: 1360: 1347: 1342: 1329: 1324: 1311: 1306: 1301: 1299: 1298: 1294: 1293: 1286: 1274: 1270: 1266: 1264: 1249: 1244: 1231: 1226: 1213: 1208: 1195: 1190: 1185: 1183: 1179: 1177: 1174: 1173: 1156: 1152: 1150: 1147: 1146: 1130: 1127: 1126: 1110: 1107: 1106: 1092: 1091: 1076: 1066: 1065: 1061: 1058: 1043: 1030: 1020: 1019: 1015: 1012: 1011: 996: 980: 971: 953: 943: 941: 938: 937: 919: 918: 914: 912: 909: 908: 887: 886: 882: 880: 877: 876: 857: 854: 853: 834: 831: 830: 803: 802: 793: 789: 775: 771: 755: 740: 736: 732: 723: 722: 713: 709: 694: 690: 686: 673: 659: 655: 645: 643: 640: 639: 612: 608: 592: 588: 582: 578: 576: 558: 544: 540: 528: 527: 523: 510: 502: 499: 498: 472: 470: 467: 466: 446: 442: 429: 423: 419: 405: 401: 389: 388: 384: 370: 367: 366: 350: 347: 346: 330: 327: 326: 313: 224:Platonic solids 201: 200: 199: 198: 197: 190: 181: 180: 179: 173: 165: 164: 161: 150: 73:, six straight 37: 17: 12: 11: 5: 17719: 17709: 17708: 17703: 17698: 17693: 17688: 17683: 17678: 17662: 17661: 17646: 17645: 17636: 17632: 17625: 17618: 17614: 17605: 17588: 17579: 17568: 17567: 17565: 17563: 17558: 17549: 17544: 17538: 17537: 17535: 17533: 17528: 17519: 17514: 17508: 17507: 17505: 17501: 17494: 17487: 17483: 17478: 17469: 17464: 17458: 17457: 17455: 17451: 17444: 17437: 17433: 17428: 17419: 17414: 17408: 17407: 17405: 17401: 17394: 17390: 17385: 17376: 17371: 17365: 17364: 17362: 17360: 17355: 17346: 17341: 17335: 17334: 17325: 17320: 17315: 17306: 17301: 17295: 17294: 17285: 17283: 17278: 17269: 17264: 17258: 17257: 17252: 17247: 17242: 17237: 17232: 17226: 17225: 17221: 17217: 17212: 17201: 17190: 17181: 17172: 17165: 17159: 17149: 17143: 17137: 17131: 17125: 17119: 17113: 17112: 17101: 17099: 17098: 17091: 17084: 17076: 17068: 17067: 17060: 17057: 17056: 17054: 17053: 17048: 17043: 17038: 17033: 17028: 17023: 17018: 17013: 17008: 17003: 16997: 16995: 16991: 16990: 16987: 16986: 16984: 16983: 16978: 16972: 16970: 16966: 16965: 16963: 16962: 16957: 16951: 16945: 16941: 16940: 16938: 16937: 16930: 16922: 16920: 16916: 16915: 16913: 16912: 16907: 16902: 16897: 16892: 16887: 16882: 16877: 16872: 16867: 16862: 16857: 16852: 16846: 16844: 16837:Catalan solids 16835: 16832: 16831: 16829: 16828: 16823: 16818: 16813: 16808: 16803: 16798: 16793: 16788: 16783: 16778: 16776:truncated cube 16773: 16768: 16762: 16760: 16743: 16740: 16739: 16737: 16736: 16731: 16726: 16721: 16716: 16710: 16708: 16695: 16694: 16686: 16685: 16678: 16671: 16663: 16654: 16653: 16651: 16650: 16648:parallelepiped 16645: 16640: 16635: 16630: 16625: 16620: 16614: 16612: 16605: 16604: 16602: 16601: 16595: 16589: 16582: 16580: 16576: 16575: 16573: 16572: 16566: 16560: 16554: 16551:Platonic solid 16547: 16545: 16541: 16540: 16538: 16537: 16536: 16535: 16529: 16523: 16511: 16506: 16501: 16495: 16493: 16489: 16488: 16486: 16485: 16479: 16473: 16467: 16461: 16455: 16449: 16443: 16436: 16434: 16430: 16429: 16427: 16426: 16421: 16416: 16414:Octadecahedron 16411: 16406: 16404:Hexadecahedron 16401: 16396: 16391: 16386: 16381: 16375: 16373: 16369: 16368: 16366: 16365: 16360: 16355: 16350: 16345: 16340: 16335: 16330: 16325: 16320: 16314: 16312: 16308: 16307: 16304: 16301: 16300: 16293: 16292: 16285: 16278: 16270: 16264: 16263: 16251: 16246: 16225: 16224:External links 16222: 16220: 16219: 16213: 16200: 16178: 16150:(2): 379–400. 16135: 16122: 16108: 16094: 16088: 16073: 16036: 16030: 16011: 15993: 15979: 15966: 15960: 15942: 15940: 15937: 15934: 15933: 15908: 15874: 15852: 15830: 15823: 15803:Holmes, Arthur 15794: 15758: 15747:on 7 June 2011 15721: 15680: 15655: 15620: 15613: 15593: 15575: 15550: 15528: 15515: 15497: 15470: 15444: 15432: 15397: 15357: 15338: 15320: 15286: 15277: 15248:(8): 679–693. 15223: 15211: 15199: 15184: 15172: 15156: 15140: 15099: 15083: 15071: 15066: 15055: 15040: 15027:(1–3): 59–71. 15004: 15002:, p. 181. 14992: 14978: 14966: 14963: 14962: 14957: 14950: 14943: 14927: 14914: 14897: 14881: 14873:Shavinina 2013 14865: 14853: 14836: 14807: 14806: 14804: 14801: 14798: 14797: 14773: 14755: 14754: 14752: 14749: 14748: 14747: 14742: 14737: 14731: 14726: 14721: 14716: 14711: 14705: 14691: 14686: 14681: 14674: 14671: 14668: 14667: 14652:Platonic solid 14643: 14635: 14632:complete graph 14618: 14617: 14611: 14610: 14581: 14577: 14576: 14573: 14567: 14566: 14563: 14557: 14556: 14553: 14547: 14546: 14543: 14537: 14536: 14533: 14527: 14526: 14523: 14517: 14516: 14513: 14507: 14506: 14498: 14497: 14490: 14487: 14448: 14447: 14438: 14437: 14429: 14428: 14427: 14418: 14417: 14409: 14408: 14407: 14406: 14405: 14403: 14400: 14384:Main article: 14381: 14378: 14352:Especially in 14332: 14329: 14321:Main article: 14318: 14315: 14269: 14266: 14240:phase diagrams 14228: 14184: 14168: 14140:Main article: 14114: 14111: 14102: 14099: 14090: 14087: 14043:polygonal mesh 14022: 14019: 14017: 14014: 13956: 13955: 13950: 13943: 13941: 13936: 13929: 13927: 13922: 13915: 13913: 13907: 13906: 13903: 13900: 13897: 13894: 13889: 13884: 13879: 13874: 13869: 13864: 13859: 13853: 13852: 13845: 13838: 13831: 13824: 13817: 13810: 13803: 13796: 13789: 13782: 13775: 13767: 13766: 13763: 13760: 13757: 13752: 13748: 13747: 13745: 13744: 13737: 13730: 13722: 13705:vertex figures 13699: 13698: 13695: 13692: 13689: 13686: 13681: 13676: 13671: 13666: 13661: 13656: 13651: 13645: 13644: 13637: 13630: 13623: 13616: 13609: 13602: 13595: 13588: 13581: 13574: 13567: 13559: 13558: 13555: 13552: 13549: 13546: 13542: 13541: 13539: 13538: 13531: 13524: 13516: 13485: 13484: 13479: 13474: 13469: 13464: 13459: 13454: 13449: 13443: 13442: 13435: 13428: 13421: 13414: 13407: 13400: 13393: 13385: 13384: 13380: 13379: 13374: 13369: 13364: 13359: 13354: 13349: 13344: 13338: 13337: 13308: 13279: 13250: 13221: 13192: 13163: 13134: 13104: 13103: 13096: 13089: 13082: 13075: 13068: 13061: 13054: 13046: 13045: 13042: 13032: 13031: 13010: 13009: 13006: 13003: 13000: 12997: 12994: 12991: 12988: 12982: 12981: 12974: 12968: 12961: 12954: 12947: 12940: 12933: 12926: 12925: 12923: 12920: 12913: 12906: 12899: 12892: 12885: 12878: 12877: 12872: 12869: 12864: 12859: 12854: 12849: 12840: 12812: 12811: 12808: 12805: 12802: 12799: 12796: 12793: 12790: 12787: 12785:Vertex config. 12781: 12780: 12773: 12770: 12763: 12756: 12749: 12742: 12735: 12728: 12724: 12723: 12721: 12718: 12711: 12704: 12697: 12690: 12683: 12676: 12672: 12671: 12666: 12663: 12658: 12653: 12648: 12641: 12634: 12629: 12622: 12621: 12614: 12607: 12599: 12569: 12568: 12565: 12558: 12551: 12544: 12537: 12530: 12522: 12521: 12518: 12511: 12504: 12497: 12490: 12483: 12475: 12474: 12471: 12468: 12465: 12461: 12460: 12457: 12454: 12449: 12444: 12439: 12434: 12428: 12427: 12411: 12408: 12379:with areas of 12365:Main article: 12362: 12359: 12354: 12351: 12338: 12335: 12331: 12330: 12314: 12305: 12301: 12295: 12291: 12287: 12284: 12279: 12275: 12269: 12265: 12261: 12258: 12257: 12252: 12248: 12242: 12238: 12234: 12231: 12226: 12222: 12216: 12212: 12208: 12205: 12204: 12199: 12195: 12189: 12185: 12181: 12178: 12173: 12169: 12163: 12159: 12155: 12152: 12151: 12148: 12142: 12139: 12134: 12131: 12128: 12123: 12116: 12111: 12108: 12099: 12094: 12088: 12084: 12080: 12075: 12071: 12066: 12061: 12060: 12055: 12050: 12044: 12040: 12036: 12031: 12027: 12022: 12017: 12016: 12011: 12006: 12000: 11996: 11992: 11987: 11983: 11978: 11973: 11972: 11969: 11965: 11962: 11959: 11954: 11947: 11945: 11940: 11937: 11933: 11929: 11926: 11924: 11922: 11919: 11918: 11902: 11893: 11884: 11875: 11860: 11857: 11856: 11855: 11844: 11838: 11835: 11830: 11827: 11824: 11821: 11818: 11815: 11812: 11809: 11806: 11803: 11800: 11797: 11794: 11791: 11788: 11785: 11782: 11779: 11776: 11773: 11770: 11767: 11764: 11761: 11758: 11755: 11752: 11749: 11746: 11743: 11740: 11737: 11734: 11731: 11728: 11725: 11722: 11719: 11716: 11713: 11710: 11704: 11701: 11654: 11651: 11638: 11633: 11629: 11623: 11620: 11615: 11612: 11607: 11603: 11597: 11594: 11589: 11586: 11581: 11577: 11571: 11568: 11563: 11560: 11555: 11551: 11545: 11542: 11537: 11534: 11522: 11521: 11504: 11500: 11496: 11491: 11487: 11483: 11478: 11474: 11470: 11465: 11461: 11455: 11452: 11446: 11443: 11422: 11413: 11404: 11395: 11386: 11385: 11374: 11367: 11363: 11359: 11354: 11347: 11342: 11338: 11334: 11329: 11322: 11317: 11313: 11309: 11304: 11297: 11292: 11288: 11284: 11279: 11272: 11267: 11263: 11259: 11236: 11221: 11218: 11217: 11216: 11205: 11202: 11199: 11196: 11193: 11190: 11187: 11184: 11181: 11178: 11175: 11172: 11169: 11166: 11163: 11160: 11157: 11154: 11151: 11148: 11145: 11142: 11139: 11136: 11133: 11130: 11127: 11073: 11072: 11061: 11058: 11055: 11052: 11046: 11041: 11037: 11034: 11031: 11028: 11022: 11017: 11013: 11010: 11007: 11004: 10998: 10993: 10989: 10986: 10983: 10980: 10974: 10969: 10965: 10962: 10959: 10944: 10937: 10930: 10923: 10892: 10891:Interior point 10889: 10877: 10872: 10869: 10865: 10861: 10858: 10853: 10849: 10843: 10839: 10835: 10830: 10827: 10823: 10819: 10816: 10811: 10807: 10801: 10797: 10793: 10788: 10785: 10781: 10777: 10774: 10769: 10765: 10759: 10755: 10751: 10748: 10745: 10740: 10735: 10731: 10727: 10722: 10717: 10713: 10709: 10704: 10699: 10695: 10691: 10686: 10681: 10677: 10665:law of cosines 10650: 10646: 10640: 10636: 10613: 10610: 10606: 10583: 10579: 10556: 10552: 10529: 10525: 10502: 10498: 10475: 10471: 10448: 10444: 10428:Main article: 10425: 10422: 10401: 10400: 10388: 10385: 10382: 10379: 10376: 10373: 10370: 10367: 10364: 10361: 10358: 10355: 10352: 10349: 10346: 10343: 10340: 10337: 10334: 10331: 10328: 10325: 10322: 10319: 10316: 10313: 10310: 10307: 10304: 10301: 10298: 10295: 10292: 10289: 10286: 10283: 10280: 10277: 10274: 10271: 10268: 10265: 10227:Main article: 10217: 10214: 10141: 10138: 10116:coincide with 10053: 10048: 10047: 10035: 10032: 10027: 10021: 10018: 10015: 10012: 10007: 10003: 9999: 9995: 9992: 9989: 9986: 9981: 9977: 9973: 9969: 9966: 9963: 9960: 9955: 9951: 9947: 9943: 9940: 9937: 9936: 9932: 9927: 9923: 9919: 9915: 9912: 9909: 9907: 9904: 9901: 9898: 9893: 9889: 9885: 9881: 9878: 9875: 9872: 9867: 9863: 9859: 9855: 9852: 9849: 9848: 9844: 9839: 9835: 9831: 9827: 9824: 9821: 9818: 9813: 9809: 9805: 9801: 9798: 9795: 9793: 9790: 9787: 9784: 9779: 9775: 9771: 9767: 9764: 9761: 9760: 9756: 9751: 9747: 9743: 9739: 9736: 9733: 9730: 9725: 9721: 9717: 9713: 9710: 9707: 9704: 9699: 9695: 9691: 9687: 9684: 9681: 9679: 9676: 9673: 9672: 9670: 9564:Spieker center 9558: 9555: 9543: 9538: 9533: 9529: 9525: 9522: 9519: 9516: 9513: 9509: 9505: 9501: 9497: 9493: 9487: 9484: 9464: 9451:as calculated 9439: 9435: 9431: 9410: 9390: 9334: 9331: 9328: 9325: 9322: 9319: 9316: 9313: 9309: 9306: 9303: 9300: 9297: 9294: 9291: 9288: 9285: 9281: 9279: 9276: 9273: 9270: 9267: 9264: 9261: 9258: 9255: 9252: 9249: 9246: 9243: 9240: 9237: 9234: 9231: 9228: 9227: 9224: 9221: 9218: 9215: 9212: 9209: 9206: 9203: 9200: 9197: 9194: 9191: 9188: 9185: 9182: 9179: 9176: 9172: 9170: 9167: 9164: 9161: 9158: 9155: 9152: 9149: 9146: 9143: 9140: 9137: 9134: 9131: 9128: 9125: 9122: 9119: 9118: 9115: 9112: 9109: 9106: 9103: 9100: 9097: 9094: 9091: 9088: 9085: 9082: 9079: 9076: 9073: 9070: 9067: 9063: 9061: 9058: 9055: 9052: 9049: 9046: 9043: 9040: 9037: 9034: 9031: 9028: 9025: 9022: 9019: 9016: 9013: 9010: 9009: 8985: 8980: 8977: 8974: 8969: 8966: 8962: 8960: 8955: 8952: 8949: 8944: 8941: 8938: 8937: 8934: 8929: 8926: 8923: 8918: 8915: 8911: 8909: 8904: 8901: 8898: 8893: 8890: 8887: 8886: 8863: 8859: 8855: 8851: 8846: 8843: 8840: 8837: 8834: 8831: 8828: 8825: 8822: 8818: 8815: 8812: 8809: 8806: 8803: 8800: 8797: 8794: 8790: 8787: 8784: 8781: 8778: 8775: 8772: 8769: 8766: 8762: 8759: 8756: 8753: 8750: 8747: 8744: 8741: 8738: 8735: 8728: 8725: 8705: 8685: 8665: 8645: 8625: 8605: 8585: 8565: 8545: 8525: 8505: 8485: 8452: 8448: 8443: 8439: 8435: 8431: 8427: 8422: 8418: 8414: 8410: 8406: 8401: 8397: 8393: 8389: 8385: 8382: 8378: 8374: 8371: 8367: 8363: 8360: 8357: 8354: 8351: 8344: 8340: 8337: 8334: 8328: 8325: 8301: 8296: 8292: 8288: 8283: 8279: 8275: 8270: 8266: 8262: 8259: 8257: 8255: 8252: 8251: 8248: 8243: 8239: 8235: 8230: 8226: 8222: 8217: 8213: 8209: 8206: 8204: 8202: 8199: 8198: 8195: 8190: 8186: 8182: 8177: 8173: 8169: 8164: 8160: 8156: 8153: 8151: 8149: 8146: 8145: 8123: 8119: 8116: 8113: 8110: 8105: 8101: 8095: 8091: 8087: 8082: 8078: 8072: 8068: 8064: 8059: 8055: 8049: 8045: 8041: 8036: 8032: 8026: 8022: 8016: 8012: 8008: 8002: 7999: 7979: 7959: 7939: 7919: 7899: 7879: 7859: 7830: 7805:∈ {1, 2, 3, 4} 7794: 7793: 7780: 7774: 7771: 7767: 7762: 7758: 7754: 7750: 7745: 7741: 7737: 7733: 7728: 7724: 7720: 7718: 7715: 7714: 7709: 7704: 7700: 7696: 7694: 7691: 7687: 7682: 7678: 7674: 7670: 7665: 7661: 7657: 7655: 7652: 7651: 7646: 7641: 7637: 7633: 7629: 7624: 7620: 7616: 7614: 7611: 7607: 7602: 7598: 7594: 7592: 7589: 7588: 7583: 7578: 7574: 7570: 7566: 7561: 7557: 7553: 7549: 7544: 7540: 7536: 7534: 7531: 7529: 7526: 7525: 7522: 7519: 7517: 7514: 7512: 7509: 7507: 7504: 7502: 7499: 7498: 7496: 7491: 7486: 7482: 7478: 7475: 7457: 7456: 7445: 7440: 7435: 7428: 7425: 7423: 7420: 7418: 7415: 7413: 7410: 7409: 7404: 7400: 7396: 7392: 7388: 7384: 7380: 7376: 7372: 7368: 7364: 7360: 7359: 7354: 7350: 7346: 7342: 7338: 7334: 7330: 7326: 7322: 7318: 7314: 7310: 7309: 7304: 7300: 7296: 7292: 7288: 7284: 7280: 7276: 7272: 7268: 7264: 7260: 7259: 7256: 7252: 7248: 7244: 7241: 7238: 7235: 7158: 7157: 7145: 7139: 7135: 7130: 7126: 7122: 7118: 7114: 7109: 7105: 7101: 7097: 7093: 7088: 7084: 7080: 7076: 7072: 7069: 7065: 7061: 7058: 7054: 7050: 7047: 7044: 7041: 7038: 7031: 7027: 7024: 7021: 7015: 7012: 7000:, which gives 6987: 6980: 6976: 6975: 6973: 6968: 6965: 6943: 6936: 6932: 6931: 6929: 6924: 6921: 6899: 6892: 6888: 6887: 6885: 6880: 6877: 6866: 6865: 6852: 6847: 6843: 6839: 6836: 6833: 6830: 6827: 6823: 6819: 6815: 6811: 6810: 6807: 6803: 6799: 6796: 6793: 6790: 6787: 6783: 6779: 6775: 6771: 6770: 6767: 6763: 6759: 6756: 6753: 6750: 6747: 6743: 6739: 6735: 6731: 6730: 6728: 6698: 6689: 6685: 6680: 6677: 6673: 6669: 6665: 6662: 6658: 6654: 6650: 6649: 6645: 6641: 6637: 6633: 6628: 6624: 6619: 6616: 6612: 6608: 6604: 6603: 6599: 6595: 6591: 6587: 6584: 6580: 6576: 6572: 6567: 6563: 6558: 6557: 6555: 6550: 6545: 6541: 6537: 6534: 6520: 6519: 6502: 6497: 6494: 6489: 6485: 6481: 6476: 6472: 6468: 6463: 6459: 6455: 6452: 6448: 6444: 6443: 6440: 6437: 6432: 6428: 6424: 6419: 6415: 6411: 6406: 6402: 6398: 6395: 6391: 6387: 6386: 6383: 6380: 6375: 6371: 6367: 6362: 6358: 6354: 6349: 6345: 6341: 6338: 6334: 6330: 6329: 6327: 6297: 6290: 6286: 6285: 6281: 6277: 6276: 6272: 6268: 6267: 6265: 6260: 6257: 6254: 6251: 6223: 6216: 6212: 6209: 6205: 6202: 6198: 6197: 6195: 6190: 6187: 6184: 6181: 6148:parallelepiped 6131: 6127: 6123: 6119: 6115: 6111: 6107: 6103: 6099: 6095: 6091: 6087: 6083: 6080: 6075: 6072: 6043: 6040: 6035: 6031: 6027: 6022: 6018: 6014: 6009: 6005: 6001: 5998: 5995: 5993: 5990: 5986: 5985: 5982: 5979: 5974: 5970: 5966: 5961: 5957: 5953: 5948: 5944: 5940: 5937: 5934: 5932: 5929: 5925: 5924: 5921: 5918: 5913: 5909: 5905: 5900: 5896: 5892: 5887: 5883: 5879: 5876: 5873: 5871: 5868: 5864: 5863: 5860: 5857: 5852: 5848: 5844: 5839: 5835: 5831: 5826: 5822: 5818: 5815: 5812: 5810: 5807: 5803: 5802: 5778: 5754: 5734: 5731: 5728: 5723: 5720: 5715: 5712: 5700: 5697: 5695: 5692: 5677: 5673: 5669: 5638: 5635: 5632: 5631: 5628: 5625: 5623: 5620: 5614: 5613: 5608: 5597: 5581: 5579: 5566: 5562: 5561: 5556: 5551: 5548: 5545: 5538: 5532: 5531: 5527: 5517: 5505: 5503: 5490: 5484: 5483: 5479: 5478: 5475: 5472: 5470: 5467: 5461: 5460: 5427: 5420: 5413: 5398: 5396: 5389: 5383: 5382: 5377: 5372: 5369: 5366: 5361: 5355: 5354: 5321: 5308: 5306: 5299: 5293: 5292: 5285: 5284: 5281: 5278: 5276: 5273: 5269: 5268: 5258: 5256: 5249: 5248:(No symmetry) 5243: 5242: 5239: 5236: 5234: 5231: 5223: 5215: 5209: 5208: 5204: 5193: 5182: 5180: 5173: 5169: 5168: 5163: 5158: 5155: 5152: 5147: 5141: 5140: 5136: 5125: 5118:triangle sides 5109: 5107: 5100: 5096: 5095: 5090: 5085: 5082: 5075: 5069: 5068: 5029: 5018: 5007: 5005: 4998: 4994: 4993: 4988: 4983: 4978: 4972: 4971: 4965: 4964: 4961: 4954: 4944: 4940: 4932: 4929: 4882:symmetry group 4865: 4862: 4822: 4819: 4801: 4798: 4768: 4765: 4735: 4732: 4702: 4699: 4669: 4666: 4636: 4633: 4603: 4600: 4570: 4567: 4541: 4511: 4508: 4478: 4475: 4449: 4419: 4416: 4386: 4383: 4353: 4350: 4316: 4313: 4287: 4261: 4258: 4240: 4239: 4237: 4235: 4220: 4216: 4196: 4191: 4189: 4163: 4162: 4160: 4158: 4156: 4154: 4151: 4150: 4148: 4146: 4144: 4142: 4130: 4127: 4120: 4117: 4102: 4090: 4086: 4082: 4077: 4073: 4061: 4060: 4058: 4056: 4054: 4052: 4040: 4037: 4030: 4027: 4012: 4000: 3996: 3992: 3987: 3983: 3971: 3970: 3968: 3966: 3964: 3962: 3950: 3947: 3940: 3937: 3922: 3910: 3906: 3902: 3897: 3893: 3881: 3880: 3878: 3876: 3874: 3872: 3869: 3868: 3853: 3850: 3836: 3831: 3814: 3808: 3805: 3791: 3786: 3774: 3771: 3764: 3761: 3746: 3742: 3741: 3726: 3723: 3709: 3704: 3687: 3681: 3678: 3664: 3659: 3647: 3636: 3632: 3631: 3616: 3613: 3599: 3594: 3577: 3566: 3561: 3549: 3546: 3539: 3536: 3521: 3517: 3516: 3514: 3512: 3510: 3508: 3505: 3504: 3487: 3484: 3481: 3470: 3465: 3448: 3445: 3442: 3431: 3426: 3414: 3403: 3399: 3398: 3395: 3392: 3389: 3386: 3385: 3171: 3168: 3143:Main article: 3140: 3137: 3103:Main article: 3100: 3097: 3046: 3045: 3036: 3023: 3020: 2964: 2961: 2958: 2957: 2952: 2946: 2945: 2938: 2912: 2909: 2857: 2854: 2851: 2850: 2848: 2846: 2840: 2839: 2832: 2825: 2821: 2820: 2817: 2814: 2796: 2784: 2781: 2780: 2779: 2774:is mapped to − 2768: 2765: 2764: 2763: 2742: 2708: 2683: 2679: 2654: 2627: 2623: 2592: 2586: 2570:symmetry group 2534: 2531: 2518: 2515: 2512: 2509: 2506: 2502: 2433: 2428: 2404: 2401: 2398: 2395: 2392: 2389: 2386: 2383: 2380: 2377: 2373: 2371: 2369: 2366: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2339: 2338: 2335: 2332: 2329: 2326: 2323: 2320: 2317: 2314: 2311: 2308: 2304: 2302: 2300: 2297: 2294: 2291: 2288: 2285: 2282: 2279: 2276: 2275: 2242: 2236: 2231: 2204: 2201: 2198: 2195: 2192: 2189: 2186: 2182: 2180: 2178: 2174: 2168: 2165: 2160: 2157: 2151: 2148: 2142: 2139: 2133: 2130: 2124: 2120: 2116: 2115: 2112: 2108: 2102: 2099: 2094: 2091: 2085: 2082: 2076: 2070: 2067: 2061: 2057: 2052: 2050: 2046: 2040: 2037: 2032: 2029: 2026: 2023: 2017: 2014: 2007: 2003: 2002: 1982: 1979: 1952: 1945: 1941: 1936: 1933: 1930: 1927: 1924: 1920: 1906: 1899: 1895: 1890: 1887: 1884: 1881: 1878: 1875: 1871: 1854: 1851: 1843:square degrees 1822: 1819: 1815: 1810: 1807: 1802: 1798: 1795: 1792: 1789: 1786: 1784: 1782: 1778: 1773: 1770: 1765: 1761: 1758: 1755: 1752: 1747: 1744: 1739: 1736: 1734: 1731: 1726: 1723: 1718: 1714: 1711: 1708: 1707: 1623: 1618: 1613: 1607: 1602: 1598: 1594: 1589: 1584: 1580: 1576: 1571: 1566: 1562: 1558: 1553: 1548: 1544: 1540: 1535: 1531: 1526: 1521: 1518: 1516: 1513: 1507: 1502: 1498: 1494: 1489: 1484: 1480: 1476: 1471: 1466: 1462: 1458: 1453: 1448: 1444: 1440: 1435: 1431: 1426: 1422: 1419: 1418: 1415: 1410: 1405: 1399: 1393: 1389: 1385: 1379: 1374: 1368: 1363: 1359: 1355: 1350: 1345: 1341: 1337: 1332: 1327: 1323: 1319: 1314: 1309: 1305: 1297: 1292: 1289: 1287: 1283: 1277: 1273: 1269: 1263: 1258: 1252: 1247: 1243: 1239: 1234: 1229: 1225: 1221: 1216: 1211: 1207: 1203: 1198: 1193: 1189: 1182: 1181: 1159: 1155: 1134: 1114: 1090: 1084: 1080: 1075: 1069: 1064: 1059: 1057: 1051: 1047: 1042: 1037: 1034: 1029: 1023: 1018: 1014: 1013: 1010: 1004: 1000: 995: 992: 987: 984: 979: 976: 972: 970: 967: 962: 958: 952: 949: 946: 945: 922: 917: 890: 885: 861: 838: 801: 796: 792: 788: 784: 779: 774: 770: 767: 764: 761: 758: 756: 753: 747: 744: 739: 735: 731: 728: 725: 724: 721: 716: 712: 708: 704: 698: 693: 689: 685: 682: 679: 676: 674: 671: 666: 663: 658: 654: 651: 648: 647: 636:dihedral angle 620: 615: 611: 607: 604: 596: 591: 585: 581: 575: 572: 567: 563: 557: 553: 547: 543: 537: 533: 526: 522: 517: 514: 509: 506: 486: 481: 477: 454: 449: 445: 441: 438: 433: 426: 422: 418: 414: 408: 404: 398: 394: 387: 383: 380: 377: 374: 354: 334: 312: 309: 191: 184: 183: 182: 174: 167: 166: 162: 155: 154: 153: 152: 151: 149: 146: 15: 9: 6: 4: 3: 2: 17718: 17707: 17704: 17702: 17699: 17697: 17694: 17692: 17689: 17687: 17684: 17682: 17679: 17677: 17674: 17673: 17671: 17660: 17656: 17652: 17647: 17644: 17640: 17637: 17635: 17628: 17621: 17615: 17613: 17609: 17606: 17604: 17600: 17596: 17592: 17589: 17587: 17583: 17580: 17578: 17574: 17570: 17569: 17566: 17564: 17562: 17559: 17557: 17553: 17550: 17548: 17545: 17543: 17540: 17539: 17536: 17534: 17532: 17529: 17527: 17523: 17520: 17518: 17515: 17513: 17510: 17509: 17506: 17504: 17497: 17490: 17484: 17482: 17479: 17477: 17473: 17470: 17468: 17465: 17463: 17460: 17459: 17456: 17454: 17447: 17440: 17434: 17432: 17429: 17427: 17423: 17420: 17418: 17415: 17413: 17410: 17409: 17406: 17404: 17397: 17391: 17389: 17386: 17384: 17380: 17377: 17375: 17372: 17370: 17367: 17366: 17363: 17361: 17359: 17356: 17354: 17350: 17347: 17345: 17342: 17340: 17337: 17336: 17333: 17329: 17326: 17324: 17321: 17319: 17318:Demitesseract 17316: 17314: 17310: 17307: 17305: 17302: 17300: 17297: 17296: 17293: 17289: 17286: 17284: 17282: 17279: 17277: 17273: 17270: 17268: 17265: 17263: 17260: 17259: 17256: 17253: 17251: 17248: 17246: 17243: 17241: 17238: 17236: 17233: 17231: 17228: 17227: 17224: 17218: 17215: 17211: 17204: 17200: 17193: 17189: 17184: 17180: 17175: 17171: 17166: 17164: 17162: 17158: 17148: 17144: 17142: 17140: 17136: 17132: 17130: 17128: 17124: 17120: 17118: 17115: 17114: 17109: 17105: 17097: 17092: 17090: 17085: 17083: 17078: 17077: 17074: 17064: 17058: 17052: 17049: 17047: 17044: 17042: 17039: 17037: 17034: 17032: 17029: 17027: 17024: 17022: 17019: 17017: 17014: 17012: 17009: 17007: 17004: 17002: 16999: 16998: 16996: 16992: 16982: 16979: 16977: 16974: 16973: 16971: 16967: 16961: 16958: 16956: 16953: 16952: 16949: 16946: 16942: 16936: 16935: 16931: 16929: 16928: 16924: 16923: 16921: 16917: 16911: 16908: 16906: 16903: 16901: 16898: 16896: 16893: 16891: 16888: 16886: 16883: 16881: 16878: 16876: 16873: 16871: 16868: 16866: 16863: 16861: 16858: 16856: 16853: 16851: 16848: 16847: 16845: 16838: 16833: 16827: 16824: 16822: 16819: 16817: 16814: 16812: 16809: 16807: 16804: 16802: 16799: 16797: 16794: 16792: 16789: 16787: 16784: 16782: 16779: 16777: 16774: 16772: 16771:cuboctahedron 16769: 16767: 16764: 16763: 16761: 16756: 16752: 16746: 16741: 16735: 16732: 16730: 16727: 16725: 16722: 16720: 16717: 16715: 16712: 16711: 16709: 16705: 16700: 16696: 16692: 16684: 16679: 16677: 16672: 16670: 16665: 16664: 16661: 16649: 16646: 16644: 16641: 16639: 16636: 16634: 16631: 16629: 16626: 16624: 16621: 16619: 16616: 16615: 16613: 16610: 16606: 16599: 16596: 16593: 16590: 16587: 16584: 16583: 16581: 16577: 16570: 16569:Johnson solid 16567: 16564: 16563:Catalan solid 16561: 16558: 16555: 16552: 16549: 16548: 16546: 16542: 16533: 16530: 16527: 16524: 16521: 16518: 16517: 16515: 16512: 16510: 16507: 16505: 16502: 16500: 16497: 16496: 16494: 16490: 16483: 16480: 16477: 16474: 16471: 16468: 16465: 16462: 16459: 16458:Hexoctahedron 16456: 16453: 16450: 16447: 16444: 16441: 16438: 16437: 16435: 16431: 16425: 16422: 16420: 16417: 16415: 16412: 16410: 16407: 16405: 16402: 16400: 16397: 16395: 16392: 16390: 16389:Tridecahedron 16387: 16385: 16382: 16380: 16379:Hendecahedron 16377: 16376: 16374: 16370: 16364: 16361: 16359: 16356: 16354: 16351: 16349: 16346: 16344: 16341: 16339: 16336: 16334: 16331: 16329: 16326: 16324: 16321: 16319: 16316: 16315: 16313: 16309: 16302: 16298: 16291: 16286: 16284: 16279: 16277: 16272: 16271: 16268: 16261: 16257: 16256: 16252: 16250: 16247: 16242: 16241: 16236: 16235:"Tetrahedron" 16233: 16228: 16227: 16216: 16210: 16207:. Routledge. 16206: 16201: 16197: 16193: 16192: 16184: 16179: 16175: 16171: 16167: 16163: 16158: 16153: 16149: 16145: 16141: 16136: 16132: 16128: 16123: 16119: 16118: 16113: 16109: 16102: 16101: 16095: 16091: 16085: 16081: 16080: 16074: 16070: 16066: 16062: 16058: 16054: 16050: 16046: 16042: 16037: 16033: 16027: 16023: 16019: 16018: 16012: 16008: 16004: 16003: 15998: 15994: 15990: 15989: 15984: 15980: 15976: 15972: 15967: 15963: 15957: 15953: 15949: 15944: 15943: 15928: 15927: 15922: 15919: 15912: 15904: 15900: 15896: 15892: 15888: 15885:(June 1903). 15884: 15878: 15862: 15856: 15848: 15844: 15840: 15834: 15826: 15824:9780177612992 15820: 15816: 15811: 15810: 15804: 15798: 15790: 15786: 15782: 15778: 15774: 15773: 15768: 15762: 15743: 15739: 15732: 15725: 15706: 15702: 15698: 15691: 15684: 15669: 15665: 15659: 15651: 15647: 15643: 15639: 15635: 15631: 15624: 15616: 15614:9780160876110 15610: 15606: 15605: 15597: 15590: 15589: 15584: 15579: 15571: 15567: 15564:, Solutions, 15560: 15559:"Problem 930" 15554: 15547: 15543: 15539: 15535: 15527: 15519: 15511:, p. 129 15510: 15509: 15501: 15485: 15481: 15474: 15467: 15464: 15462: 15455: 15453: 15451: 15449: 15441: 15436: 15428: 15424: 15420: 15416: 15412: 15408: 15401: 15393: 15389: 15385: 15381: 15377: 15373: 15372: 15367: 15361: 15353: 15349: 15342: 15331: 15324: 15316: 15310: 15299: 15298: 15290: 15281: 15273: 15269: 15265: 15261: 15256: 15251: 15247: 15243: 15242: 15234: 15227: 15220: 15215: 15208: 15203: 15197:, p. 11. 15196: 15191: 15189: 15181: 15176: 15169: 15165: 15160: 15153: 15149: 15144: 15136: 15132: 15127: 15122: 15118: 15114: 15110: 15103: 15096: 15092: 15087: 15080: 15075: 15064: 15059: 15052: 15047: 15045: 15035: 15030: 15026: 15022: 15018: 15014: 15008: 15001: 14996: 14988: 14982: 14975: 14970: 14961: 14958: 14955: 14952: 14951: 14947: 14940: 14936: 14931: 14924: 14923:"Tetrahedron" 14918: 14912:, Table I(i). 14911: 14906: 14904: 14902: 14894: 14890: 14889:Cromwell 1997 14885: 14878: 14874: 14869: 14862: 14857: 14849: 14848: 14840: 14831: 14830: 14825: 14824:"Tetrahedron" 14822: 14815: 14813: 14808: 14794: 14790: 14786: 14780: 14778: 14770: 14766: 14760: 14756: 14746: 14743: 14741: 14738: 14735: 14732: 14730: 14727: 14725: 14722: 14720: 14717: 14715: 14712: 14709: 14706: 14703: 14699: 14695: 14694:Demihypercube 14692: 14690: 14687: 14685: 14682: 14680: 14677: 14676: 14663: 14659: 14655: 14653: 14649: 14641: 14633: 14629: 14625: 14616: 14612: 14609: 14605: 14601: 14597: 14593: 14589: 14585: 14582: 14578: 14574: 14572: 14568: 14564: 14562: 14561:Automorphisms 14558: 14554: 14552: 14548: 14544: 14542: 14538: 14534: 14532: 14528: 14524: 14522: 14518: 14514: 14512: 14508: 14504: 14499: 14494: 14486: 14482: 14479: 14475: 14471: 14470:Marvin Minsky 14467: 14466: 14461: 14457: 14442: 14433: 14422: 14413: 14399: 14397: 14393: 14387: 14377: 14375: 14371: 14367: 14363: 14359: 14355: 14350: 14348: 14341: 14337: 14328: 14324: 14314: 14312: 14308: 14304: 14300: 14299:semiconductor 14296: 14291: 14289: 14285: 14282:If six equal 14279: 14275: 14265: 14263: 14259: 14255: 14250: 14248: 14243: 14241: 14236: 14221: 14217: 14199: 14198:central angle 14195: 14194: 14177: 14161: 14157: 14153: 14152:sp-hybridized 14149: 14143: 14136: 14131: 14124: 14119: 14110: 14108: 14098: 14096: 14086: 14084: 14080: 14076: 14072: 14068: 14064: 14060: 14056: 14052: 14048: 14045:of irregular 14044: 14040: 14036: 14027: 14013: 14011: 14007: 14002: 14000: 13995: 13993: 13989: 13985: 13984:mirror images 13981: 13977: 13973: 13969: 13965: 13961: 13953: 13947: 13942: 13939: 13933: 13928: 13925: 13919: 13914: 13911: 13910: 13904: 13901: 13898: 13895: 13893: 13890: 13888: 13885: 13883: 13880: 13878: 13875: 13873: 13870: 13868: 13865: 13863: 13860: 13858: 13855: 13854: 13850: 13846: 13843: 13839: 13836: 13832: 13829: 13825: 13822: 13818: 13815: 13811: 13808: 13804: 13801: 13797: 13794: 13790: 13787: 13783: 13780: 13776: 13773: 13769: 13768: 13761: 13756: 13753: 13749: 13743: 13738: 13736: 13731: 13729: 13724: 13723: 13720: 13716: 13711: 13708: 13706: 13696: 13693: 13690: 13687: 13685: 13682: 13680: 13677: 13675: 13672: 13670: 13667: 13665: 13662: 13660: 13657: 13655: 13652: 13650: 13647: 13646: 13642: 13638: 13635: 13631: 13628: 13624: 13621: 13617: 13614: 13610: 13607: 13603: 13600: 13596: 13593: 13589: 13586: 13582: 13579: 13575: 13572: 13568: 13565: 13561: 13560: 13553: 13547: 13543: 13537: 13532: 13530: 13525: 13523: 13518: 13517: 13514: 13510: 13505: 13502: 13500: 13496: 13492: 13483: 13480: 13478: 13475: 13473: 13470: 13468: 13465: 13463: 13460: 13458: 13455: 13453: 13450: 13448: 13445: 13444: 13440: 13436: 13433: 13429: 13426: 13422: 13419: 13415: 13412: 13408: 13405: 13401: 13398: 13394: 13391: 13387: 13386: 13381: 13378: 13375: 13373: 13370: 13368: 13365: 13363: 13360: 13358: 13355: 13353: 13350: 13348: 13345: 13343: 13340: 13339: 13309: 13280: 13251: 13222: 13193: 13164: 13135: 13106: 13105: 13101: 13097: 13094: 13090: 13087: 13083: 13080: 13076: 13073: 13069: 13066: 13062: 13059: 13055: 13052: 13048: 13047: 13043: 13040: 13037: 13033: 13030: 13026: 13023: 13021: 13017: 13007: 13004: 13001: 12998: 12995: 12992: 12989: 12987: 12984: 12979: 12975: 12972: 12969: 12966: 12962: 12959: 12955: 12952: 12948: 12945: 12941: 12938: 12934: 12931: 12928: 12924: 12921: 12918: 12914: 12911: 12907: 12904: 12900: 12897: 12893: 12890: 12886: 12883: 12880: 12876: 12873: 12870: 12868: 12865: 12863: 12860: 12858: 12855: 12853: 12850: 12847: 12841: 12838: 12837:Trapezohedron 12835: 12830: 12824: 12822: 12821: 12820:trapezohedron 12809: 12806: 12803: 12800: 12797: 12794: 12791: 12788: 12786: 12783: 12782: 12778: 12774: 12771: 12768: 12764: 12761: 12757: 12754: 12750: 12747: 12743: 12740: 12736: 12733: 12729: 12726: 12725: 12722: 12719: 12716: 12712: 12709: 12705: 12702: 12698: 12695: 12691: 12688: 12684: 12681: 12677: 12674: 12673: 12670: 12667: 12664: 12662: 12659: 12657: 12654: 12652: 12649: 12647: 12642: 12640: 12635: 12633: 12630: 12627: 12626: 12620: 12615: 12613: 12608: 12606: 12601: 12600: 12598: 12594: 12591: 12585: 12583: 12579: 12578: 12566: 12563: 12559: 12556: 12552: 12549: 12545: 12542: 12538: 12535: 12531: 12528: 12524: 12523: 12519: 12516: 12512: 12509: 12505: 12502: 12498: 12495: 12491: 12488: 12484: 12481: 12477: 12476: 12466: 12463: 12462: 12458: 12455: 12453: 12450: 12448: 12445: 12443: 12440: 12438: 12435: 12433: 12430: 12429: 12426: 12422: 12419: 12417: 12407: 12404: 12378: 12374: 12368: 12358: 12350: 12348: 12344: 12334: 12312: 12303: 12293: 12289: 12282: 12277: 12267: 12263: 12250: 12240: 12236: 12229: 12224: 12214: 12210: 12197: 12187: 12183: 12176: 12171: 12161: 12157: 12146: 12140: 12137: 12132: 12129: 12106: 12097: 12092: 12086: 12082: 12078: 12073: 12069: 12064: 12053: 12048: 12042: 12038: 12034: 12029: 12025: 12020: 12009: 12004: 11998: 11994: 11990: 11985: 11981: 11976: 11967: 11963: 11960: 11943: 11938: 11935: 11931: 11927: 11925: 11920: 11909: 11908: 11907: 11901: 11892: 11883: 11874: 11868: 11842: 11836: 11833: 11825: 11822: 11819: 11816: 11813: 11810: 11807: 11804: 11801: 11792: 11789: 11786: 11783: 11780: 11777: 11774: 11771: 11762: 11759: 11756: 11753: 11750: 11747: 11744: 11741: 11732: 11729: 11726: 11723: 11720: 11717: 11714: 11711: 11702: 11699: 11692: 11691: 11690: 11688: 11684: 11680: 11676: 11672: 11668: 11664: 11660: 11650: 11636: 11631: 11627: 11621: 11618: 11613: 11610: 11605: 11601: 11595: 11592: 11587: 11584: 11579: 11575: 11569: 11566: 11561: 11558: 11553: 11549: 11543: 11540: 11535: 11532: 11502: 11498: 11494: 11489: 11485: 11481: 11476: 11472: 11468: 11463: 11459: 11453: 11450: 11444: 11441: 11434: 11433: 11432: 11430: 11425: 11421: 11416: 11412: 11407: 11403: 11398: 11394: 11389: 11372: 11365: 11361: 11357: 11352: 11345: 11340: 11336: 11332: 11327: 11320: 11315: 11311: 11307: 11302: 11295: 11290: 11286: 11282: 11277: 11270: 11265: 11261: 11257: 11248: 11247: 11246: 11244: 11239: 11235: 11231: 11227: 11203: 11197: 11194: 11191: 11188: 11185: 11182: 11179: 11176: 11173: 11170: 11167: 11161: 11158: 11155: 11152: 11149: 11146: 11143: 11140: 11137: 11134: 11131: 11128: 11125: 11118: 11117: 11116: 11114: 11110: 11106: 11102: 11098: 11094: 11090: 11086: 11082: 11078: 11075:For vertices 11059: 11056: 11053: 11050: 11039: 11035: 11032: 11029: 11026: 11015: 11011: 11008: 11005: 11002: 10991: 10987: 10984: 10981: 10978: 10967: 10963: 10960: 10957: 10950: 10949: 10948: 10943: 10936: 10929: 10922: 10918: 10914: 10910: 10906: 10902: 10898: 10888: 10870: 10867: 10863: 10859: 10856: 10851: 10841: 10833: 10828: 10825: 10821: 10817: 10814: 10809: 10799: 10791: 10786: 10783: 10779: 10775: 10772: 10767: 10757: 10746: 10743: 10738: 10733: 10725: 10720: 10715: 10707: 10702: 10697: 10689: 10684: 10679: 10666: 10648: 10644: 10638: 10634: 10611: 10608: 10604: 10581: 10577: 10554: 10527: 10523: 10500: 10496: 10473: 10469: 10446: 10442: 10431: 10421: 10418: 10412: 10409: 10404: 10386: 10383: 10380: 10377: 10371: 10368: 10365: 10362: 10359: 10356: 10350: 10347: 10344: 10341: 10338: 10335: 10329: 10326: 10323: 10320: 10317: 10314: 10308: 10305: 10302: 10299: 10296: 10293: 10287: 10284: 10281: 10278: 10275: 10272: 10266: 10263: 10256: 10255: 10254: 10252: 10248: 10244: 10240: 10236: 10230: 10225: 10223: 10213: 10211: 10206: 10202: 10200: 10196: 10192: 10187: 10185: 10181: 10177: 10173: 10168: 10166: 10161: 10157: 10155: 10151: 10147: 10137: 10135: 10131: 10127: 10123: 10119: 10115: 10111: 10107: 10103: 10099: 10095: 10091: 10087: 10083: 10079: 10075: 10071: 10066: 10064: 10060: 10056: 10033: 10030: 10025: 10019: 10016: 10005: 10001: 9993: 9990: 9979: 9975: 9967: 9964: 9953: 9949: 9941: 9938: 9925: 9921: 9913: 9910: 9905: 9902: 9891: 9887: 9879: 9876: 9865: 9861: 9853: 9850: 9837: 9833: 9825: 9822: 9811: 9807: 9799: 9796: 9791: 9788: 9777: 9773: 9765: 9762: 9749: 9745: 9737: 9734: 9723: 9719: 9711: 9708: 9697: 9693: 9685: 9682: 9677: 9674: 9668: 9659: 9658: 9657: 9654: 9651: 9648: 9644: 9640: 9635: 9633: 9629: 9625: 9620: 9618: 9614: 9610: 9606: 9602: 9598: 9594: 9590: 9585: 9581: 9579: 9575: 9571: 9570:Gaspard Monge 9567: 9565: 9554: 9541: 9536: 9507: 9491: 9485: 9482: 9462: 9454: 9433: 9408: 9388: 9380: 9375: 9373: 9369: 9365: 9361: 9356: 9354: 9349: 9332: 9326: 9323: 9320: 9317: 9314: 9304: 9301: 9298: 9295: 9292: 9286: 9283: 9277: 9271: 9268: 9265: 9262: 9259: 9250: 9247: 9244: 9241: 9238: 9232: 9229: 9222: 9216: 9213: 9210: 9207: 9204: 9195: 9192: 9189: 9186: 9183: 9177: 9174: 9168: 9162: 9159: 9156: 9153: 9150: 9141: 9138: 9135: 9132: 9129: 9123: 9120: 9113: 9107: 9104: 9101: 9098: 9095: 9086: 9083: 9080: 9077: 9074: 9068: 9065: 9059: 9053: 9050: 9047: 9044: 9041: 9032: 9029: 9026: 9023: 9020: 9014: 9011: 8983: 8978: 8975: 8972: 8967: 8964: 8958: 8953: 8950: 8947: 8942: 8939: 8932: 8927: 8924: 8921: 8916: 8913: 8907: 8902: 8899: 8896: 8891: 8888: 8861: 8857: 8853: 8849: 8841: 8838: 8835: 8832: 8829: 8826: 8823: 8813: 8810: 8807: 8804: 8801: 8798: 8795: 8785: 8782: 8779: 8776: 8773: 8770: 8767: 8757: 8754: 8751: 8748: 8745: 8742: 8739: 8736: 8726: 8723: 8703: 8683: 8663: 8643: 8623: 8603: 8583: 8563: 8543: 8523: 8503: 8483: 8470: 8466: 8450: 8446: 8441: 8437: 8433: 8429: 8425: 8420: 8416: 8412: 8408: 8404: 8399: 8395: 8391: 8387: 8383: 8380: 8376: 8372: 8369: 8365: 8361: 8358: 8355: 8352: 8349: 8342: 8338: 8335: 8332: 8326: 8323: 8299: 8294: 8290: 8286: 8281: 8277: 8273: 8268: 8264: 8260: 8258: 8253: 8246: 8241: 8237: 8233: 8228: 8224: 8220: 8215: 8211: 8207: 8205: 8200: 8193: 8188: 8184: 8180: 8175: 8171: 8167: 8162: 8158: 8154: 8152: 8147: 8121: 8117: 8114: 8111: 8108: 8103: 8099: 8093: 8089: 8085: 8080: 8076: 8070: 8066: 8062: 8057: 8053: 8047: 8043: 8039: 8034: 8030: 8024: 8020: 8014: 8010: 8006: 8000: 7997: 7977: 7957: 7937: 7917: 7897: 7877: 7857: 7848: 7846: 7842: 7838: 7833: 7824: 7820: 7816: 7812: 7804: 7800: 7778: 7772: 7765: 7760: 7756: 7748: 7743: 7739: 7731: 7726: 7722: 7716: 7707: 7702: 7698: 7692: 7685: 7680: 7676: 7668: 7663: 7659: 7653: 7644: 7639: 7635: 7627: 7622: 7618: 7612: 7605: 7600: 7596: 7590: 7581: 7576: 7572: 7564: 7559: 7555: 7547: 7542: 7538: 7532: 7527: 7520: 7515: 7510: 7505: 7500: 7494: 7489: 7484: 7480: 7476: 7473: 7466: 7465: 7464: 7462: 7443: 7438: 7433: 7426: 7421: 7416: 7411: 7402: 7398: 7390: 7386: 7378: 7374: 7366: 7362: 7352: 7348: 7340: 7336: 7328: 7324: 7316: 7312: 7302: 7298: 7290: 7286: 7278: 7274: 7266: 7262: 7254: 7246: 7242: 7239: 7236: 7233: 7226: 7225: 7224: 7222: 7217: 7215: 7211: 7207: 7203: 7199: 7195: 7191: 7187: 7183: 7179: 7175: 7171: 7167: 7163: 7143: 7137: 7133: 7128: 7124: 7120: 7116: 7112: 7107: 7103: 7099: 7095: 7091: 7086: 7082: 7078: 7074: 7070: 7067: 7063: 7059: 7056: 7052: 7048: 7045: 7042: 7039: 7036: 7029: 7025: 7022: 7019: 7013: 7010: 7003: 7002: 7001: 6966: 6963: 6922: 6919: 6878: 6875: 6845: 6841: 6837: 6834: 6831: 6828: 6825: 6817: 6805: 6801: 6797: 6794: 6791: 6788: 6785: 6777: 6765: 6761: 6757: 6754: 6751: 6748: 6745: 6737: 6726: 6696: 6671: 6656: 6639: 6610: 6593: 6578: 6553: 6548: 6543: 6539: 6535: 6532: 6525: 6524: 6523: 6495: 6487: 6483: 6479: 6474: 6470: 6466: 6461: 6457: 6450: 6438: 6430: 6426: 6422: 6417: 6413: 6409: 6404: 6400: 6393: 6381: 6373: 6369: 6365: 6360: 6356: 6352: 6347: 6343: 6336: 6325: 6258: 6255: 6252: 6249: 6188: 6185: 6182: 6179: 6172: 6171: 6170: 6167: 6149: 6145: 6121: 6113: 6105: 6097: 6089: 6073: 6070: 6061: 6041: 6033: 6029: 6025: 6020: 6016: 6012: 6007: 6003: 5996: 5994: 5980: 5972: 5968: 5964: 5959: 5955: 5951: 5946: 5942: 5935: 5933: 5919: 5911: 5907: 5903: 5898: 5894: 5890: 5885: 5881: 5874: 5872: 5858: 5850: 5846: 5842: 5837: 5833: 5829: 5824: 5820: 5813: 5811: 5791: 5776: 5768: 5752: 5732: 5729: 5726: 5721: 5718: 5713: 5710: 5691: 5675: 5671: 5667: 5655: 5652: 5647: 5645: 5629: 5626: 5624: 5619: 5616: 5615: 5612: 5607: 5603: 5596: 5589: 5585: 5577: 5571: 5563: 5557: 5552: 5549: 5544: 5537: 5534: 5533: 5526: 5523: 5516: 5509: 5501: 5495: 5489: 5485: 5480: 5476: 5473: 5471: 5466: 5463: 5462: 5459: 5426: 5419: 5412: 5409: 5402: 5394: 5388: 5384: 5378: 5373: 5370: 5360: 5357: 5356: 5353: 5320: 5312: 5304: 5298: 5294: 5290: 5286: 5282: 5279: 5277: 5271: 5270: 5267: 5265: 5264:trivial group 5254: 5244: 5240: 5237: 5235: 5230: 5222: 5214: 5211: 5210: 5203: 5199: 5192: 5186: 5178: 5170: 5164: 5159: 5156: 5146: 5143: 5142: 5135: 5131: 5124: 5117: 5113: 5105: 5097: 5091: 5086: 5083: 5081: 5074: 5071: 5070: 5066: 5035: 5028: 5024: 5017: 5011: 5003: 4995: 4992: 4989: 4987: 4984: 4982: 4979: 4977: 4974: 4973: 4970: 4966: 4951: 4948: 4938: 4928: 4926: 4922: 4917: 4912: 4910: 4906: 4902: 4898: 4893: 4891: 4887: 4883: 4879: 4870: 4861: 4859: 4855: 4851: 4847: 4842: 4840: 4836: 4832: 4828: 4818: 4799: 4796: 4766: 4763: 4733: 4730: 4700: 4697: 4667: 4664: 4634: 4631: 4601: 4598: 4568: 4565: 4539: 4530: 4509: 4506: 4476: 4473: 4447: 4438: 4417: 4414: 4384: 4381: 4351: 4348: 4335: 4314: 4311: 4285: 4259: 4256: 4218: 4214: 4211:arc sec 4164: 4161: 4159: 4157: 4155: 4153: 4152: 4128: 4125: 4118: 4115: 4088: 4084: 4080: 4075: 4072: 4062: 4038: 4035: 4028: 4025: 3998: 3994: 3990: 3985: 3982: 3972: 3948: 3945: 3938: 3935: 3908: 3904: 3900: 3895: 3892: 3882: 3879: 3877: 3875: 3873: 3871: 3870: 3851: 3848: 3812: 3806: 3803: 3772: 3769: 3762: 3759: 3743: 3724: 3721: 3685: 3679: 3676: 3645: 3633: 3614: 3611: 3575: 3547: 3544: 3537: 3534: 3518: 3515: 3513: 3511: 3509: 3507: 3506: 3485: 3482: 3479: 3446: 3443: 3440: 3412: 3400: 3387: 3382: 3379: 3321: 3316: 3314: 3310: 3229: 3223: 3221: 3217: 3212: 3208: 3204: 3200: 3195: 3193: 3189: 3185: 3184:3-orthoscheme 3176: 3167: 3164: 3156: 3151: 3146: 3136: 3134: 3130: 3128: 3124: 3120: 3111: 3106: 3096: 3094: 3090: 3085: 3083: 3079: 3075: 3071: 3066: 3064: 3060: 3056: 3055:perpendicular 3051: 3041: 3037: 3032: 3028: 3027: 3019: 3017: 3013: 3009: 3005: 3001: 2997: 2992: 2990: 2986: 2978: 2974: 2969: 2956: 2951: 2948: 2947: 2943: 2939: 2936: 2932: 2928: 2926: 2922: 2918: 2908: 2906: 2897: 2893: 2891: 2887: 2883: 2879: 2871: 2867: 2862: 2849: 2847: 2842: 2837: 2833: 2830: 2826: 2823: 2818: 2815: 2812: 2811: 2808: 2803: 2801: 2800:Coxeter plane 2794: 2790: 2777: 2773: 2769: 2766: 2760: 2756: 2752: 2746:3 ((1 2)(3 4) 2743: 2730: 2726: 2722: 2709: 2706: 2705: 2703: 2699: 2681: 2677: 2669: 2643: 2642: 2641: 2625: 2621: 2613: 2609: 2575: 2571: 2562: 2558: 2556: 2552: 2548: 2539: 2530: 2513: 2510: 2507: 2492: 2461: 2457: 2453: 2449: 2431: 2426: 2402: 2396: 2393: 2390: 2387: 2384: 2381: 2378: 2372: 2367: 2361: 2358: 2355: 2352: 2349: 2346: 2343: 2333: 2327: 2324: 2321: 2318: 2315: 2312: 2309: 2303: 2298: 2292: 2289: 2286: 2283: 2280: 2265: 2261: 2256: 2240: 2234: 2229: 2199: 2196: 2193: 2190: 2187: 2181: 2176: 2172: 2166: 2163: 2158: 2155: 2149: 2146: 2140: 2137: 2131: 2128: 2122: 2118: 2110: 2106: 2100: 2097: 2092: 2089: 2083: 2080: 2074: 2068: 2065: 2059: 2055: 2048: 2044: 2038: 2035: 2030: 2027: 2024: 2021: 2015: 2012: 2005: 1980: 1977: 1969: 1964: 1950: 1943: 1939: 1934: 1931: 1928: 1925: 1922: 1918: 1904: 1897: 1893: 1888: 1885: 1882: 1879: 1876: 1873: 1869: 1860: 1850: 1848: 1845:, or 0.04387 1844: 1840: 1820: 1817: 1813: 1808: 1805: 1800: 1796: 1793: 1790: 1787: 1785: 1776: 1771: 1768: 1763: 1759: 1756: 1753: 1750: 1745: 1742: 1737: 1735: 1729: 1724: 1721: 1716: 1712: 1709: 1697: 1692: 1690: 1685: 1681: 1677: 1673: 1669: 1665: 1661: 1643: 1638: 1621: 1616: 1611: 1605: 1600: 1596: 1592: 1587: 1582: 1578: 1574: 1569: 1564: 1560: 1556: 1551: 1546: 1542: 1538: 1533: 1529: 1524: 1519: 1517: 1511: 1505: 1500: 1496: 1492: 1487: 1482: 1478: 1474: 1469: 1464: 1460: 1456: 1451: 1446: 1442: 1438: 1433: 1429: 1424: 1420: 1413: 1408: 1403: 1397: 1391: 1387: 1383: 1377: 1372: 1366: 1361: 1357: 1353: 1348: 1343: 1339: 1335: 1330: 1325: 1321: 1317: 1312: 1307: 1303: 1295: 1290: 1288: 1281: 1275: 1271: 1267: 1261: 1256: 1250: 1245: 1241: 1237: 1232: 1227: 1223: 1219: 1214: 1209: 1205: 1201: 1196: 1191: 1187: 1157: 1153: 1132: 1112: 1088: 1082: 1078: 1073: 1062: 1055: 1049: 1045: 1040: 1035: 1032: 1027: 1016: 1008: 1002: 998: 993: 990: 985: 982: 977: 974: 968: 965: 960: 956: 950: 947: 915: 907: 883: 875: 859: 852: 836: 829: 820: 816: 799: 794: 790: 786: 782: 777: 772: 768: 765: 762: 759: 757: 751: 745: 742: 737: 733: 729: 726: 719: 714: 710: 706: 702: 696: 691: 687: 683: 680: 677: 675: 669: 664: 661: 656: 652: 649: 637: 632: 618: 613: 609: 605: 602: 594: 589: 583: 579: 573: 570: 565: 561: 555: 551: 545: 541: 535: 531: 524: 520: 515: 512: 507: 504: 484: 479: 475: 452: 447: 443: 439: 436: 431: 424: 420: 416: 412: 406: 402: 396: 392: 385: 381: 378: 375: 372: 352: 332: 318: 308: 306: 302: 298: 294: 290: 285: 283: 279: 275: 271: 266: 264: 260: 256: 252: 248: 244: 239: 237: 233: 229: 225: 220: 218: 214: 210: 206: 195: 188: 178: 171: 159: 145: 143: 139: 135: 130: 128: 124: 119: 117: 113: 109: 104: 102: 98: 95: 91: 86: 84: 80: 76: 72: 68: 64: 60: 56: 48: 44: 39: 35: 33: 28: 27: 22: 17638: 17607: 17598: 17590: 17581: 17572: 17552:10-orthoplex 17288:Dodecahedron 17280: 17266: 17209: 17198: 17187: 17178: 17169: 17160: 17156: 17146: 17138: 17134: 17126: 17122: 17062: 16981:trapezohedra 16932: 16925: 16729:dodecahedron 16482:Apeirohedron 16433:>20 faces 16384:Dodecahedron 16332: 16260:kaleidocycle 16254: 16238: 16204: 16195: 16189: 16147: 16143: 16130: 16126: 16116: 16099: 16078: 16044: 16040: 16016: 16001: 15987: 15974: 15970: 15947: 15939:Bibliography 15924: 15911: 15894: 15890: 15877: 15865:. Retrieved 15855: 15846: 15833: 15808: 15797: 15771: 15761: 15749:. Retrieved 15742:the original 15737: 15724: 15714:15 September 15712:. Retrieved 15705:the original 15700: 15696: 15683: 15671:. Retrieved 15667: 15658: 15633: 15629: 15623: 15603: 15596: 15586: 15578: 15569: 15565: 15553: 15537: 15533: 15525: 15518: 15513:( Art. 163 ) 15507: 15500: 15488:. Retrieved 15483: 15473: 15458: 15435: 15410: 15406: 15400: 15375: 15369: 15360: 15351: 15347: 15341: 15323: 15296: 15289: 15280: 15245: 15239: 15226: 15214: 15207:Bottema 1969 15202: 15175: 15159: 15143: 15116: 15112: 15102: 15094: 15091:Coxeter 1973 15086: 15079:Coxeter 1973 15074: 15063:Coxeter 1973 15058: 15051:Coxeter 1973 15024: 15020: 15007: 14995: 14981: 14969: 14960:Brittin 1945 14954:Coxeter 1948 14946: 14930: 14917: 14910:Coxeter 1948 14884: 14868: 14856: 14846: 14839: 14827: 14793:golden ratio 14788: 14785:Coxeter 1973 14759: 14701: 14621: 14608:planar graph 14483: 14463: 14454: 14389: 14366:Rubik's Cube 14351: 14344: 14340:4-sided dice 14326: 14292: 14281: 14262:tetrahedrane 14257: 14251: 14244: 14237: 14218: 14191: 14145: 14104: 14092: 14067:aerodynamics 14039:approximated 14032: 14016:Applications 14003: 13996: 13980:right-handed 13972:dodecahedron 13957: 13861: 13718: 13714: 13702: 13653: 13512: 13508: 13494: 13488: 13466: 13446: 13361: 13341: 13013: 12971:Plane tiling 12845: 12828: 12817: 12815: 12643:(Tetragonal) 12592: 12574: 12572: 12470:Equilateral 12436: 12413: 12405: 12370: 12356: 12340: 12332: 11899: 11890: 11881: 11872: 11866: 11862: 11859:Circumcenter 11686: 11682: 11678: 11674: 11670: 11666: 11662: 11658: 11656: 11653:Circumradius 11523: 11431:is given by 11428: 11423: 11419: 11414: 11410: 11405: 11401: 11396: 11392: 11390: 11387: 11242: 11237: 11233: 11225: 11223: 11112: 11108: 11104: 11100: 11096: 11092: 11088: 11084: 11080: 11076: 11074: 10941: 10934: 10927: 10920: 10916: 10912: 10908: 10904: 10900: 10896: 10894: 10433: 10413: 10407: 10405: 10402: 10250: 10246: 10242: 10238: 10235:law of sines 10232: 10219: 10207: 10203: 10188: 10180:rhombohedron 10169: 10162: 10158: 10143: 10133: 10129: 10125: 10121: 10117: 10113: 10109: 10105: 10101: 10093: 10089: 10085: 10077: 10067: 10062: 10058: 10051: 10049: 9655: 9652: 9646: 9642: 9638: 9636: 9632:Euler points 9631: 9627: 9621: 9612: 9604: 9596: 9592: 9586: 9582: 9573: 9568: 9560: 9475:is given by 9376: 9357: 9350: 8475: 7849: 7828: 7822: 7818: 7814: 7810: 7802: 7798: 7795: 7458: 7220: 7218: 7213: 7209: 7205: 7201: 7197: 7193: 7192:. The angle 7189: 7185: 7181: 7177: 7176:. The angle 7173: 7169: 7165: 7161: 7159: 6867: 6521: 6168: 5792: 5702: 5656: 5650: 5648: 5643: 5640: 5617: 5605: 5602:cyclic group 5594: 5592: 5587: 5583: 5542: 5535: 5524: 5514: 5507: 5464: 5424: 5417: 5410: 5405: 5400: 5358: 5318: 5315: 5310: 5261: 5228: 5220: 5212: 5201: 5198:cyclic group 5190: 5184: 5144: 5133: 5122: 5115: 5111: 5079: 5072: 5026: 5015: 5009: 4963:Description 4934: 4913: 4908: 4904: 4897:kaleidoscope 4894: 4875: 4843: 4841:to a cube.) 4831:mirror image 4826: 4824: 4436: 4333: 4243: 3317: 3308: 3224: 3219: 3215: 3205:that is the 3196: 3183: 3181: 3170:Orthoschemes 3160: 3131: 3127:right angles 3122: 3116: 3088: 3086: 3069: 3067: 3062: 3052: 3049: 2993: 2982: 2914: 2902: 2877: 2875: 2865: 2816:Face/vertex 2813:Centered by 2788: 2787:The regular 2786: 2775: 2771: 2758: 2754: 2750: 2728: 2724: 2720: 2567: 2544: 2257: 1965: 1856: 1693: 1683: 1679: 1671: 1659: 1644:of a face (2 1639: 828:circumsphere 825: 633: 324: 286: 267: 254: 240: 221: 204: 202: 134:circumsphere 131: 120: 105: 100: 87: 62: 59:tetrahedrons 58: 54: 46: 40: 38: 31: 25: 21:tetrahedroid 17561:10-demicube 17522:9-orthoplex 17472:8-orthoplex 17422:7-orthoplex 17379:6-orthoplex 17349:5-orthoplex 17304:Pentachoron 17292:Icosahedron 17267:Tetrahedron 16751:semiregular 16734:icosahedron 16714:tetrahedron 16424:Icosahedron 16372:11–20 faces 16358:Enneahedron 16348:Heptahedron 16338:Pentahedron 16333:Tetrahedron 15867:20 February 15751:11 November 15164:Fekete 1985 15000:Kepler 1619 14956:Table I(i). 14745:Orthoscheme 14708:Pentachoron 14640:wheel graph 14584:Hamiltonian 14374:Pyramorphix 14358:4-sided die 14354:roleplaying 14323:Color space 14317:Color space 14278:Electronics 14274:Electricity 14238:Quaternary 14193:Tetrahedron 14135:dot product 14095:spaceframes 13976:left-handed 12846:Tetrahedron 12456:Heptagonal 11095:, and feet 9637:The center 9574:Monge point 6060:determinant 5769:' area and 5399:Four equal 5309:Four equal 5289:Disphenoids 5112:equilateral 5010:equilateral 4958:equivalence 4943:, ), and (S 3207:convex hull 3199:orthoscheme 3125:vertex are 2975:within the 2789:tetrahedron 2456:alternating 1968:unit sphere 1696:solid angle 311:Measurement 217:deltahedron 77:, and four 47:tetrahedron 32:Tetrahedron 17706:Tetrahedra 17676:Deltahedra 17670:Categories 17547:10-simplex 17531:9-demicube 17481:8-demicube 17431:7-demicube 17388:6-demicube 17358:5-demicube 17272:Octahedron 17046:prismatoid 16976:bipyramids 16960:antiprisms 16934:hosohedron 16724:octahedron 16609:prismatoid 16594:(infinite) 16363:Decahedron 16353:Octahedron 16343:Hexahedron 16318:Monohedron 16311:1–10 faces 16198:: 227–232. 15636:(3): 145. 15195:Kahan 2012 15166:, p. 15150:, p. 15119:: 128631. 14937:, p. 14891:, p. 14875:, p. 14861:Cundy 1952 14803:References 14580:Properties 14162:molecule ( 14047:tetrahedra 13751:Spherical 13545:Spherical 13482:V3.3.3.3.3 13020:octahedron 12882:Polyhedron 12827:Family of 12636:(Trigonal) 12597:antiprisms 12588:Family of 12473:Isosceles 12447:Pentagonal 12437:Triangular 10253:, we have 10195:disphenoid 9617:Euler line 9613:Euler line 9601:concurrent 9379:skew lines 5183:Two equal 4916:honeycombs 4858:octahedron 3429:109°28′16″ 3192:disphenoid 3163:disphenoid 3145:Disphenoid 3139:Disphenoid 3082:concurrent 2843:Projective 2712:8 ((1 2 3) 2608:isomorphic 2551:symmetries 1839:steradians 1660:horizontal 289:tessellate 270:truncation 263:rectifying 259:octahedron 67:polyhedron 55:tetrahedra 26:Tetraedron 17595:orthoplex 17517:9-simplex 17467:8-simplex 17417:7-simplex 17374:6-simplex 17344:5-simplex 17313:Tesseract 17041:birotunda 17031:bifrustum 16796:snub cube 16691:polyhedra 16623:antiprism 16328:Trihedron 16297:Polyhedra 16240:MathWorld 16166:1019-8385 16133:(1): 1–6. 16069:250435684 16017:Polyhedra 15926:MathWorld 15309:cite book 15255:1304.0179 15135:0096-3003 14974:Park 2016 14829:MathWorld 14592:symmetric 14284:resistors 14113:Chemistry 14107:airfields 13755:Euclidean 13041:, (*332) 13008:V∞.3.3.3 13002:V6.3.3.3 12999:V5.3.3.3 12996:V4.3.3.3 12993:V3.3.3.3 12990:V2.3.3.3 12577:antiprism 12464:Improper 12452:Hexagonal 12300:‖ 12286:‖ 12283:− 12274:‖ 12260:‖ 12247:‖ 12233:‖ 12230:− 12221:‖ 12207:‖ 12194:‖ 12180:‖ 12177:− 12168:‖ 12154:‖ 12079:− 12035:− 11991:− 11936:− 11802:− 11778:− 11757:− 11353:≤ 11159:≥ 11051:≥ 11036:⋅ 11012:⋅ 10988:⋅ 10964:⋅ 10864:θ 10860:⁡ 10848:Δ 10838:Δ 10822:θ 10818:⁡ 10806:Δ 10796:Δ 10780:θ 10776:⁡ 10764:Δ 10754:Δ 10744:− 10730:Δ 10712:Δ 10694:Δ 10676:Δ 10605:θ 10551:Δ 10375:∠ 10372:⁡ 10366:⋅ 10354:∠ 10351:⁡ 10345:⋅ 10333:∠ 10330:⁡ 10312:∠ 10309:⁡ 10303:⋅ 10291:∠ 10288:⁡ 10282:⋅ 10270:∠ 10267:⁡ 10176:Aristotle 10017:− 10002:α 9994:⁡ 9976:α 9968:⁡ 9950:α 9942:⁡ 9922:α 9914:⁡ 9903:− 9888:α 9880:⁡ 9862:α 9854:⁡ 9834:α 9826:⁡ 9808:α 9800:⁡ 9789:− 9774:α 9766:⁡ 9746:α 9738:⁡ 9720:α 9712:⁡ 9694:α 9686:⁡ 9675:− 9518:− 9508:× 9434:− 9318:− 9296:− 9242:− 9208:− 9187:− 9133:− 9099:− 9078:− 9024:− 8839:− 8805:− 8771:− 8737:− 8696:opposite 8656:opposite 8616:opposite 8451:γ 8447:⁡ 8434:− 8430:β 8426:⁡ 8413:− 8409:α 8405:⁡ 8392:− 8388:γ 8384:⁡ 8377:β 8373:⁡ 8366:α 8362:⁡ 8287:− 8234:− 8181:− 8086:− 8063:− 8040:− 7477:⋅ 7237:⋅ 7223:= 0 then 7138:γ 7134:⁡ 7121:− 7117:β 7113:⁡ 7100:− 7096:α 7092:⁡ 7079:− 7075:γ 7071:⁡ 7064:β 7060:⁡ 7053:α 7049:⁡ 6842:β 6838:⁡ 6818:⋅ 6802:α 6798:⁡ 6778:⋅ 6762:γ 6758:⁡ 6738:⋅ 6672:⋅ 6657:⋅ 6640:⋅ 6611:⋅ 6594:⋅ 6579:⋅ 6536:⋅ 6253:⋅ 6183:⋅ 6122:− 6106:− 6090:− 5590:triangles 5588:isosceles 5510:triangles 5508:isosceles 5403:triangles 5313:triangles 5311:isosceles 5116:isosceles 5012:triangles 4194:35°15′52″ 4175:𝜿 4126:≈ 4036:≈ 3946:≈ 3849:π 3817:𝜿 3813:− 3804:π 3770:≈ 3722:π 3690:𝜿 3686:− 3677:π 3612:π 3580:𝜿 3564:70°31′44″ 3545:≈ 3490:𝟁 3483:− 3480:π 3468:70°31′44″ 3451:𝜿 3444:− 3441:π 3397:dihedral 3078:incenters 2925:conformal 2882:rectangle 2845:symmetry 2572:known as 2388:− 2379:− 2359:− 2344:− 2325:− 2316:− 2159:− 2141:− 2123:− 2093:− 2060:− 2031:− 1929:± 1889:− 1874:± 1841:, 1809.8 1821:π 1818:− 1797:⁡ 1760:⁡ 1751:− 1743:π 1713:⁡ 874:midsphere 795:∘ 787:≈ 769:⁡ 738:− 730:⁡ 715:∘ 707:≈ 684:⁡ 653:⁡ 603:≈ 556:⋅ 521:⋅ 437:≈ 382:⋅ 121:Like all 101:3-simplex 94:Euclidean 34:(journal) 17649:Topics: 17612:demicube 17577:polytope 17571:Uniform 17332:600-cell 17328:120-cell 17281:Demicube 17255:Pentagon 17235:Triangle 17021:bicupola 17001:pyramids 16927:dihedron 16323:Dihedron 16114:(1619). 15999:(1973). 15985:(1948). 15805:(1965). 15769:(1875). 15490:7 August 15440:Lee 1997 15427:15558830 15015:(1989). 14673:See also 14624:skeleton 14541:Diameter 14511:Vertices 14478:HAL 9000 14460:monolith 14370:Pyraminx 14311:crystals 14301:used in 14288:soldered 14176:ammonium 14174:) or an 14123:ammonium 14105:At some 14101:Aviation 13964:compound 13896:{12i,3} 13762:Paraco. 13554:Paraco. 13548:Euclid. 13472:V3.4.3.4 13457:V3.3.3.3 13044:, (332) 13036:Symmetry 12818:digonal 12810:∞.3.3.3 12804:7.3.3.3 12801:6.3.3.3 12798:5.3.3.3 12795:4.3.3.3 12792:3.3.3.3 12789:2.3.3.3 12575:digonal 12467:Regular 12347:Centroid 12337:Centroid 11228:and the 11220:Inradius 10596:and let 10193:and the 9605:centroid 9597:bimedian 9589:centroid 8716:. Then, 7204:, while 6986:‖ 6972:‖ 6942:‖ 6928:‖ 6898:‖ 6884:‖ 6296:‖ 6264:‖ 6222:‖ 6194:‖ 4969:Symmetry 4960:diagram 3789:54°44′8″ 3662:54°44′8″ 3016:3-sphere 3012:600-cell 2977:600-cell 2748:, etc.; 2714:, etc.; 2533:Symmetry 2452:demicube 1676:centroid 906:exsphere 851:insphere 303:and the 282:kleetope 247:compound 138:insphere 116:triangle 79:vertices 43:geometry 17586:simplex 17556:10-cube 17323:24-cell 17309:16-cell 17250:Hexagon 17104:regular 17063:italics 17051:scutoid 17036:rotunda 17026:frustum 16755:uniform 16704:regular 16689:Convex 16643:pyramid 16628:frustum 16174:2154824 16061:3608204 15977:: 6–10. 15789:3571917 15777:Bibcode 15638:Bibcode 15392:2689983 15272:3647851 14698:simplex 14689:Caltrop 14588:regular 14380:Geology 14307:valence 14295:silicon 14214:⁠ 14202:⁠ 14160:methane 13968:origami 13962:. This 13905:{3i,3} 13902:{6i,3} 13899:{9i,3} 13377:sr{3,3} 13372:tr{3,3} 13367:rr{3,3} 12595:-gonal 12590:uniform 12432:Digonal 12416:pyramid 11661:. Let 11230:inradii 10947:. Then 10197:of the 10146:simplex 9353:bisects 6164:⁠ 6152:⁠ 5765:is the 5584:scalene 5401:scalene 5185:scalene 5067:{3,3}. 4852:). The 3309:smaller 3302:√ 3239:√ 3232:√ 3203:simplex 3074:cevians 3008:16-cell 2739:⁠ 2716:⁠ 2610:to the 1674:is the 1653:√ 1646:√ 791:109.471 142:tangent 112:polygon 108:pyramid 97:simplex 65:, is a 17526:9-cube 17476:8-cube 17426:7-cube 17383:6-cube 17353:5-cube 17240:Square 17117:Family 17016:cupola 16969:duals: 16955:prisms 16633:cupola 16509:vertex 16211: 16172: 16164: 16086: 16067: 16059: 16028: 15958: 15821: 15787: 15673:26 May 15611: 15425: 15390: 15270: 15133: 14638:, and 14531:Radius 14293:Since 14196:. The 14041:by, a 13477:V4.6.6 13467:V3.3.3 13462:V3.6.6 13452:V3.6.6 13447:V3.3.3 13357:t{3,3} 13352:r{3,3} 13347:t{3,3} 12973:image 12932:image 12884:image 12582:digons 12442:Square 12125: 12113: 11956: 11107:, and 11087:, and 10940:, and 10915:, and 10663:. The 10132:, but 10050:where 9593:median 9366:, the 8877:where 8676:, and 8576:, and 8136:where 7950:, and 7890:, and 7160:where 6956:, and 6868:where 6716: 6712: 6522:Hence 6516: 6315: 6311: 6241: 6237: 5745:where 5699:Volume 4976:Schön. 3133:Kepler 3004:5-cell 2985:chiral 2890:wedges 2886:square 2870:square 2824:Image 1794:arccos 1757:arcsin 1710:arccos 1668:median 904:, and 766:arctan 727:arccos 711:70.529 681:arctan 650:arccos 213:convex 17245:p-gon 16638:wedge 16618:prism 16478:(132) 16186:(PDF) 16104:(PDF) 16065:S2CID 16057:JSTOR 15745:(PDF) 15734:(PDF) 15708:(PDF) 15693:(PDF) 15562:(PDF) 15388:JSTOR 15333:(PDF) 15301:(PDF) 15268:JSTOR 15250:arXiv 15236:(PDF) 14751:Notes 14628:graph 14551:Girth 14521:Edges 14331:Games 14220:Water 14178:ion ( 13892:{∞,3} 13887:{8,3} 13882:{7,3} 13877:{6,3} 13872:{5,3} 13867:{4,3} 13862:{3,3} 13857:{2,3} 13362:{3,3} 13342:{3,3} 12839:name 12353:Faces 11950:where 6714:where 6313:where 6144:graph 5008:Four 4921:above 4905:nodes 4880:of a 4129:0.408 4039:0.707 3949:1.225 3773:0.577 3548:1.155 3391:edge 3209:of a 3117:In a 2868:is a 2819:Edge 1847:spats 1689:proof 1642:slope 936:are: 606:0.118 440:1.732 232:Plato 75:edges 71:faces 29:, or 17603:cube 17276:Cube 17106:and 16719:cube 16600:(57) 16571:(92) 16565:(13) 16559:(13) 16528:(16) 16504:edge 16499:face 16472:(90) 16466:(60) 16460:(48) 16454:(32) 16448:(30) 16442:(24) 16209:ISBN 16162:ISSN 16084:ISBN 16026:ISBN 15956:ISBN 15869:2012 15819:ISBN 15785:OCLC 15753:2009 15716:2006 15675:2024 15609:ISBN 15492:2018 15423:PMID 15315:link 15131:ISSN 14696:and 14622:The 14390:The 14372:and 14345:The 14286:are 14276:and 14121:The 14004:The 13997:The 13978:and 13721:,3} 13005:... 12922:... 12871:... 12807:... 12720:... 12665:... 12567:... 12520:... 12459:... 11418:and 11241:for 10895:Let 10434:Let 10152:cf. 10112:and 10068:The 10061:and 9622:The 9453:here 9421:and 9000:and 7990:is: 7850:Let 7827:and 7212:and 7200:and 7188:and 5767:base 5087:*332 5063:and 4991:Ord. 4986:Orb. 4981:Cox. 4956:Edge 3394:arc 3211:tree 3010:and 2719:1 ± 2547:cube 2489:and 2448:dual 2260:cube 1694:Its 1664:apex 634:Its 243:dual 236:fire 175:The 127:nets 45:, a 17152:(p) 16753:or 16588:(4) 16553:(5) 16522:(9) 16484:(∞) 16152:doi 16049:doi 15899:doi 15646:doi 15542:doi 15415:doi 15380:doi 15260:doi 15246:110 15121:doi 15117:472 15029:doi 14877:333 14642:, W 14634:, K 14462:in 14057:of 14033:In 13649:3.3 13493:{3, 12400:600 12397:185 12395:124 12390:120 12383:800 12381:436 12119:and 11391:If 10857:cos 10815:cos 10773:cos 10369:sin 10348:sin 10327:sin 10306:sin 10285:sin 10264:sin 10156:). 9991:cos 9965:cos 9939:cos 9911:cos 9877:cos 9851:cos 9823:cos 9797:cos 9763:cos 9735:cos 9709:cos 9683:cos 8850:192 8438:cos 8417:cos 8396:cos 8381:cos 8370:cos 8359:cos 7474:288 7251:det 7125:cos 7104:cos 7083:cos 7068:cos 7057:cos 7046:cos 6835:cos 6795:cos 6755:cos 6079:det 5658:to 5586:or 5553:*22 5474:222 5416:or 5374:2*2 5160:*33 5110:An 5094:12 5089:332 3834:60° 3745:𝟁 3707:60° 3635:𝝉 3597:60° 3520:𝟀 3402:𝒍 3197:An 3123:one 3087:An 3068:An 2994:In 2704:): 1912:and 1691:). 253:or 57:or 51:pl. 41:In 17672:: 17657:• 17653:• 17633:21 17629:• 17626:k1 17622:• 17619:k2 17597:• 17554:• 17524:• 17502:21 17498:• 17495:41 17491:• 17488:42 17474:• 17452:21 17448:• 17445:31 17441:• 17438:32 17424:• 17402:21 17398:• 17395:22 17381:• 17351:• 17330:• 17311:• 17290:• 17274:• 17206:/ 17195:/ 17185:/ 17176:/ 17154:/ 16611:‌s 16237:. 16196:16 16194:. 16188:. 16170:MR 16168:. 16160:. 16148:13 16146:. 16142:. 16129:. 16063:. 16055:. 16045:36 16043:. 16024:. 16020:. 15975:24 15973:. 15954:. 15950:. 15923:. 15895:55 15893:. 15889:. 15845:. 15817:. 15815:32 15783:. 15736:. 15701:75 15699:. 15695:. 15666:. 15644:. 15634:22 15632:. 15585:, 15570:11 15568:, 15538:29 15536:, 15482:. 15465:, 15447:^ 15421:. 15411:10 15409:. 15386:. 15376:54 15374:. 15350:. 15311:}} 15307:{{ 15266:. 15258:. 15244:. 15238:. 15187:^ 15168:68 15152:67 15129:. 15115:. 15111:. 15069:". 15043:^ 15025:17 15023:. 15019:. 14939:68 14900:^ 14893:55 14826:. 14811:^ 14776:^ 14700:– 14654:. 14606:, 14602:, 14598:, 14594:, 14590:, 14586:, 14565:24 14376:. 14264:. 14222:, 14180:NH 14164:CH 14097:. 14081:, 14077:, 14073:, 14069:, 14065:, 13707:. 13697:3 13694:3 13691:3 13688:3 13515:} 13501:. 13038:: 12848:) 12584:. 12418:. 12403:. 12388:47 12349:. 11834:24 11681:, 11677:, 11669:, 11665:, 11649:. 11409:, 11400:, 11103:, 11099:, 11083:, 11079:, 10933:, 10926:, 10911:, 10907:, 10515:, 10488:, 10461:, 10249:, 10245:, 10241:, 10186:. 10100:, 10082:sr 10065:. 10054:ij 10006:34 9980:24 9954:14 9926:34 9892:23 9866:13 9838:24 9812:23 9778:12 9750:14 9724:13 9698:12 9580:. 8636:, 8556:. 8536:, 8516:, 8496:, 8122:12 7930:, 7870:, 7831:ij 7821:, 7817:, 7813:, 7801:, 7761:34 7744:24 7727:14 7703:34 7681:23 7664:13 7640:24 7623:23 7601:12 7577:14 7560:13 7543:12 7463:: 7216:. 7168:, 7164:, 6912:, 6533:36 6239:or 5649:A 5630:2 5627:22 5611:. 5604:, 5560:2 5555:22 5539:2v 5518:2v 5477:4 5381:4 5376:2× 5362:2d 5322:2d 5283:1 5241:2 5232:1v 5224:1h 5207:. 5200:, 5167:3 5162:33 5148:3v 5132:, 5126:3v 5092:24 5025:, 4927:. 4892:. 4825:A 4817:. 4784:, 4751:, 4685:, 4652:, 4586:, 4553:, 4494:, 4461:, 4402:, 4369:, 4299:, 4277:, 3315:. 3182:A 3161:A 3084:. 3065:. 3006:, 2991:. 2892:. 2802:. 2757:, 2753:, 2727:± 2723:± 2557:. 2529:. 2255:. 1849:. 1725:27 1722:23 1268:16 1003:24 872:, 849:, 284:. 203:A 140:) 129:. 103:. 85:. 53:: 23:, 17641:- 17639:n 17631:k 17624:2 17617:1 17610:- 17608:n 17601:- 17599:n 17593:- 17591:n 17584:- 17582:n 17575:- 17573:n 17500:4 17493:2 17486:1 17450:3 17443:2 17436:1 17400:2 17393:1 17222:n 17220:H 17213:2 17210:G 17202:4 17199:F 17191:8 17188:E 17182:7 17179:E 17173:6 17170:E 17161:n 17157:D 17150:2 17147:I 17139:n 17135:B 17127:n 17123:A 17095:e 17088:t 17081:v 17065:. 16757:) 16749:( 16706:) 16702:( 16682:e 16675:t 16668:v 16289:e 16282:t 16275:v 16262:. 16243:. 16217:. 16176:. 16154:: 16131:4 16092:. 16071:. 16051:: 16034:. 16009:. 15964:. 15929:. 15905:. 15901:: 15871:. 15827:. 15791:. 15779:: 15755:. 15718:. 15677:. 15652:. 15648:: 15640:: 15618:. 15544:: 15529:p 15526:L 15494:. 15468:. 15463:” 15442:. 15429:. 15417:: 15394:. 15382:: 15352:8 15317:) 15274:. 15262:: 15252:: 15221:. 15209:. 15170:. 15154:. 15137:. 15123:: 15097:. 15067:3 15037:. 15031:: 14989:. 14976:. 14941:. 14895:. 14879:. 14863:. 14832:. 14783:( 14702:n 14644:4 14636:4 14575:4 14555:3 14545:1 14535:1 14525:6 14515:4 14258:t 14233:O 14229:2 14224:H 14211:3 14208:/ 14205:1 14185:4 14169:4 13741:e 13734:t 13727:v 13719:n 13715:n 13713:* 13684:3 13679:3 13674:3 13669:3 13664:3 13659:3 13654:3 13535:e 13528:t 13521:v 13513:n 13509:n 13507:* 13495:n 12844:( 12829:n 12618:e 12611:t 12604:v 12593:n 12313:) 12304:2 12294:0 12290:x 12278:2 12268:3 12264:x 12251:2 12241:0 12237:x 12225:2 12215:2 12211:x 12198:2 12188:0 12184:x 12172:2 12162:1 12158:x 12147:( 12141:2 12138:1 12133:= 12130:B 12107:) 12098:T 12093:] 12087:0 12083:x 12074:3 12070:x 12065:[ 12054:T 12049:] 12043:0 12039:x 12030:2 12026:x 12021:[ 12010:T 12005:] 11999:0 11995:x 11986:1 11982:x 11977:[ 11968:( 11964:= 11961:A 11944:B 11939:1 11932:A 11928:= 11921:C 11903:3 11900:x 11897:, 11894:2 11891:x 11888:, 11885:1 11882:x 11879:, 11876:0 11873:x 11867:C 11843:. 11837:V 11829:) 11826:C 11823:c 11820:+ 11817:B 11814:b 11811:+ 11808:A 11805:a 11799:( 11796:) 11793:C 11790:c 11787:+ 11784:B 11781:b 11775:A 11772:a 11769:( 11766:) 11763:C 11760:c 11754:B 11751:b 11748:+ 11745:A 11742:a 11739:( 11736:) 11733:C 11730:c 11727:+ 11724:B 11721:b 11718:+ 11715:A 11712:a 11709:( 11703:= 11700:R 11687:V 11683:C 11679:B 11675:A 11671:c 11667:b 11663:a 11659:R 11637:r 11632:4 11628:A 11622:3 11619:1 11614:+ 11611:r 11606:3 11602:A 11596:3 11593:1 11588:+ 11585:r 11580:2 11576:A 11570:3 11567:1 11562:+ 11559:r 11554:1 11550:A 11544:3 11541:1 11536:= 11533:V 11520:. 11503:4 11499:A 11495:+ 11490:3 11486:A 11482:+ 11477:2 11473:A 11469:+ 11464:1 11460:A 11454:V 11451:3 11445:= 11442:r 11429:r 11424:4 11420:A 11415:3 11411:A 11406:2 11402:A 11397:1 11393:A 11373:, 11366:2 11362:r 11358:2 11346:2 11341:4 11337:r 11333:1 11328:+ 11321:2 11316:3 11312:r 11308:1 11303:+ 11296:2 11291:2 11287:r 11283:1 11278:+ 11271:2 11266:1 11262:r 11258:1 11243:i 11238:i 11234:r 11226:r 11204:. 11201:) 11198:M 11195:P 11192:+ 11189:L 11186:P 11183:+ 11180:K 11177:P 11174:+ 11171:J 11168:P 11165:( 11162:3 11156:D 11153:P 11150:+ 11147:C 11144:P 11141:+ 11138:B 11135:P 11132:+ 11129:A 11126:P 11113:P 11109:M 11105:L 11101:K 11097:J 11093:P 11089:D 11085:C 11081:B 11077:A 11060:. 11057:V 11054:9 11045:d 11040:F 11033:D 11030:P 11027:+ 11021:c 11016:F 11009:C 11006:P 11003:+ 10997:b 10992:F 10985:B 10982:P 10979:+ 10973:a 10968:F 10961:A 10958:P 10945:d 10942:F 10938:c 10935:F 10931:b 10928:F 10924:a 10921:F 10917:D 10913:C 10909:B 10905:A 10901:V 10897:P 10876:) 10871:j 10868:i 10852:l 10842:k 10834:+ 10829:k 10826:i 10810:l 10800:j 10792:+ 10787:l 10784:i 10768:k 10758:j 10750:( 10747:2 10739:2 10734:l 10726:+ 10721:2 10716:k 10708:+ 10703:2 10698:j 10690:= 10685:2 10680:i 10649:j 10645:P 10639:i 10635:P 10612:j 10609:i 10582:i 10578:P 10555:i 10528:4 10524:P 10501:3 10497:P 10474:2 10470:P 10447:1 10443:P 10408:O 10387:. 10384:A 10381:B 10378:O 10363:B 10360:C 10357:O 10342:C 10339:A 10336:O 10324:= 10321:A 10318:C 10315:O 10300:C 10297:B 10294:O 10279:B 10276:A 10273:O 10251:C 10247:B 10243:A 10239:O 10134:O 10130:v 10126:M 10122:v 10118:v 10114:M 10110:O 10106:v 10102:M 10094:O 10090:O 10086:O 10078:O 10063:j 10059:i 10052:α 10034:0 10031:= 10026:| 10020:1 10011:) 9998:( 9985:) 9972:( 9959:) 9946:( 9931:) 9918:( 9906:1 9897:) 9884:( 9871:) 9858:( 9843:) 9830:( 9817:) 9804:( 9792:1 9783:) 9770:( 9755:) 9742:( 9729:) 9716:( 9703:) 9690:( 9678:1 9669:| 9647:T 9643:M 9639:T 9542:. 9537:6 9532:| 9528:) 9524:) 9521:c 9515:b 9512:( 9504:a 9500:( 9496:| 9492:d 9486:= 9483:V 9463:V 9438:c 9430:b 9409:a 9389:d 9333:. 9330:) 9327:W 9324:+ 9321:v 9315:u 9312:( 9308:) 9305:v 9302:+ 9299:u 9293:W 9290:( 9287:= 9284:z 9278:, 9275:) 9272:v 9269:+ 9266:u 9263:+ 9260:W 9257:( 9254:) 9251:u 9248:+ 9245:W 9239:v 9236:( 9233:= 9230:Z 9223:, 9220:) 9217:V 9214:+ 9211:u 9205:w 9202:( 9199:) 9196:u 9193:+ 9190:w 9184:V 9181:( 9178:= 9175:y 9169:, 9166:) 9163:u 9160:+ 9157:w 9154:+ 9151:V 9148:( 9145:) 9142:w 9139:+ 9136:V 9130:u 9127:( 9124:= 9121:Y 9114:, 9111:) 9108:U 9105:+ 9102:w 9096:v 9093:( 9090:) 9087:w 9084:+ 9081:v 9075:U 9072:( 9069:= 9066:x 9060:, 9057:) 9054:w 9051:+ 9048:v 9045:+ 9042:U 9039:( 9036:) 9033:v 9030:+ 9027:U 9021:w 9018:( 9015:= 9012:X 8984:, 8979:z 8976:y 8973:x 8968:= 8965:s 8959:, 8954:Y 8951:X 8948:z 8943:= 8940:r 8933:, 8928:X 8925:Z 8922:y 8917:= 8914:q 8908:, 8903:Z 8900:Y 8897:x 8892:= 8889:p 8862:w 8858:v 8854:u 8845:) 8842:s 8836:r 8833:+ 8830:q 8827:+ 8824:p 8821:( 8817:) 8814:s 8811:+ 8808:r 8802:q 8799:+ 8796:p 8793:( 8789:) 8786:s 8783:+ 8780:r 8777:+ 8774:q 8768:p 8765:( 8761:) 8758:s 8755:+ 8752:r 8749:+ 8746:q 8743:+ 8740:p 8734:( 8727:= 8724:V 8704:W 8684:w 8664:V 8644:v 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7387:c 7379:3 7375:b 7367:3 7363:a 7353:2 7349:d 7341:2 7337:c 7329:2 7325:b 7317:2 7313:a 7303:1 7299:d 7291:1 7287:c 7279:1 7275:b 7267:1 7263:a 7255:( 7247:| 7243:= 7240:V 7234:6 7221:d 7214:b 7210:a 7206:γ 7202:c 7198:a 7194:β 7190:c 7186:b 7182:d 7178:α 7174:d 7170:γ 7166:β 7162:α 7144:, 7129:2 7108:2 7087:2 7043:2 7040:+ 7037:1 7030:6 7026:c 7023:b 7020:a 7014:= 7011:V 6979:c 6967:= 6964:c 6935:b 6923:= 6920:b 6891:a 6879:= 6876:a 6846:. 6832:c 6829:a 6826:= 6822:c 6814:a 6806:, 6792:c 6789:b 6786:= 6782:c 6774:b 6766:, 6752:b 6749:a 6746:= 6742:b 6734:a 6727:{ 6697:| 6688:2 6684:c 6676:c 6668:b 6661:c 6653:a 6644:c 6636:b 6627:2 6623:b 6615:b 6607:a 6598:c 6590:a 6583:b 6575:a 6566:2 6562:a 6554:| 6549:= 6544:2 6540:V 6496:, 6493:) 6488:3 6484:c 6480:, 6475:2 6471:c 6467:, 6462:1 6458:c 6454:( 6451:= 6447:c 6439:, 6436:) 6431:3 6427:b 6423:, 6418:2 6414:b 6410:, 6405:1 6401:b 6397:( 6394:= 6390:b 6382:, 6379:) 6374:3 6370:a 6366:, 6361:2 6357:a 6353:, 6348:1 6344:a 6340:( 6337:= 6333:a 6326:{ 6289:c 6280:b 6271:a 6259:= 6256:V 6250:6 6215:c 6208:b 6201:a 6189:= 6186:V 6180:6 6161:6 6158:/ 6155:1 6130:) 6126:d 6118:c 6114:, 6110:d 6102:b 6098:, 6094:d 6086:a 6082:( 6074:6 6071:1 6042:. 6039:) 6034:3 6030:d 6026:, 6021:2 6017:d 6013:, 6008:1 6004:d 6000:( 5997:= 5989:d 5981:, 5978:) 5973:3 5969:c 5965:, 5960:2 5956:c 5952:, 5947:1 5943:c 5939:( 5936:= 5928:c 5920:, 5917:) 5912:3 5908:b 5904:, 5899:2 5895:b 5891:, 5886:1 5882:b 5878:( 5875:= 5867:b 5859:, 5856:) 5851:3 5847:a 5843:, 5838:2 5834:a 5830:, 5825:1 5821:a 5817:( 5814:= 5806:a 5777:h 5753:A 5733:. 5730:h 5727:A 5722:3 5719:1 5714:= 5711:V 5676:2 5672:/ 5668:3 5621:2 5618:C 5609:2 5606:Z 5598:2 5595:C 5558:4 5546:2 5543:C 5536:C 5528:4 5525:V 5515:C 5468:2 5465:D 5428:2 5425:D 5421:2 5418:Z 5414:4 5411:V 5379:8 5367:4 5365:S 5359:D 5319:D 5280:1 5274:1 5272:C 5238:* 5229:C 5227:= 5221:C 5219:= 5216:s 5213:C 5205:2 5202:Z 5194:s 5191:C 5165:6 5153:3 5151:C 5145:C 5137:3 5134:S 5123:C 5080:T 5076:d 5073:T 5030:4 5027:S 5019:d 5016:T 4945:4 4941:3 4800:6 4797:1 4767:2 4764:3 4734:3 4731:4 4701:6 4698:1 4668:2 4665:1 4635:3 4632:1 4602:2 4599:1 4569:2 4566:3 4540:1 4510:6 4507:1 4477:3 4474:1 4448:1 4418:6 4415:1 4385:2 4382:1 4352:2 4349:3 4315:3 4312:1 4286:1 4260:3 4257:4 4219:2 4215:3 4119:6 4116:1 4089:l 4085:/ 4081:R 4076:2 4029:2 4026:1 3999:l 3995:/ 3991:R 3986:1 3939:2 3936:3 3909:l 3905:/ 3901:R 3896:0 3852:3 3807:2 3763:3 3760:1 3725:3 3680:2 3646:1 3615:3 3576:2 3538:3 3535:4 3486:2 3447:2 3413:2 3304:3 3241:3 3234:2 2872:. 2797:2 2776:x 2772:x 2762:) 2759:k 2755:j 2751:i 2741:) 2736:2 2733:/ 2729:k 2725:j 2721:i 2682:4 2678:A 2653:T 2626:4 2622:S 2591:d 2585:T 2517:} 2514:3 2511:, 2508:4 2505:{ 2501:h 2432:2 2427:2 2403:. 2400:) 2397:1 2394:, 2391:1 2385:, 2382:1 2376:( 2368:, 2365:) 2362:1 2356:, 2353:1 2350:, 2347:1 2341:( 2334:, 2331:) 2328:1 2322:, 2319:1 2313:, 2310:1 2307:( 2299:, 2296:) 2293:1 2290:, 2287:1 2284:, 2281:1 2278:( 2241:3 2235:6 2230:2 2203:) 2200:1 2197:, 2194:0 2191:, 2188:0 2185:( 2177:, 2173:) 2167:3 2164:1 2156:, 2150:3 2147:2 2138:, 2132:9 2129:2 2119:( 2111:, 2107:) 2101:3 2098:1 2090:, 2084:3 2081:2 2075:, 2069:9 2066:2 2056:( 2049:, 2045:) 2039:3 2036:1 2028:, 2025:0 2022:, 2016:9 2013:8 2006:( 1981:y 1978:x 1951:) 1944:2 1940:1 1935:, 1932:1 1926:, 1923:0 1919:( 1905:) 1898:2 1894:1 1886:, 1883:0 1880:, 1877:1 1870:( 1814:) 1809:3 1806:1 1801:( 1791:3 1788:= 1777:) 1772:3 1769:1 1764:( 1754:3 1746:2 1738:= 1730:) 1717:( 1684:C 1680:C 1672:C 1655:2 1648:2 1622:. 1617:2 1612:) 1606:2 1601:4 1597:d 1593:+ 1588:2 1583:3 1579:d 1575:+ 1570:2 1565:2 1561:d 1557:+ 1552:2 1547:1 1543:d 1539:+ 1534:2 1530:a 1525:( 1520:= 1512:) 1506:4 1501:4 1497:d 1493:+ 1488:4 1483:3 1479:d 1475:+ 1470:4 1465:2 1461:d 1457:+ 1452:4 1447:1 1443:d 1439:+ 1434:4 1430:a 1425:( 1421:4 1414:, 1409:2 1404:) 1398:3 1392:2 1388:R 1384:2 1378:+ 1373:4 1367:2 1362:4 1358:d 1354:+ 1349:2 1344:3 1340:d 1336:+ 1331:2 1326:2 1322:d 1318:+ 1313:2 1308:1 1304:d 1296:( 1291:= 1282:9 1276:4 1272:R 1262:+ 1257:4 1251:4 1246:4 1242:d 1238:+ 1233:4 1228:3 1224:d 1220:+ 1215:4 1210:2 1206:d 1202:+ 1197:4 1192:1 1188:d 1158:i 1154:d 1133:R 1113:a 1089:. 1083:6 1079:a 1074:= 1068:E 1063:r 1056:, 1050:8 1046:a 1041:= 1036:R 1033:r 1028:= 1022:M 1017:r 1009:, 999:a 994:= 991:R 986:3 983:1 978:= 975:r 969:, 966:a 961:4 957:6 951:= 948:R 921:E 916:r 889:M 884:r 860:r 837:R 800:. 783:) 778:2 773:( 763:2 760:= 752:) 746:3 743:1 734:( 720:, 703:) 697:2 692:2 688:( 678:= 670:) 665:3 662:1 657:( 619:. 614:3 610:a 595:2 590:6 584:3 580:a 574:= 571:a 566:3 562:6 552:) 546:2 542:a 536:4 532:3 525:( 516:3 513:1 508:= 505:V 485:a 480:3 476:6 453:. 448:2 444:a 432:3 425:2 421:a 417:= 413:) 407:2 403:a 397:4 393:3 386:( 379:4 376:= 373:A 353:A 333:a 49:( 36:.

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Index