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198:
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181:
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169:
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10051:{\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\,}
1114:
14443:
7480:
2007:
9587:: 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
7465:
6067:
12334:{\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}}}
10897:
1642:{\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}}}
4844:) 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
1846:
2907:
6874:
652:
9358:
6715:{\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}}}
950:
14120:, 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.
7797:{\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}}}
3224:
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
10421:
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,
10170:
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
5668:
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
5652:
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
640:
10425:
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
5664:
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
10215:
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
9660:
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
7166:
5807:
10681:
6524:
8475:
7240:
2223:{\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}}}
10430:
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.
5657:, 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.
1712:
10409:
9009:
11394:
5800:
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.
5327:
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
9014:
5665:
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.
1973:
8325:
3176:
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.
11081:
10233:
6731:
3236:
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
2899:. 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
6151:
2428:
9661:
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.
820:{\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}}}
9645:, 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.
11658:
6245:
474:
511:
8886:
7845:
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
6319:
4538:, 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
14246:, 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.
8145:
14491:
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.
1109:{\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}}}
10159:. 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
8330:
11874:
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
11864:
1697:
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
4929:
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
10095:, 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,
8891:
7460:{\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|\,.}
7017:
9563:
6330:
1875:
11535:
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
9572:
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,
8150:
14495:
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.
9594:
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.
5523:
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
4934:. 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
10270:
6062:{\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}}}
5131:
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
7009:
6965:
6921:
10892:{\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})}
12344:
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.
11262:
2564:
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
11529:
4958:, ) 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.
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2012:
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657:
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1841:{\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}}}
4243:
11225:
2575:
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
6869:{\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}}}
3839:
3712:
9353:{\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}}}
379:
2781:
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 (
2615:
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913:
3241:, 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
5699:
15602:, 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.
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508:. 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:
10578:
4197:
14201:), 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
13276:
13247:
13237:
13208:
13199:
13179:
13170:
13141:
13131:
10672:
9385:, 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.
5359:
5070:
5060:
4910:. 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
3385:
3375:
3365:
3357:
3347:
3322:
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
3307:
3297:
3287:
3279:
3269:
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13033:
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.
5604:
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
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14782:. 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.
14338:
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).
2002:
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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-
15700:
9489:
203:
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
16691:
10083:
of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle.
272:, 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.,
4859:. Regular tetrahedra, however, cannot fill space by themselves (moreover, it is not scissors-congruent to any other polyhedra which can fill the space, see
635:{\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}.}
15325:
829:
13544:
11706:
10678:
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:
9652:
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
649:—the angle between two planar—and its angle between lines from the center of a regular tetrahedron between two vertices is respectively:
12417:
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.
7161:{\displaystyle V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}},\,}
2895:. 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
6519:{\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}}}
8470:{\displaystyle V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}}}
3124:
Kepler's drawing of a regular tetrahedron inscribed in a cube, and one of the four trirectangular tetrahedra that surround it, filling the cube.
15306:
15095:
13750:
4918:
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
4899:. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as
2273:
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
1699:
14023:(itself is the dual of Szilassi polyhedron) and the tetrahedron are the only two known polyhedra in which every diagonal lies on the sides.
12627:
4946:
The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a
15357:
Lindelof, L. (1867). "Sur les maxima et minima d'une fonction des rayons vecteurs menés d'un point mobile à plusieurs centres fixes".
14806:
constant ≈ 1.618, for which
Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
10404:{\displaystyle \sin \angle OAB\cdot \sin \angle OBC\cdot \sin \angle OCA=\sin \angle OAC\cdot \sin \angle OCB\cdot \sin \angle OBA.\,}
9004:{\displaystyle {\begin{aligned}p={\sqrt {xYZ}},&\quad q={\sqrt {yZX}},\\r={\sqrt {zXY}},&\quad s={\sqrt {xyz}},\end{aligned}}}
16684:
3389:
by its planes of symmetry. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.
11389:{\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}}},}
6970:
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6882:
17669:
14040:
An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.
14005:. A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull.
13537:
2721:
rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together
15741:
3330:
2560:
can be grouped into two groups of four, each forming a regular tetrahedron, showing one of the two tetrahedra in the cube. The
1968:{\displaystyle \left(\pm 1,0,-{\frac {1}{\sqrt {2}}}\right)\quad {\mbox{and}}\quad \left(0,\pm 1,{\frac {1}{\sqrt {2}}}\right)}
14019:
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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14263:
There are molecules with the shape based on four nearby atoms whose bonds form the sides of a tetrahedral structure, such as
11448:
8320:{\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}}}
3201:. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a
10414:
One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.
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14104:
A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as
13039:
11076:{\displaystyle PA\cdot F_{\mathrm {a} }+PB\cdot F_{\mathrm {b} }+PC\cdot F_{\mathrm {c} }+PD\cdot F_{\mathrm {d} }\geq 9V.}
3807:
3680:
3088:
4883:
For
Euclidean 3-space, there are 3 simple and related Goursat tetrahedra. They can be seen as points on and within a cube.
4118:
4028:
3938:
3762:
3537:
3333:. There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron
17104:
16298:
15962:
15067:
9610:
of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all
5717:
5714:
The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume:
14036:
2938:, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
13743:
13530:
4542:
which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges
4212:
2891:
define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a
15833:
15623:
13046:
11132:
8327:
The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.
6180:
The absolute value of the scalar triple product can be represented as the following absolute values of determinants:
4979:
17:
14301:
together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.
15715:
15340:
13694:
10209:
10194:
9622:). The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the
4947:
4864:
4856:
2712:
245:, who associated those four solids with nature. The regular tetrahedron was considered as the classical element of
13942:
9577:
and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes.
8607:
are the lengths of the tetrahedron's edges as in the following image. Here, the first three form a triangle, with
15152:
12620:
10427:
3061:
Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess.
2806:, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A
2589:
15295:
Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong
University Press, 1994, pp. 53–54
14939:
13956:
222:. In other words, all of its faces are the same size and shape (congruent) and all edges are the same length. A
16007:
15993:
15023:
14997:
14877:
14779:
14152:
14069:
9435:
1872:, defining the four vertices of a tetrahedron with edge length 2, centered at the origin, and two-level edges:
12357:
6146:{\textstyle {\frac {1}{6}}\det(\mathbf {a} -\mathbf {d} ,\mathbf {b} -\mathbf {d} ,\mathbf {c} -\mathbf {d} )}
3486:
3447:
299:
129:(any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
17706:
16596:
15251:
14893:
13736:
10440:
10239:
9637:
of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the
7471:
4799:
4766:
4733:
4700:
4667:
4634:
4601:
4568:
4509:
4476:
4417:
4384:
4351:
4314:
4259:
125:
base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a
12386:. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces are
9641:
and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute
3029:
by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
2423:{\displaystyle {\begin{aligned}(1,1,1),&\quad (1,-1,-1),\\(-1,1,-1),&\quad (-1,-1,1).\end{aligned}}}
17711:
17701:
15674:
15168:
14638:
14475:
14073:
6154:
5417:
It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the
4935:
4255:
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths
2233:
16890:
16831:
15314:. Dept of Mathematics, Chulalongkorn University, Bangkok. Archived from the original on 27 February 2009.
13970:
13948:
3582:
3202:
2506:
479:
9381:
of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the
4180:
3852:
3725:
3615:
17696:
16920:
16880:
16486:
16032:
15783:
Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography
15569:
14750:
14689:
13998:
13962:
13902:
12613:
10200:
On otherhand, several irregular tetrahedra are known, of which copies can tile space, for instance the
9463:
4860:
3129:
3115:
2999:
2983:
307:
10189:
claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a
3102:
has concurrent cevians that join the vertices to the points of contact of the opposite faces with the
921:
889:
16915:
16910:
14610:
14257:
13689:
13684:
11653:{\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}
9664:
The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron.
9382:
6240:{\displaystyle 6\cdot V={\begin{Vmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{Vmatrix}}}
2584:
469:{\displaystyle A=4\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)=a^{2}{\sqrt {3}}\approx 1.732a^{2}.}
6740:
6339:
17691:
17127:
16480:
16011:
15997:
14614:
14470:
14264:
13714:
The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3
12872:
12867:
12382:
There exist tetrahedra having integer-valued edge lengths, face areas and volume. These are called
9588:
5672:
3069:
2965:
2931:
2711:(in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the
273:
246:
15613:
14320:
of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how
10610:
3146:
discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.
2658:
2433:
17097:
17021:
17016:
16895:
16801:
16608:
16536:
16456:
16291:
15913:
15849:
14856:
14551:
14541:
14484:
14313:
14061:
14020:
12885:
12877:
10556:
9619:
9618:
of the tetrahedron. In addition the four medians are divided in a 3:1 ratio by the centroid (see
2960:
2817:
2561:
15120:"Finite number of similarity classes in Longest Edge Bisection of nearly equilateral tetrahedra"
14694:
12742:
10640:
9392:, and the distance between the edges is defined as the distance between the two skew lines. Let
3209:
311:
16885:
16826:
16816:
16761:
16542:
15825:
15777:
15593:
14606:
14406:
14402:
14396:
13977:
of five tetrahedra has been known for hundreds of years. It comes up regularly in the world of
13928:
12862:
10087:
found that, corresponding to any given tetrahedron is a point now known as an isogonic center,
8881:{\displaystyle V={\frac {\sqrt {\,(-p+q+r+s)\,(p-q+r+s)\,(p+q-r+s)\,(p+q+r-s)}}{192\,u\,v\,w}}}
7847:
7470:
Given the distances between the vertices of a tetrahedron the volume can be computed using the
6314:{\displaystyle 6\cdot V={\begin{Vmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{Vmatrix}}}
4911:
3006:
2466:
291:, a regular tetrahedron with four triangular pyramids attached to each of its faces. i.e., its
15897:
14887:
4986:
4906:
For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a
17641:
17634:
17627:
16905:
16821:
16776:
16264:
16259:
15893:
15598:
15490:
15178:
15162:
14949:
14521:
14203:
14162:
14081:
13367:
13357:
12679:
8140:{\displaystyle V={\frac {\sqrt {4a^{2}b^{2}c^{2}-a^{2}X^{2}-b^{2}Y^{2}-c^{2}Z^{2}+XYZ}}{12}}}
7851:
5307:
4078:
3988:
3898:
2915:
2803:
2274:
1869:
284:
280:
41:
17166:
17144:
17132:
15853:
15243:
17298:
17245:
16865:
16791:
16739:
16184:
15787:
15648:
15381:
14744:
14724:
14561:
14089:
13482:
13382:
13050:
12795:
12671:
12661:
12383:
12377:
10583:
10529:
10502:
10475:
10448:
10182:
4926:
4900:
4855:
A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in the
3323:
2871:
2683:
2627:
1159:
916:
261:
219:
10226:
9362:
Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron
4450:
of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is
8:
17716:
17686:
17653:
17552:
17302:
17031:
16900:
16875:
16860:
16796:
16744:
16201:
15471:
15057:
15055:
14933:
14739:
14531:
14317:
14016:
13974:
13500:
This polyhedron is topologically related as a part of sequence of regular polyhedra with
13487:
13472:
13462:
12666:
10175:
10091:, at which the solid angles subtended by the faces are equal, having a common value of π
9667:
There is a relation among the angles made by the faces of a general tetrahedron given by
4896:
3221:
1857:
288:
257:
15791:
15652:
14903:
1984:
17522:
17472:
17422:
17379:
17349:
17309:
17272:
17090:
17046:
17011:
16870:
16765:
16714:
16653:
16530:
16524:
16284:
16193:
16075:
16067:
16017:
15818:
15398:
15319:
15278:
15260:
14734:
14571:
14093:
14065:
14045:
12996:
12600:
12457:
12426:
12387:
10220:
9469:
9415:
9395:
8710:
8690:
8670:
8650:
8630:
8610:
8590:
8570:
8550:
8530:
8510:
8490:
7984:
7964:
7944:
7924:
7904:
7884:
7864:
7855:
5783:
5759:
5498:
5397:
4888:
4849:
4546:
4454:
4292:
3652:
3419:
3010:
2952:
1139:
1119:
866:
843:
359:
339:
303:
118:
104:
15931:
15190:
15083:
15052:
13501:
10084:
5075:
3198:
3018:
3014:
2501:
17661:
17026:
16836:
16811:
16755:
16643:
16567:
16519:
16492:
16462:
16269:
that also includes a description of a "rotating ring of tetrahedra", also known as a
16242:
16219:
16172:
16094:
16079:
16036:
15966:
15928:
15829:
15795:
15752:
15619:
15433:
15376:
15141:
15044:
15027:
14831:
14775:
14594:
14451:
14422:
14085:
14009:
13892:
13679:
13387:
13026:
12968:
12961:
12462:
12368:
The sum of the areas of any three faces is greater than the area of the fourth face.
9634:
9374:
5114:
4996:
4539:
3022:
2678:
1678:
100:
89:
12975:
10637:
be the dihedral angle between the two faces of the tetrahedron adjacent to the edge
5277:. An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ).
17665:
17230:
17219:
17208:
17197:
17188:
17179:
17118:
17114:
16965:
16648:
16628:
16450:
16162:
16126:
16059:
15909:
15656:
15552:
15425:
15416:
Rassat, André; Fowler, Patrick W. (2004). "Is There a "Most Chiral
Tetrahedron"?".
15390:
15270:
15131:
15039:
14729:
14581:
14409:
to explain the formation of the Earth, was popular through the early 20th century.
14357:
14002:
13934:
13897:
13887:
13509:
12954:
12940:
12656:
11859:{\displaystyle R={\frac {\sqrt {(aA+bB+cC)(aA+bB-cC)(aA-bB+cC)(-aA+bB+cC)}}{24V}}.}
10160:
10108:
10080:
9611:
9370:
8487:
The volume of a tetrahedron can be ascertained by using the Heron formula. Suppose
5532:
5418:
4991:
3103:
3092:
2927:
2900:
315:
238:
187:
137:
133:
16109:
15477:
8479:
197:
17255:
17240:
16602:
16514:
16509:
16474:
16429:
16419:
16409:
16404:
16245:
16180:
16122:
16088:
15781:
14834:
14602:
14466:
14376:
14372:
12777:
12763:
12435:
12353:
10164:
5777:
5586:
5140:
5044:
5033:
4845:
3143:
3120:
2708:
2622:
2565:
2470:
2458:
2270:
1674:
253:
241:. Known since antiquity, the Platonic solid is named after the Greek philosopher
93:
85:
81:
16026:
15854:"William Lowthian Green and his Theory of the Evolution of the Earth's Features"
14375:, with the number rolled appearing around the bottom or on the top vertex. Some
14140:
13782:
12770:
10426:
triangles, there are four such constraints on sums of angles, and the number of
6177:
of the volume of any parallelepiped that shares three converging edges with it.
5580:
5313:
4922:
which is multiplied by mirror reflections into the vertices of the polyhedron.)
4879:
2906:
2571:
2548:
17605:
16786:
16709:
16658:
16561:
16424:
16414:
16167:
16150:
16111:
What has the Volume of a
Tetrahedron to do with Computer Programming Languages?
14802:𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the
14662:
14658:
14642:
13765:
12749:
12572:
12558:
12452:
10675:
9574:
9378:
6158:
5001:
3238:
3165:
2580:
646:
310:, which is a tessellation. Some tetrahedra that are not regular, including the
249:, because of his interpretation of its sharpest corner being most penetrating.
234:
204:
15136:
15119:
14253:
of mixtures of chemical substances are represented graphically as tetrahedra.
14012:
is another polyhedron with four faces, but it does not have triangular faces.
12947:
12565:
10115:, of the vertices. In the event that the solid angle at one of the vertices,
5504:
5263:
5012:
230:. There are eight convex deltahedra, one of which is the regular tetrahedron.
17680:
17622:
17510:
17503:
17496:
17460:
17453:
17446:
17410:
17403:
16991:
16847:
16781:
16579:
16573:
16468:
16399:
16389:
16176:
15871:
15813:
15145:
14704:
14598:
14480:
14309:
14250:
14208:
14158:
13715:
13377:
13110:
13103:
12847:
12830:
12827:
A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual
12756:
12544:
9580:
6153:, or any other combination of pairs of vertices that form a simply connected
5510:
5403:
5274:
4892:
3065:
2935:
2810:
2462:
1853:
180:
15556:
13609:
13096:
13082:
13068:
12988:
12537:
3160:
3050:
3041:
136:, a tetrahedron can be folded from a single sheet of paper. It has two such
17562:
16394:
16270:
15799:
15437:
15429:
14803:
14618:
14350:
14272:
14077:
14053:
13994:
13990:
13982:
13882:
13810:
13789:
13602:
13492:
13089:
13075:
13061:
12981:
12920:
12913:
12551:
10245:
10190:
9606:
and a line segment joining the midpoints of two opposite edges is called a
5612:
5208:
4907:
4841:
838:
144:
31:
12927:
2755:
rotation by an angle of 180° such that an edge maps to the opposite edge:
168:
17571:
17532:
17482:
17432:
17389:
17359:
16434:
16368:
16358:
16348:
14755:
14718:
14650:
14384:
14368:
14364:
14333:
14288:
14284:
14145:
14049:
14048:, complicated three-dimensional shapes are commonly broken down into, or
13986:
13674:
6070:
4848:, a family of space-filling tetrahedra. All space-filling tetrahedra are
3217:
3137:
3072:. When only one pair of opposite edges are perpendicular, it is called a
2618:
1978:
1706:
227:
14431:
13803:
13796:
13595:
13588:
13581:
12906:
12899:
12584:
A regular tetrahedron can be seen as a degenerate polyhedron, a uniform
9558:{\displaystyle V={\frac {d|(\mathbf {a} \times \mathbf {(b-c)} )|}{6}}.}
7854:
in the 15th century, as a three-dimensional analogue of the 1st century
2945:
2789:): the rotations correspond to those of the cube about face-to-face axes
17557:
17541:
17491:
17441:
17398:
17368:
17282:
17056:
16944:
16734:
16701:
16619:
16373:
16363:
16353:
16338:
16328:
16307:
16071:
15639:
Brittin, W. E. (1945). "Valence angle of the tetrahedral carbon atom".
15402:
15282:
14634:
14105:
14057:
13867:
13669:
13362:
13030:
12892:
12711:
12649:
12642:
10227:
A law of sines for tetrahedra and the space of all shapes of tetrahedra
9627:
9389:
5653:
computer graphics. One of the commonly used subdivision methods is the
5299:
4868:
3173:
3155:
642:
Its volume can also be obtained by dissecting a cube into three parts.
269:
223:
77:
35:
15660:
15118:
Trujillo-Pino, Agustín; Suárez, Jose Pablo; Padrón, Miguel A. (2024).
13435:
12718:
12525:
10178:
gives two more regular compounds, containing five and ten tetrahedra.
226:
polyhedron in which all of its faces are equilateral triangles is the
17613:
17527:
17477:
17427:
17384:
17354:
17323:
17051:
17041:
16986:
16970:
16806:
16633:
16250:
15936:
14839:
14166:
13442:
13421:
13407:
13025:
A truncation process applied to the tetrahedron produces a series of
12787:
12697:
12607:
12587:
12511:
12490:
10186:
10092:
9363:
3164:
A space-filling tetrahedral disphenoid inside a cube. Two edges have
2892:
2707:—the identity and 11 proper rotations—with the following
1868:
One way to construct a regular tetrahedron is by using the following
1849:
884:
16063:
15394:
15274:
14128:
14072:. These methods have wide applications in practical applications in
13838:
13817:
13630:
13574:
12690:
12518:
3185:
2978:
2793:
17587:
17342:
17338:
17265:
16937:
16669:
16333:
15517:
14488:
14380:
14346:
14294:
14186:
14133:
14117:
13859:
13852:
13845:
13659:
13651:
13644:
13637:
12704:
12442:
11240:
10099:
lies inside the tetrahedron, and because the sum of distances from
9599:
6157:. Comparing this formula with that used to compute the volume of a
5187:
4867:
fills space with alternating regular tetrahedron cells and regular
3329:
Every regular polytope, including the regular tetrahedron, has its
3026:
2987:
2910:
A tetragonal disphenoid viewed orthogonally to the two green edges.
1686:
861:
292:
148:
126:
53:
15265:
15104:, p. 63, §4.3 Rotation groups in two dimensions; notion of a
14672:
14360:, dating from 2600 BC, was played with a set of tetrahedral dice.
13623:
9583:
found a center that exists in every tetrahedron, now known as the
3361:
is subdivided into 24 instances of its characteristic tetrahedron
2839:
17596:
17566:
17333:
17328:
17319:
17260:
17061:
17036:
16638:
15980:
Bottema, O. (1969). "A Theorem of
Bobillier on the Tetrahedron".
14708:
14699:
14321:
14305:
14170:
13978:
12504:
10156:
5273:
Its only isometry is the identity, and the symmetry group is the
3213:
152:
122:
107:
15959:
A Mathematical Space
Odyssey: Solid Geometry in the 21st Century
13616:
9412:
be the distance between the skew lines formed by opposite edges
5701:, the iterated LEB produces no more than 37 similarity classes.
4343:
around its exterior right-triangle face (the edges opposite the
3255:, so all its edges are edges or diagonals of the cube. The cube
1183:
from an arbitrary point in 3-space to its four vertices, it is:
306:
in the ratio of two tetrahedra to one octahedron, they form the
174:
Regular tetrahedron, described as the classical element of fire.
17536:
17486:
17436:
17393:
17363:
17314:
17250:
15611:
14916:
14914:
14912:
14798:) uses the greek letter 𝝓 (phi) to represent one of the three
14637:
of the tetrahedron (comprising the vertices and edges) forms a
14513:
14298:
13449:
3084:
2995:
2896:
2880:
2846:
15860:. Vol. XXV. Geological Publishing Company. pp. 1–10.
13831:
13824:
13428:
13400:
12725:
12497:
9567:
7981:
be those of the opposite edges. The volume of the tetrahedron
5541:. A digonal disphenoid has Schläfli symbol { }∨{ }.
2457:, centered at the origin. For the other tetrahedron (which is
327:
16136:
Lee, Jung Rye (1997). "The Law of
Cosines in a Tetrahedron".
14230:
14211:
between any two vertices of a perfect tetrahedron is arccos(−
13414:
12592:
11684:
be the lengths of the three edges that meet at a vertex, and
10185:
by themselves, although this result seems likely enough that
6069:
The volume of a tetrahedron can be ascertained in terms of a
1652:
242:
16276:
16216:
The
Routledge International Handbook of Innovation Education
14909:
14641:, with 4 vertices, and 6 edges. It is a special case of the
14001:, in which the ten tetrahedra are arranged as five pairs of
7004:{\displaystyle c={\begin{Vmatrix}\mathbf {c} \end{Vmatrix}}}
6960:{\displaystyle b={\begin{Vmatrix}\mathbf {b} \end{Vmatrix}}}
6916:{\displaystyle a={\begin{Vmatrix}\mathbf {a} \end{Vmatrix}}}
17286:
16729:
16260:
Free paper models of a tetrahedron and many other polyhedra
15519:
Spherical Trigonometry: For the Use of Colleges and Schools
13877:
13467:
11126:
to the faces, and suppose the faces have equal areas, then
9656:
towards the circumcenter. Also, an orthogonal line through
7191:, is the angle between the two edges connecting the vertex
6862:
6512:
6161:, we conclude that the volume of a tetrahedron is equal to
2557:
15926:
15244:"Altitudes of a tetrahedron and traceless quadratic forms"
14169:) at the four corners of a tetrahedron. For instance in a
9598:
A line segment joining a vertex of a tetrahedron with the
4925:
Among the Goursat tetrahedra which generate 3-dimensional
3064:
If all three pairs of opposite edges of a tetrahedron are
1981:, centroid at the origin, with lower face parallel to the
147:) on which all four vertices lie, and another sphere (the
16151:"On the volume of a hyperbolic and spherical tetrahedron"
15341:"Déterminants sphérique et hyperbolique de Cayley-Menger"
15117:
14790:
14788:
14271:-butyltetrahedrane, known derivative of the hypothetical
11399:
with equality if and only if the tetrahedron is regular.
5150:. A triangular pyramid has Schläfli symbol {3}∨( ).
279:
The tetrahedron is yet related to another two solids: By
15742:"Radial and Pruned Tetrahedral Interpolation Techniques"
15223:
14442:
14260:
are represented graphically on a two-dimensional plane.
12834:, containing 6 vertices, in two sets of colinear edges.
12352:
The tetrahedron's center of mass can be computed as the
10232:
7921:
be the lengths of three edges that meet at a point, and
233:
The regular tetrahedron is also one of the five regular
15872:"Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron"
15618:, U. S. Government Printing Office, p. 13-10,
15543:
centroidal Voronoi tessellation and its applications",
11524:{\displaystyle r={\frac {3V}{A_{1}+A_{2}+A_{3}+A_{4}}}}
9466:. Then another formula for the volume of a tetrahedron
3311:
four different ways, with all six surrounding the same
2866:
2579:
The regular tetrahedron has 24 isometries, forming the
237:, a set of polyhedrons in which all of their faces are
15201:
15199:
14785:
14747:– constructed by joining two tetrahedra along one face
12161:
11982:
9682:
7508:
7269:
6985:
6941:
6897:
6567:
6277:
6207:
6078:
5804:
Given the vertices of a tetrahedron in the following:
4805:
4772:
4739:
4706:
4673:
4640:
4607:
4574:
4515:
4482:
4423:
4390:
4357:
4320:
4265:
4217:
4124:
4034:
3944:
3857:
3812:
3768:
3730:
3685:
3620:
3543:
3318:
cube diagonal. The cube can also be dissected into 48
3189:
A cube dissected into six characteristic orthoschemes.
2236:
1921:
482:
207:
is left, where the five edge angles do not quite meet.
143:
For any tetrahedron there exists a sphere (called the
92:. The tetrahedron is the simplest of all the ordinary
15004:
11926:
11709:
11542:
11451:
11265:
11135:
10967:
10684:
10643:
10613:
10586:
10559:
10532:
10505:
10478:
10451:
10273:
9676:
9492:
9472:
9438:
9418:
9398:
9017:
8894:
8733:
8713:
8693:
8673:
8653:
8633:
8613:
8593:
8573:
8553:
8533:
8513:
8493:
8333:
8153:
8007:
7987:
7967:
7947:
7927:
7907:
7887:
7867:
7483:
7243:
7020:
6973:
6929:
6885:
6734:
6542:
6333:
6259:
6189:
5810:
5786:
5762:
5720:
5675:
5647:
5493:
Generalized disphenoids (2 pairs of equal triangles)
4941:
4802:
4769:
4736:
4703:
4670:
4637:
4604:
4571:
4549:
4512:
4479:
4457:
4420:
4387:
4354:
4317:
4295:
4262:
4215:
4183:
4121:
4081:
4031:
3991:
3941:
3901:
3855:
3810:
3765:
3728:
3683:
3655:
3618:
3585:
3540:
3489:
3450:
3422:
3055:
Tetrahedral symmetries shown in tetrahedral diagrams
2686:
2661:
2630:
2592:
2509:
2436:
2283:
2010:
1987:
1878:
1715:
1189:
1162:
1142:
1122:
953:
924:
892:
869:
846:
833:
Regular tetrahedron ABCD and its circumscribed sphere
655:
514:
382:
362:
342:
16240:
14829:
10930:, and for which the areas of the opposite faces are
4936:
dissected into characteristic tetrahedra of the cube
4145:{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}
4055:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}
3965:{\displaystyle {\sqrt {\tfrac {3}{2}}}\approx 1.225}
3789:{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}
3564:{\displaystyle {\sqrt {\tfrac {4}{3}}}\approx 1.155}
2994:
Regular tetrahedra can be stacked face-to-face in a
336:
Given that the regular tetrahedron with edge length
15389:(5). Mathematical Association of America: 227–243.
15211:
15196:
15191:"Simplex Volumes and the Cayley-Menger Determinant"
13981:. Joining the twenty vertices would form a regular
9388:Any two opposite edges of a tetrahedron lie on two
3233:tetrahedron because it contains four right angles.
3168:
of 90°, and four edges have dihedral angles of 60°.
376:is four times the area of an equilateral triangle:
16266:An Amazing, Space Filling, Non-regular Tetrahedron
15817:
15491:"Einige Bemerkungen über die dreiseitige Pyramide"
13969:An interesting polyhedron can be constructed from
12425:A regular tetrahedron can be seen as a triangular
12420:
12333:
11858:
11652:
11523:
11388:
11219:
11075:
10891:
10666:
10629:
10599:
10572:
10545:
10518:
10491:
10464:
10403:
10050:
9557:
9478:
9454:
9424:
9404:
9352:
9003:
8880:
8719:
8699:
8679:
8659:
8639:
8619:
8599:
8579:
8559:
8539:
8519:
8499:
8469:
8319:
8139:
7993:
7973:
7953:
7933:
7913:
7893:
7873:
7796:
7459:
7160:
7003:
6959:
6915:
6868:
6714:
6518:
6313:
6239:
6145:
6061:
5792:
5768:
5748:
5693:
4840:packs with directly congruent or enantiomorphous (
4819:
4786:
4753:
4720:
4687:
4654:
4621:
4588:
4555:
4529:
4496:
4463:
4437:
4404:
4371:
4334:
4301:
4279:
4237:
4191:
4144:
4104:
4054:
4014:
3964:
3924:
3870:
3833:
3788:
3743:
3706:
3661:
3633:
3596:
3563:
3506:
3467:
3428:
2778:reflections in a plane perpendicular to an edge: 6
2699:
2669:
2643:
2609:
2532:
2449:
2422:
2258:
2222:
1996:
1967:
1840:
1641:
1175:
1148:
1128:
1108:
939:
907:
875:
852:
819:
634:
500:
468:
368:
348:
260:figure comprising two such dual tetrahedra form a
252:The regular tetrahedron is self-dual, meaning its
16138:J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math
15495:Sammlung mathematischer Aufsätze u. Bemerkungen 1
15092:, pp. 33–34, §3.1 Congruent transformations.
15064:, pp. 71–72, §4.7 Characteristic tetrahedra.
10910:be any interior point of a tetrahedron of volume
9534:
9522:
2794:Orthogonal projections of the regular tetrahedron
17678:
15324:: CS1 maint: bot: original URL status unknown (
15076:, pp. 292–293, Table I(i); "Tetrahedron, 𝛼
14978:
14865:
13997:of each other. Superimposing both forms gives a
10434:
10417:Putting any of the four vertices in the role of
7261:
6089:
4238:{\displaystyle {\tfrac {{\text{arc sec }}3}{2}}}
15852:(January 1900). Winchell, Newton Horace (ed.).
15444:
14855:Ford, Walter Burton; Ammerman, Charles (1913),
14060:in the process of setting up the equations for
13029:. Truncating edges down to points produces the
11220:{\displaystyle PA+PB+PC+PD\geq 3(PJ+PK+PL+PM).}
5335:. A tetragonal disphenoid has Coxeter diagram
2887:The two skew perpendicular opposite edges of a
2269:A regular tetrahedron can be embedded inside a
14278:
12397:and the other two are isosceles with areas of
3834:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}}
3707:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}}
3283:can be dissected into six such 3-orthoschemes
3205:with right triangle or obtuse triangle faces.
17098:
16685:
16292:
15898:"The tetrahedral principle in kite structure"
15497:(in German). Berlin: Maurer. pp. 105–132
14324:of silicon form and what shapes they assume.
13744:
13538:
12621:
10212:. The complete list remains an open problem.
7219:, is defined by the position of the vertices
3109:
2926:The tetrahedron can also be represented as a
218:is a tetrahedron in which all four faces are
16148:
15956:
15914:10.1038/scientificamerican06131903-22947supp
15632:
15415:
15241:
15229:
15158:
14945:
14854:
14778:at a vertex. In chemistry, it is called the
13522:32 symmetry mutation of regular tilings: {3,
12310:
12296:
12284:
12270:
12257:
12243:
12231:
12217:
12204:
12190:
12178:
12164:
11917:can be formulated as matrix-vector product:
11668:Denote the circumradius of a tetrahedron as
11438:denote the area of each faces, the value of
10171:added to make a cube, which has 8 vertices.
9626:of the tetrahedron that is analogous to the
5441:. A rhombic disphenoid has Coxeter diagram
2918:when applied to the two special edge pairs.
2527:
2515:
356:. The surface area of a regular tetrahedron
16128:Harmonices Mundi (The Harmony of the World)
14379:-like puzzles are tetrahedral, such as the
14157:The tetrahedron shape is seen in nature in
10131:. If however, a tetrahedron has a vertex,
9568:Properties analogous to those of a triangle
4914:of the generated polyhedron contains three
4831:
3395:Characteristics of the regular tetrahedron
2610:{\displaystyle \mathrm {T} _{\mathrm {d} }}
2430:This yields a tetrahedron with edge-length
1977:Expressed symmetrically as 4 points on the
1693:to a vertex of the base is twice that from
1116:For a regular tetrahedron with side length
17105:
17091:
16692:
16678:
16299:
16285:
15615:Pilot's Handbook of Aeronautical Knowledge
15587:
13751:
13737:
13728:32 symmetry mutation of regular tilings: {
13545:
13531:
12628:
12614:
11235:Denoting the inradius of a tetrahedron as
3197:is a tetrahedron where all four faces are
3036:
16213:
16166:
15848:
15786:. Vol. Part I. London: E. Stanford.
15515:
15308:The Various Kinds of Centres of Simplices
15264:
15135:
15043:
14883:
14099:
10400:
10174:Inscribing tetrahedra inside the regular
10047:
9455:{\displaystyle \mathbf {b} -\mathbf {c} }
9321:
8871:
8867:
8863:
8830:
8802:
8774:
8743:
7453:
7183:are the plane angles occurring in vertex
7157:
1136:, the radius of its circumscribed sphere
16724:
16024:
15612:Federal Aviation Administration (2009),
15379:(1981). "Which tetrahedra fill space?".
15375:
15356:
14899:
14345:
14144:Calculation of the central angle with a
14139:
14127:
14035:
11700:be the volume of the tetrahedron. Then
10580:be the area of the face opposite vertex
8478:
4878:
3407:
3404:
3184:
3159:
3119:
3046:Tetrahedral symmetry subgroup relations
2977:
2905:
2870:
2570:
2547:
1863:
828:
325:
17670:List of regular polytopes and compounds
16155:Communications in Analysis and Geometry
16050:Cundy, H. Martyn (1952). "Deltahedra".
16006:
15992:
15979:
15739:
15638:
15533:
15465:
15463:
15461:
15459:
15217:
15101:
15089:
15073:
15061:
15022:
14970:
14964:
14920:
14795:
13040:Family of uniform tetrahedral polyhedra
11696:the length of the opposite edges. Let
10248:is that in a tetrahedron with vertices
6529:are expressed as row or column vectors.
4820:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
4787:{\displaystyle {\sqrt {\tfrac {3}{2}}}}
4754:{\displaystyle {\sqrt {\tfrac {4}{3}}}}
4721:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
4688:{\displaystyle {\sqrt {\tfrac {1}{2}}}}
4655:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
4622:{\displaystyle {\sqrt {\tfrac {1}{2}}}}
4589:{\displaystyle {\sqrt {\tfrac {3}{2}}}}
4530:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
4497:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
4438:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
4405:{\displaystyle {\sqrt {\tfrac {1}{2}}}}
4372:{\displaystyle {\sqrt {\tfrac {3}{2}}}}
4335:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
4280:{\displaystyle {\sqrt {\tfrac {4}{3}}}}
3032:
3025:) can be constructed as tilings of the
2655:It has rotational tetrahedral symmetry
476:The height of a regular tetrahedron is
14:
17679:
16149:Murakami, Jun; Yano, Masakazu (2005).
16121:
16086:
15812:
15488:
15304:
15174:
15010:
14483:, a cognitive scientist and expert on
14256:However, quaternary phase diagrams in
10202:characteristic orthoscheme of the cube
10150:
10135:, with solid angle greater than π sr,
5591:
5515:
5408:
5318:
5268:
5192:
5119:
5017:
4887:An irregular tetrahedron which is the
4874:
2651:. They can be categorized as follows:
2259:{\textstyle {\frac {2{\sqrt {6}}}{3}}}
1677:along an edge is twice that along the
1673:distance covered from the base to the
1669:), corresponding to the fact that the
158:
103:case of the more general concept of a
16673:
16280:
16241:
16107:
16049:
15927:
15776:
15698:
15338:
15242:Havlicek, Hans; Weiß, Gunter (2003).
15205:
14871:
14830:
14031:
12371:
5704:
2930:, and projected onto the plane via a
302:(fill space), but if alternated with
16699:
16191:
15892:
15456:
15359:Acta Societatis Scientiarum Fennicae
14984:
14825:
14823:
14499:
10553:be the points of a tetrahedron. Let
10219:The tetrahedron is unique among the
7850:, is essentially due to the painter
4901:Wythoff's kaleidoscopic construction
2867:Cross section of regular tetrahedron
501:{\textstyle {\frac {\sqrt {6}}{3}}a}
256:is another regular tetrahedron. The
121:, which is a polyhedron with a flat
16135:
15963:Mathematical Association of America
15450:
15124:Applied Mathematics and Computation
10422:the result is the fourth identity.
10216:tetrahedron have the same volume.)
5749:{\displaystyle V={\frac {1}{3}}Ah.}
2973:
2921:
2533:{\displaystyle \mathrm {h} \{4,3\}}
1709:at a vertex subtended by a face is
1651:With respect to the base plane the
61:
24:
16316:Listed by number of faces and type
15957:Alsina, C.; Nelsen, R. B. (2015).
14479:to be a tetrahedron, according to
14412:
14165:atoms are surrounded by atoms (or
14094:naval architecture and engineering
12591:, where base polygons are reduced
11055:
11031:
11007:
10983:
10858:
10848:
10816:
10806:
10774:
10764:
10740:
10722:
10704:
10686:
10561:
10385:
10364:
10343:
10322:
10301:
10280:
5648:Subdivision and similarity classes
4950:is formed. Two other isometries (C
4942:Isometries of irregular tetrahedra
4729:, and a right triangle with edges
3871:{\displaystyle {\tfrac {\pi }{3}}}
3744:{\displaystyle {\tfrac {\pi }{3}}}
3634:{\displaystyle {\tfrac {\pi }{3}}}
2990:, seen in stereographic projection
2663:
2601:
2595:
2511:
1078:
1032:
931:
899:
25:
17728:
16234:
15751:. HPL-98-95: 1–32. Archived from
14820:
10901:
10223:in possessing no parallel faces.
10201:
9602:of the opposite face is called a
5198:triangles with a common base edge
3507:{\displaystyle \pi -2{\text{𝟁}}}
3468:{\displaystyle \pi -2{\text{𝜿}}}
2677:. This symmetry is isomorphic to
332:3D model of a regular tetrahedron
15534:Lévy, Bruno; Liu, Yang (2010), "
15305:Outudee, Somluck; New, Stephen.
14936:, Mathematische Basteleien, 2001
14671:
14512:
14450:
14441:
14430:
14421:
13955:
13941:
13927:
13858:
13851:
13844:
13837:
13830:
13823:
13816:
13809:
13802:
13795:
13788:
13781:
13650:
13643:
13636:
13629:
13622:
13615:
13608:
13601:
13594:
13587:
13580:
13573:
13448:
13441:
13434:
13427:
13420:
13413:
13406:
13399:
13342:
13337:
13332:
13327:
13322:
13313:
13308:
13303:
13298:
13293:
13284:
13279:
13274:
13269:
13264:
13255:
13250:
13245:
13240:
13235:
13226:
13221:
13216:
13211:
13206:
13197:
13192:
13187:
13182:
13177:
13168:
13163:
13158:
13153:
13148:
13139:
13134:
13129:
13124:
13119:
13109:
13102:
13095:
13088:
13081:
13074:
13067:
13060:
12987:
12974:
12967:
12960:
12953:
12946:
12926:
12919:
12912:
12905:
12898:
12786:
12776:
12769:
12762:
12755:
12748:
12741:
12724:
12717:
12710:
12703:
12696:
12689:
12571:
12564:
12557:
12550:
12543:
12536:
12524:
12517:
12510:
12503:
12496:
12489:
10231:
10210:disphenoid tetrahedral honeycomb
10195:tetrahedral-octahedral honeycomb
9531:
9528:
9525:
9514:
9448:
9440:
6989:
6945:
6901:
6832:
6824:
6792:
6784:
6752:
6744:
6698:
6694:
6686:
6678:
6671:
6663:
6654:
6646:
6637:
6633:
6625:
6617:
6608:
6600:
6593:
6585:
6576:
6572:
6457:
6400:
6343:
6299:
6290:
6281:
6225:
6218:
6211:
6136:
6128:
6120:
6112:
6104:
6096:
5999:
5938:
5877:
5816:
5585:
5579:
5509:
5503:
5463:
5458:
5453:
5448:
5443:
5402:
5357:
5352:
5347:
5342:
5337:
5312:
5262:
5186:
5113:
5068:
5063:
5058:
5053:
5048:
5011:
4865:tetrahedral-octahedral honeycomb
4857:disphenoid tetrahedral honeycomb
3383:
3378:
3373:
3368:
3363:
3355:
3350:
3345:
3340:
3335:
3305:
3300:
3295:
3290:
3285:
3277:
3272:
3267:
3262:
3257:
3049:
3040:
2951:
2944:
2845:
2838:
2494:
2489:
2484:
2479:
2474:
940:{\displaystyle r_{\mathrm {E} }}
908:{\displaystyle r_{\mathrm {M} }}
298:Regular tetrahedra alone do not
287:. The dual of this solid is the
196:
179:
167:
110:, and may thus also be called a
15949:
15920:
15886:
15864:
15842:
15806:
15770:
15740:Vondran, Gary L. (April 1998).
15733:
15701:"Resistance-Distance Sum Rules"
15692:
15667:
15605:
15562:
15527:
15509:
15482:
15409:
15369:
15350:
15332:
15298:
15289:
15235:
15184:
15111:
15016:
14990:
14955:
14026:
12421:Related polyhedra and compounds
11881:of a tetrahedron with vertices
11869:
11663:
10205:
10068:is the angle between the faces
9366:the volume of the tetrahedron.
9293:
9184:
9075:
8974:
8923:
8483:Six edge-lengths of Tetrahedron
4931:
3180:
2914:This property also applies for
2385:
2316:
2194:
2064:
1927:
1919:
1689:of the base, the distance from
1071:
984:
326:
117:The tetrahedron is one kind of
15820:Principles of physical geology
14926:
14848:
14768:
14626:Table of graphs and parameters
14327:
14153:Tetrahedral molecular geometry
14070:partial differential equations
11839:
11809:
11806:
11779:
11776:
11749:
11746:
11719:
11211:
11175:
10886:
10760:
10147:lies outside the tetrahedron.
10119:, measures exactly π sr, then
10103:to the vertices is a minimum,
10021:
10008:
9995:
9982:
9969:
9956:
9941:
9928:
9907:
9894:
9881:
9868:
9853:
9840:
9827:
9814:
9793:
9780:
9765:
9752:
9739:
9726:
9713:
9700:
9542:
9538:
9510:
9506:
9340:
9322:
9318:
9300:
9285:
9267:
9264:
9246:
9230:
9212:
9209:
9191:
9176:
9158:
9155:
9137:
9121:
9103:
9100:
9082:
9067:
9049:
9046:
9028:
8855:
8831:
8827:
8803:
8799:
8775:
8771:
8744:
6996:
6982:
6952:
6938:
6908:
6894:
6503:
6464:
6446:
6407:
6389:
6350:
6306:
6274:
6232:
6204:
6140:
6092:
6049:
6010:
5988:
5949:
5927:
5888:
5866:
5827:
5125:triangle base and three equal
4630:, a right triangle with edges
3140:, as at the corner of a cube.
3087:that join the vertices to the
2713:unit quaternion representation
2410:
2386:
2375:
2351:
2341:
2317:
2306:
2288:
2213:
2195:
1848:This is approximately 0.55129
1681:of a face. In other words, if
321:
13:
1:
16527:(two infinite groups and 75)
16306:
16214:Shavinina, Larisa V. (2013).
15641:Journal of Chemical Education
15418:Chemistry: A European Journal
15252:American Mathematical Monthly
14861:, Macmillan, pp. 294–295
14813:
14774:It is also the angle between
14227:), or approximately 109.47°.
10441:Trigonometry of a tetrahedron
10435:Law of cosines for tetrahedra
10240:Trigonometry of a tetrahedron
5694:{\displaystyle {\sqrt {3/2}}}
5469:and Schläfli symbol sr{2,2}.
5434:, present as the point group
3149:
3074:semi-orthocentric tetrahedron
2982:A single 30-tetrahedron ring
2875:A central cross section of a
17072:Degenerate polyhedra are in
16545:(two infinite groups and 50)
16194:"Regular polytope distances"
15714:(2): 633–649. Archived from
15545:ACM Transactions on Graphics
15045:10.1016/0898-1221(89)90148-X
14669:
14123:
14074:computational fluid dynamics
13971:five intersecting tetrahedra
12994:
12938:
12890:
12845:
10630:{\displaystyle \theta _{ij}}
7858:for the area of a triangle.
5655:Longest Edge Bisection (LEB)
5363:and Schläfli symbol s{2,4}.
5043:. A regular tetrahedron has
5025:It forms the symmetry group
4249:
4247:
4208:
4203:
4201:
4160:
4158:
4156:
4154:
4114:
4070:
4068:
4066:
4064:
4024:
3980:
3978:
3976:
3974:
3934:
3848:
3843:
3803:
3798:
3758:
3721:
3716:
3676:
3671:
3648:
3611:
3606:
3597:{\displaystyle 2{\text{𝜿}}}
3578:
3573:
3533:
3482:
3477:
3443:
3438:
3415:
3401:
3013:with tetrahedral cells (the
2942:
2852:
2833:
2670:{\displaystyle \mathrm {T} }
2450:{\displaystyle 2{\sqrt {2}}}
1662:) is twice that of an edge (
205:a thin volume of empty space
155:to the tetrahedron's faces.
80:composed of four triangular
7:
16891:pentagonal icositetrahedron
16832:truncated icosidodecahedron
16025:Cromwell, Peter R. (1997).
15028:"Trisecting an Orthoscheme"
14998:"Sections of a Tetrahedron"
14683:
14487:who advised Kubrick on the
14367:, this solid is known as a
14279:Electricity and electronics
14111:
13949:Compound of five tetrahedra
13394:Duals to uniform polyhedra
12347:
11243:of its triangular faces as
11230:
11122:of the perpendiculars from
10914:for which the vertices are
10573:{\displaystyle \Delta _{i}}
10193:that can tile space as the
7207:, does so for the vertices
4192:{\displaystyle {\text{𝜿}}}
4176:
4074:
3984:
3894:
3755:
3645:
3530:
3412:
3399:
2998:aperiodic chain called the
2543:
10:
17733:
17659:
17086:
16921:pentagonal hexecontahedron
16881:deltoidal icositetrahedron
16168:10.4310/cag.2005.v13.n2.a5
16033:Cambridge University Press
16016:(3rd ed.). New York:
15699:Klein, Douglas J. (2002).
15470:Inequalities proposed in “
15339:Audet, Daniel (May 2011).
14751:Trirectangular tetrahedron
14405:, originally published by
14394:
14390:
14331:
14282:
14150:
13999:compound of ten tetrahedra
13963:Compound of ten tetrahedra
13723:
13517:
12375:
12356:of its four vertices, see
10667:{\displaystyle P_{i}P_{j}}
10438:
10237:
10181:Regular tetrahedra cannot
7230:If we do not require that
3331:characteristic orthoscheme
3239:characteristic of the cube
3153:
3130:trirectangular tetrahedron
3116:Trirectangular tetrahedron
3113:
3110:Trirectangular tetrahedron
3091:of the opposite faces are
2465:, a polyhedron that is by
308:alternated cubic honeycomb
283:the tetrahedron becomes a
29:
27:Polyhedron with four faces
17070:
17004:
16979:
16961:
16954:
16929:
16916:disdyakis triacontahedron
16911:deltoidal hexecontahedron
16845:
16753:
16708:
16618:
16597:Kepler–Poinsot polyhedron
16589:
16554:
16502:
16443:
16382:
16321:
16314:
16117:(Thesis). pp. 16–17.
15908:(1432supp): s2294–22950.
15679:American Chemical Society
15137:10.1016/j.amc.2024.128631
14661:, each a skeleton of its
14624:
14590:
14580:
14570:
14560:
14550:
14540:
14530:
14520:
14511:
14506:
14371:, one of the more common
14258:communication engineering
13775:
13769:
13761:
13567:
13561:
13555:
13393:
13045:
13038:
12886:Apeirogonal trapezohedron
12483:
12480:
12434:
10244:A corollary of the usual
7472:Cayley–Menger determinant
5709:
5578:
5575:
5502:
5497:
5492:
5401:
5396:
5311:
5306:
5298:
5261:
5256:
5207:, also isomorphic to the
5185:
5182:
5112:
5109:
5010:
5007:
4978:
4973:
4966:
4963:
4948:3-dimensional point group
4871:cells in a ratio of 2:1.
4838:space-filling tetrahedron
3394:
3227:birectangular tetrahedron
3132:the three face angles at
2964:
2617:. This symmetry group is
2585:full tetrahedral symmetry
2004:plane, the vertices are:
16052:The Mathematical Gazette
15850:Hitchcock, Charles Henry
15230:Murakami & Yano 2005
15159:Alsina & Nelsen 2015
14946:Alsina & Nelsen 2015
14858:Plane and Solid Geometry
14761:
14721:– 4-dimensional analogue
14469:originally intended the
14341:
13935:Two tetrahedra in a cube
12873:Pentagonal trapezohedron
12868:Tetragonal trapezohedron
12363:
9589:orthocentric tetrahedron
9373:or in three-dimensional
4832:Space-filling tetrahedra
3070:orthocentric tetrahedron
2966:Stereographic projection
2932:stereographic projection
2552:The cube and tetrahedron
2230:with the edge length of
30:Not to be confused with
17022:gyroelongated bipyramid
16896:rhombic triacontahedron
16802:truncated cuboctahedron
16609:Uniform star polyhedron
16537:quasiregular polyhedron
16192:Park, Poo-Sung (2016).
15982:Elemente der Mathematik
15778:Green, William Lowthian
15557:10.1145/1778765.1778856
14485:artificial intelligence
14314:solid-state electronics
14062:finite element analysis
13923:Compounds of tetrahedra
13508:}, continuing into the
12878:Hexagonal trapezohedron
12738:Spherical tiling image
7818:represent the vertices
5302:(Four equal triangles)
4861:Hilbert's third problem
4105:{\displaystyle _{2}R/l}
4015:{\displaystyle _{1}R/l}
3925:{\displaystyle _{0}R/l}
3068:, then it is called an
2961:Orthographic projection
2818:Orthographic projection
99:The tetrahedron is the
17017:truncated trapezohedra
16886:disdyakis dodecahedron
16852:(duals of Archimedean)
16827:rhombicosidodecahedron
16817:truncated dodecahedron
16543:semiregular polyhedron
16087:Fekete, A. E. (1985).
15894:Bell, Alexander Graham
15858:The American Geologist
15583:(5): 162–166, May 1985
15516:Todhunter, I. (1886),
15489:Crelle, A. L. (1821).
15430:10.1002/chem.200400869
15032:Computers Math. Applic
14780:tetrahedral bond angle
14715:-dimensional analogues
14690:Boerdijk–Coxeter helix
14407:William Lowthian Green
14403:tetrahedral hypothesis
14397:tetrahedral hypothesis
14353:
14148:
14137:
14100:Structural engineering
14096:, and related fields.
14082:electromagnetic fields
14041:
13776:Noncompact hyperbolic
13568:Noncompact hyperbolic
12863:Trigonal trapezohedron
12404:, while the volume is
12335:
11860:
11654:
11525:
11390:
11256:= 1, 2, 3, 4, we have
11221:
11077:
10893:
10668:
10631:
10601:
10574:
10547:
10520:
10493:
10466:
10405:
10176:compound of five cubes
10052:
9614:at a point called the
9559:
9480:
9456:
9426:
9406:
9354:
9005:
8882:
8721:
8701:
8681:
8661:
8641:
8621:
8601:
8581:
8561:
8541:
8521:
8501:
8484:
8471:
8321:
8141:
7995:
7975:
7955:
7935:
7915:
7895:
7875:
7798:
7461:
7162:
7005:
6961:
6917:
6870:
6716:
6520:
6315:
6241:
6147:
6063:
5794:
5770:
5750:
5695:
5270:Four unequal triangles
4912:Coxeter-Dynkin diagram
4884:
4821:
4788:
4755:
4722:
4689:
4656:
4623:
4590:
4557:
4531:
4498:
4465:
4439:
4406:
4373:
4336:
4303:
4281:
4239:
4193:
4146:
4106:
4056:
4016:
3966:
3926:
3872:
3835:
3790:
3745:
3708:
3663:
3635:
3598:
3565:
3508:
3469:
3430:
3229:. It is also called a
3190:
3169:
3125:
3081:isodynamic tetrahedron
3000:Boerdijk–Coxeter helix
2991:
2984:Boerdijk–Coxeter helix
2916:tetragonal disphenoids
2911:
2884:
2804:orthogonal projections
2718:identity (identity; 1)
2701:
2671:
2645:
2611:
2576:
2553:
2534:
2469:a cube. This form has
2451:
2424:
2260:
2224:
1998:
1969:
1842:
1643:
1177:
1150:
1130:
1110:
941:
909:
877:
854:
834:
821:
636:
502:
470:
370:
350:
333:
16906:pentakis dodecahedron
16822:truncated icosahedron
16777:truncated tetrahedron
16590:non-convex polyhedron
16108:Kahan, W. M. (2012).
16093:. Marcel Dekker Inc.
15708:Croatica Chemica Acta
15599:Pythagorean Triangles
14800:characteristic angles
14476:2001: A Space Odyssey
14349:
14143:
14131:
14039:
12853:Digonal trapezohedron
12680:Apeirogonal antiprism
12336:
11861:
11655:
11526:
11391:
11222:
11078:
10894:
10669:
10632:
10602:
10600:{\displaystyle P_{i}}
10575:
10548:
10546:{\displaystyle P_{4}}
10521:
10519:{\displaystyle P_{3}}
10494:
10492:{\displaystyle P_{2}}
10467:
10465:{\displaystyle P_{1}}
10406:
10155:A tetrahedron is a 3-
10139:still corresponds to
10053:
9560:
9481:
9457:
9427:
9407:
9383:Murakami–Yano formula
9355:
9006:
8883:
8722:
8702:
8682:
8662:
8642:
8622:
8602:
8582:
8562:
8542:
8522:
8502:
8482:
8472:
8322:
8142:
7996:
7976:
7956:
7936:
7916:
7896:
7876:
7852:Piero della Francesca
7807:where the subscripts
7799:
7462:
7163:
7006:
6962:
6918:
6871:
6717:
6521:
6316:
6242:
6148:
6064:
5795:
5771:
5751:
5696:
5308:Tetragonal disphenoid
5257:Irregular tetrahedron
4882:
4822:
4789:
4756:
4723:
4690:
4657:
4624:
4591:
4558:
4532:
4499:
4466:
4440:
4407:
4374:
4345:characteristic angles
4337:
4304:
4282:
4240:
4194:
4147:
4107:
4057:
4017:
3967:
3927:
3873:
3836:
3791:
3746:
3709:
3664:
3636:
3599:
3566:
3509:
3470:
3431:
3188:
3163:
3123:
2981:
2934:. This projection is
2909:
2874:
2702:
2700:{\displaystyle A_{4}}
2672:
2646:
2644:{\displaystyle S_{4}}
2612:
2574:
2551:
2535:
2452:
2425:
2275:Cartesian coordinates
2261:
2225:
1999:
1970:
1870:Cartesian coordinates
1864:Cartesian coordinates
1843:
1644:
1178:
1176:{\displaystyle d_{i}}
1151:
1131:
1111:
942:
910:
878:
855:
832:
822:
637:
503:
471:
371:
351:
331:
285:truncated tetrahedron
268:. Its interior is an
220:equilateral triangles
17707:Prismatoid polyhedra
16866:rhombic dodecahedron
16792:truncated octahedron
15382:Mathematics Magazine
14745:Triangular dipyramid
14725:Synergetics (Fuller)
14695:Möbius configuration
14316:, and silicon has a
14267:allotrope and tetra-
14090:chemical engineering
12842:-gonal trapezohedra
12672:Heptagonal antiprism
12662:Pentagonal antiprism
12650:Triangular antiprism
12378:Heronian tetrahedron
11924:
11707:
11540:
11449:
11263:
11133:
10965:
10682:
10641:
10611:
10584:
10557:
10530:
10503:
10476:
10449:
10271:
9674:
9620:Commandino's theorem
9490:
9470:
9436:
9416:
9396:
9015:
8892:
8731:
8711:
8691:
8671:
8651:
8631:
8611:
8591:
8571:
8551:
8531:
8511:
8491:
8331:
8151:
8005:
7985:
7965:
7945:
7925:
7905:
7885:
7865:
7481:
7241:
7018:
6971:
6927:
6883:
6732:
6540:
6331:
6257:
6187:
6076:
5808:
5784:
5760:
5718:
5673:
5531:, isomorphic to the
5139:, isomorphic to the
5032:, isomorphic to the
5008:Regular tetrahedron
4800:
4767:
4734:
4701:
4668:
4635:
4602:
4569:
4547:
4510:
4477:
4455:
4448:characteristic radii
4446:(edges that are the
4418:
4385:
4352:
4315:
4293:
4260:
4213:
4181:
4119:
4079:
4029:
3989:
3939:
3899:
3853:
3808:
3763:
3726:
3681:
3653:
3616:
3583:
3538:
3487:
3448:
3420:
3324:Heronian tetrahedron
3106:of the tetrahedron.
3100:isogonic tetrahedron
3083:is one in which the
3033:Irregular tetrahedra
2684:
2659:
2628:
2590:
2507:
2434:
2281:
2277:of the vertices are
2234:
2008:
1985:
1876:
1713:
1187:
1160:
1140:
1120:
951:
922:
890:
867:
844:
653:
512:
480:
380:
360:
340:
312:Schläfli orthoscheme
262:stellated octahedron
17712:Pyramids (geometry)
17702:Self-dual polyhedra
17654:pentagonal polytope
17553:Uniform 10-polytope
17113:Fundamental convex
16901:triakis icosahedron
16876:tetrakis hexahedron
16861:triakis tetrahedron
16797:rhombicuboctahedron
16202:Forum Geometricorum
16090:Real Linear Algebra
15932:"Tetrahedral graph"
15902:Scientific American
15792:1875vmge.book.....G
15749:HP Technical Report
15653:1945JChEd..22..145B
15577:Crux Mathematicorum
15551:(4): 119:1–119:11,
15472:Crux Mathematicorum
14740:Tetrahedron packing
14611:distance-transitive
14460:Tetrahedral objects
14308:is the most common
14167:lone electron pairs
14017:Szilassi polyhedron
12843:
12783:Plane tiling image
12667:Hexagonal antiprism
12635:
12388:isosceles triangles
12384:Heronian tetrahedra
11360:
11335:
11310:
11285:
10753:
10735:
10717:
10699:
10151:Geometric relations
10107:coincides with the
9639:twelve-point sphere
7848:Tartaglia's formula
7780:
7763:
7746:
7722:
7700:
7683:
7659:
7642:
7620:
7596:
7579:
7562:
5593:Two pairs of equal
5576:Phyllic disphenoid
5517:Two pairs of equal
5110:Triangular pyramid
4897:Goursat tetrahedron
4895:is an example of a
4875:Fundamental domains
3011:regular 4-polytopes
2889:regular tetrahedron
2877:regular tetrahedron
2820:
1620:
1602:
1584:
1566:
1520:
1502:
1484:
1466:
1381:
1363:
1345:
1327:
1265:
1247:
1229:
1211:
289:triakis tetrahedron
276:the tetrahedron).
216:regular tetrahedron
159:Regular tetrahedron
72:), also known as a
17523:Uniform 9-polytope
17473:Uniform 8-polytope
17423:Uniform 7-polytope
17380:Uniform 6-polytope
17350:Uniform 5-polytope
17310:Uniform polychoron
17273:Uniform polyhedron
17121:in dimensions 2–10
16871:triakis octahedron
16756:Archimedean solids
16531:regular polyhedron
16525:uniform polyhedron
16487:Hectotriadiohedron
16243:Weisstein, Eric W.
16018:Dover Publications
15929:Weisstein, Eric W.
15824:. Nelson. p.
15675:"White phosphorus"
15377:Senechal, Marjorie
15365:(Part 1): 189–203.
15106:fundamental region
14832:Weisstein, Eric W.
14735:Tetrahedral number
14615:3-vertex-connected
14354:
14149:
14138:
14136:ion is tetrahedral
14066:numerical solution
14064:especially in the
14046:numerical analysis
14042:
14032:Numerical analysis
14021:Császár polyhedron
14003:stellae octangulae
12997:Face configuration
12837:
12598:
12372:Integer tetrahedra
12331:
12329:
12321:
12115:
11856:
11650:
11521:
11386:
11346:
11321:
11296:
11271:
11217:
11073:
10889:
10739:
10721:
10703:
10685:
10664:
10627:
10597:
10570:
10543:
10516:
10489:
10462:
10428:degrees of freedom
10401:
10048:
10035:
9555:
9476:
9452:
9422:
9402:
9369:For tetrahedra in
9350:
9348:
9001:
8999:
8878:
8717:
8697:
8677:
8657:
8637:
8617:
8597:
8577:
8557:
8537:
8517:
8497:
8485:
8467:
8317:
8315:
8137:
7991:
7971:
7951:
7931:
7911:
7891:
7871:
7794:
7788:
7766:
7749:
7732:
7708:
7686:
7669:
7645:
7628:
7606:
7582:
7565:
7548:
7457:
7442:
7158:
7001:
6995:
6957:
6951:
6913:
6907:
6866:
6861:
6712:
6706:
6516:
6511:
6311:
6305:
6237:
6231:
6143:
6059:
6057:
5790:
5766:
5746:
5705:General properties
5691:
5611:isomorphic to the
5499:Digonal disphenoid
5398:Rhombic disphenoid
5183:Mirrored sphenoid
4889:fundamental domain
4885:
4850:scissors-congruent
4817:
4814:
4784:
4781:
4751:
4748:
4718:
4715:
4685:
4682:
4652:
4649:
4619:
4616:
4586:
4583:
4553:
4527:
4524:
4494:
4491:
4461:
4435:
4432:
4402:
4399:
4369:
4366:
4347:𝟀, 𝝉, 𝟁), plus
4332:
4329:
4299:
4277:
4274:
4235:
4233:
4189:
4142:
4133:
4102:
4052:
4043:
4012:
3962:
3953:
3922:
3868:
3866:
3831:
3821:
3786:
3777:
3741:
3739:
3704:
3694:
3659:
3631:
3629:
3594:
3561:
3552:
3504:
3465:
3426:
3248:and one of length
3191:
3170:
3126:
2992:
2912:
2885:
2816:
2697:
2667:
2641:
2607:
2577:
2556:The vertices of a
2554:
2530:
2447:
2420:
2418:
2256:
2220:
2218:
1997:{\displaystyle xy}
1994:
1965:
1925:
1838:
1836:
1639:
1637:
1606:
1588:
1570:
1552:
1506:
1488:
1470:
1452:
1367:
1349:
1331:
1313:
1251:
1233:
1215:
1197:
1173:
1146:
1126:
1106:
1104:
937:
905:
873:
850:
835:
817:
815:
632:
498:
466:
366:
346:
334:
318:, can tessellate.
74:triangular pyramid
17697:Individual graphs
17675:
17674:
17662:Polytope families
17119:uniform polytopes
17081:
17080:
17000:
16999:
16837:snub dodecahedron
16812:icosidodecahedron
16667:
16666:
16568:Archimedean solid
16555:convex polyhedron
16463:Icosidodecahedron
16225:978-0-203-38714-6
16100:978-0-8247-7238-3
16042:978-0-521-55432-9
16013:Regular Polytopes
16002:. Methuen and Co.
15999:Regular Polytopes
15994:Coxeter, H. S. M.
15972:978-1-61444-216-5
15661:10.1021/ed022p145
15594:Wacław Sierpiński
15424:(24): 6575–6580.
14681:
14680:
14657:. It is one of 5
14631:
14630:
14507:Tetrahedral graph
14500:Tetrahedral graph
14159:covalently bonded
14086:civil engineering
14010:square hosohedron
13993:forms, which are
13985:. There are both
13920:
13919:
13712:
13711:
13498:
13497:
13027:uniform polyhedra
13023:
13022:
12825:
12824:
12686:Polyhedron image
12643:Digonal antiprism
12582:
12581:
12154:
12137:
12131:
12125:
11968:
11962:
11851:
11842:
11635:
11609:
11583:
11557:
11519:
11381:
11361:
11336:
11311:
11286:
11102:, interior point
10221:uniform polyhedra
9635:nine-point circle
9550:
9479:{\displaystyle V}
9425:{\displaystyle a}
9405:{\displaystyle d}
9375:elliptic geometry
8992:
8967:
8941:
8916:
8876:
8858:
8720:{\displaystyle W}
8700:{\displaystyle w}
8680:{\displaystyle V}
8660:{\displaystyle v}
8640:{\displaystyle U}
8620:{\displaystyle u}
8600:{\displaystyle w}
8580:{\displaystyle v}
8560:{\displaystyle u}
8540:{\displaystyle W}
8520:{\displaystyle V}
8500:{\displaystyle U}
8465:
8356:
8135:
8131:
7994:{\displaystyle V}
7974:{\displaystyle z}
7954:{\displaystyle y}
7934:{\displaystyle x}
7914:{\displaystyle c}
7894:{\displaystyle b}
7874:{\displaystyle a}
7152:
7043:
6087:
5793:{\displaystyle h}
5769:{\displaystyle A}
5735:
5689:
5645:
5644:
4964:Tetrahedron name
4827:
4815:
4813:
4794:
4782:
4780:
4761:
4749:
4747:
4728:
4716:
4714:
4695:
4683:
4681:
4662:
4650:
4648:
4629:
4617:
4615:
4596:
4584:
4582:
4563:
4556:{\displaystyle 1}
4540:60-90-30 triangle
4537:
4525:
4523:
4504:
4492:
4490:
4471:
4464:{\displaystyle 1}
4445:
4433:
4431:
4412:
4400:
4398:
4379:
4367:
4365:
4342:
4330:
4328:
4309:
4302:{\displaystyle 1}
4287:
4275:
4273:
4253:
4252:
4245:
4232:
4223:
4206:
4199:
4187:
4152:
4134:
4132:
4112:
4062:
4044:
4042:
4022:
3972:
3954:
3952:
3932:
3878:
3865:
3846:
3841:
3829:
3820:
3801:
3796:
3778:
3776:
3751:
3738:
3719:
3714:
3702:
3693:
3674:
3669:
3662:{\displaystyle 1}
3641:
3628:
3609:
3604:
3592:
3576:
3571:
3553:
3551:
3514:
3502:
3480:
3475:
3463:
3441:
3436:
3429:{\displaystyle 2}
3231:quadrirectangular
3059:
3058:
3009:, all the convex
2971:
2970:
2864:
2863:
2709:conjugacy classes
2679:alternating group
2445:
2254:
2248:
2180:
2164:
2163:
2146:
2145:
2114:
2098:
2097:
2083:
2082:
2052:
2030:
2029:
1958:
1957:
1924:
1912:
1911:
1822:
1785:
1759:
1738:
1411:
1386:
1295:
1270:
1149:{\displaystyle R}
1129:{\displaystyle a}
1097:
1096:
1064:
1063:
1049:
1017:
1016:
999:
974:
970:
876:{\displaystyle r}
853:{\displaystyle R}
837:The radii of its
791:
759:
710:
678:
611:
608:
579:
575:
549:
545:
529:
493:
489:
445:
410:
406:
369:{\displaystyle A}
349:{\displaystyle a}
304:regular octahedra
101:three-dimensional
18:Tetrahedral angle
16:(Redirected from
17724:
17666:Regular polytope
17227:
17216:
17205:
17164:
17107:
17100:
17093:
17084:
17083:
16959:
16958:
16955:Dihedral uniform
16930:Dihedral regular
16853:
16769:
16718:
16694:
16687:
16680:
16671:
16670:
16503:elemental things
16481:Enneacontahedron
16451:Icositetrahedron
16301:
16294:
16287:
16278:
16277:
16256:
16255:
16229:
16210:
16198:
16188:
16170:
16145:
16132:
16131:. Johann Planck.
16123:Kepler, Johannes
16118:
16116:
16104:
16083:
16058:(318): 263–266.
16046:
16021:
16003:
15989:
15976:
15943:
15942:
15941:
15924:
15918:
15917:
15890:
15884:
15883:
15881:
15879:
15874:. Web of Stories
15868:
15862:
15861:
15846:
15840:
15839:
15823:
15810:
15804:
15803:
15774:
15768:
15767:
15765:
15763:
15757:
15746:
15737:
15731:
15730:
15728:
15726:
15720:
15705:
15696:
15690:
15689:
15687:
15685:
15671:
15665:
15664:
15636:
15630:
15628:
15609:
15603:
15591:
15585:
15584:
15574:
15566:
15560:
15559:
15542:
15531:
15525:
15523:
15513:
15507:
15506:
15504:
15502:
15486:
15480:
15467:
15454:
15448:
15442:
15441:
15413:
15407:
15406:
15373:
15367:
15366:
15354:
15348:
15347:
15345:
15336:
15330:
15329:
15323:
15315:
15313:
15302:
15296:
15293:
15287:
15286:
15268:
15248:
15239:
15233:
15227:
15221:
15215:
15209:
15203:
15194:
15188:
15182:
15172:
15166:
15156:
15150:
15149:
15139:
15115:
15109:
15099:
15093:
15087:
15081:
15071:
15065:
15059:
15050:
15049:
15047:
15020:
15014:
15008:
15002:
15001:
14994:
14988:
14982:
14976:
14959:
14953:
14943:
14937:
14932:Köller, Jürgen,
14930:
14924:
14918:
14907:
14897:
14891:
14881:
14875:
14869:
14863:
14862:
14852:
14846:
14845:
14844:
14827:
14807:
14792:
14783:
14772:
14730:Tetrahedral kite
14677:3-fold symmetry
14675:
14668:
14667:
14607:distance-regular
14582:Chromatic number
14516:
14504:
14503:
14454:
14445:
14434:
14425:
14358:Royal Game of Ur
14265:white phosphorus
14245:
14243:
14242:
14226:
14224:
14223:
14220:
14217:
14200:
14199:
14198:
14184:
14183:
14182:
13959:
13945:
13931:
13862:
13855:
13848:
13841:
13834:
13827:
13820:
13813:
13806:
13799:
13792:
13785:
13770:Compact hyperb.
13753:
13746:
13739:
13721:
13720:
13654:
13647:
13640:
13633:
13626:
13619:
13612:
13605:
13598:
13591:
13584:
13577:
13547:
13540:
13533:
13515:
13514:
13510:hyperbolic plane
13502:Schläfli symbols
13452:
13445:
13438:
13431:
13424:
13417:
13410:
13403:
13347:
13346:
13345:
13341:
13340:
13336:
13335:
13331:
13330:
13326:
13325:
13318:
13317:
13316:
13312:
13311:
13307:
13306:
13302:
13301:
13297:
13296:
13289:
13288:
13287:
13283:
13282:
13278:
13277:
13273:
13272:
13268:
13267:
13260:
13259:
13258:
13254:
13253:
13249:
13248:
13244:
13243:
13239:
13238:
13231:
13230:
13229:
13225:
13224:
13220:
13219:
13215:
13214:
13210:
13209:
13202:
13201:
13200:
13196:
13195:
13191:
13190:
13186:
13185:
13181:
13180:
13173:
13172:
13171:
13167:
13166:
13162:
13161:
13157:
13156:
13152:
13151:
13144:
13143:
13142:
13138:
13137:
13133:
13132:
13128:
13127:
13123:
13122:
13113:
13106:
13099:
13092:
13085:
13078:
13071:
13064:
13036:
13035:
12991:
12978:
12971:
12964:
12957:
12950:
12941:Spherical tiling
12930:
12923:
12916:
12909:
12902:
12844:
12836:
12790:
12780:
12773:
12766:
12759:
12752:
12745:
12728:
12721:
12714:
12707:
12700:
12693:
12657:Square antiprism
12636:
12630:
12623:
12616:
12597:
12575:
12568:
12561:
12554:
12547:
12540:
12528:
12521:
12514:
12507:
12500:
12493:
12436:Regular pyramids
12432:
12431:
12413:
12412:
12409:
12403:
12402:
12396:
12395:
12340:
12338:
12337:
12332:
12330:
12326:
12322:
12318:
12317:
12308:
12307:
12292:
12291:
12282:
12281:
12265:
12264:
12255:
12254:
12239:
12238:
12229:
12228:
12212:
12211:
12202:
12201:
12186:
12185:
12176:
12175:
12155:
12147:
12135:
12132:
12129:
12123:
12120:
12116:
12112:
12111:
12106:
12102:
12101:
12100:
12088:
12087:
12068:
12067:
12062:
12058:
12057:
12056:
12044:
12043:
12024:
12023:
12018:
12014:
12013:
12012:
12000:
11999:
11966:
11963:
11960:
11953:
11952:
11916:
11907:
11898:
11889:
11880:
11865:
11863:
11862:
11857:
11852:
11850:
11718:
11717:
11659:
11657:
11656:
11651:
11646:
11645:
11636:
11628:
11620:
11619:
11610:
11602:
11594:
11593:
11584:
11576:
11568:
11567:
11558:
11550:
11530:
11528:
11527:
11522:
11520:
11518:
11517:
11516:
11504:
11503:
11491:
11490:
11478:
11477:
11467:
11459:
11395:
11393:
11392:
11387:
11382:
11380:
11379:
11367:
11362:
11359:
11354:
11342:
11337:
11334:
11329:
11317:
11312:
11309:
11304:
11292:
11287:
11284:
11279:
11267:
11226:
11224:
11223:
11218:
11082:
11080:
11079:
11074:
11060:
11059:
11058:
11036:
11035:
11034:
11012:
11011:
11010:
10988:
10987:
10986:
10898:
10896:
10895:
10890:
10885:
10884:
10866:
10865:
10856:
10855:
10843:
10842:
10824:
10823:
10814:
10813:
10801:
10800:
10782:
10781:
10772:
10771:
10752:
10747:
10734:
10729:
10716:
10711:
10698:
10693:
10673:
10671:
10670:
10665:
10663:
10662:
10653:
10652:
10636:
10634:
10633:
10628:
10626:
10625:
10606:
10604:
10603:
10598:
10596:
10595:
10579:
10577:
10576:
10571:
10569:
10568:
10552:
10550:
10549:
10544:
10542:
10541:
10525:
10523:
10522:
10517:
10515:
10514:
10498:
10496:
10495:
10490:
10488:
10487:
10471:
10469:
10468:
10463:
10461:
10460:
10410:
10408:
10407:
10402:
10235:
10183:tessellate space
10161:electromagnetism
10109:geometric median
10081:geometric median
10057:
10055:
10054:
10049:
10040:
10039:
10024:
10020:
10019:
9998:
9994:
9993:
9972:
9968:
9967:
9944:
9940:
9939:
9910:
9906:
9905:
9884:
9880:
9879:
9856:
9852:
9851:
9830:
9826:
9825:
9796:
9792:
9791:
9768:
9764:
9763:
9742:
9738:
9737:
9716:
9712:
9711:
9564:
9562:
9561:
9556:
9551:
9546:
9545:
9537:
9517:
9509:
9500:
9485:
9483:
9482:
9477:
9461:
9459:
9458:
9453:
9451:
9443:
9431:
9429:
9428:
9423:
9411:
9409:
9408:
9403:
9371:hyperbolic space
9359:
9357:
9356:
9351:
9349:
9010:
9008:
9007:
9002:
9000:
8993:
8982:
8968:
8957:
8942:
8931:
8917:
8906:
8887:
8885:
8884:
8879:
8877:
8875:
8742:
8741:
8726:
8724:
8723:
8718:
8706:
8704:
8703:
8698:
8686:
8684:
8683:
8678:
8666:
8664:
8663:
8658:
8646:
8644:
8643:
8638:
8626:
8624:
8623:
8618:
8606:
8604:
8603:
8598:
8586:
8584:
8583:
8578:
8566:
8564:
8563:
8558:
8546:
8544:
8543:
8538:
8526:
8524:
8523:
8518:
8506:
8504:
8503:
8498:
8476:
8474:
8473:
8468:
8466:
8464:
8456:
8455:
8443:
8435:
8434:
8422:
8414:
8413:
8401:
8390:
8379:
8359:
8357:
8352:
8341:
8326:
8324:
8323:
8318:
8316:
8309:
8308:
8296:
8295:
8283:
8282:
8256:
8255:
8243:
8242:
8230:
8229:
8203:
8202:
8190:
8189:
8177:
8176:
8146:
8144:
8143:
8138:
8136:
8118:
8117:
8108:
8107:
8095:
8094:
8085:
8084:
8072:
8071:
8062:
8061:
8049:
8048:
8039:
8038:
8029:
8028:
8016:
8015:
8000:
7998:
7997:
7992:
7980:
7978:
7977:
7972:
7960:
7958:
7957:
7952:
7940:
7938:
7937:
7932:
7920:
7918:
7917:
7912:
7900:
7898:
7897:
7892:
7880:
7878:
7877:
7872:
7837:
7817:
7803:
7801:
7800:
7795:
7793:
7792:
7779:
7774:
7762:
7757:
7745:
7740:
7721:
7716:
7699:
7694:
7682:
7677:
7658:
7653:
7641:
7636:
7619:
7614:
7595:
7590:
7578:
7573:
7561:
7556:
7499:
7498:
7466:
7464:
7463:
7458:
7452:
7448:
7447:
7443:
7417:
7416:
7405:
7404:
7393:
7392:
7381:
7380:
7367:
7366:
7355:
7354:
7343:
7342:
7331:
7330:
7317:
7316:
7305:
7304:
7293:
7292:
7281:
7280:
7195:to the vertices
7167:
7165:
7164:
7159:
7153:
7151:
7143:
7142:
7130:
7122:
7121:
7109:
7101:
7100:
7088:
7077:
7066:
7046:
7044:
7039:
7028:
7010:
7008:
7007:
7002:
7000:
6999:
6992:
6966:
6964:
6963:
6958:
6956:
6955:
6948:
6922:
6920:
6919:
6914:
6912:
6911:
6904:
6875:
6873:
6872:
6867:
6865:
6864:
6855:
6835:
6827:
6815:
6795:
6787:
6775:
6755:
6747:
6728:
6724:
6721:
6719:
6718:
6713:
6711:
6710:
6703:
6702:
6701:
6689:
6681:
6674:
6666:
6657:
6649:
6642:
6641:
6640:
6628:
6620:
6611:
6603:
6596:
6588:
6581:
6580:
6579:
6558:
6557:
6528:
6525:
6523:
6522:
6517:
6515:
6514:
6502:
6501:
6489:
6488:
6476:
6475:
6460:
6445:
6444:
6432:
6431:
6419:
6418:
6403:
6388:
6387:
6375:
6374:
6362:
6361:
6346:
6327:
6323:
6320:
6318:
6317:
6312:
6310:
6309:
6302:
6293:
6284:
6253:
6249:
6246:
6244:
6243:
6238:
6236:
6235:
6228:
6221:
6214:
6176:
6174:
6173:
6170:
6167:
6152:
6150:
6149:
6144:
6139:
6131:
6123:
6115:
6107:
6099:
6088:
6080:
6068:
6066:
6065:
6060:
6058:
6048:
6047:
6035:
6034:
6022:
6021:
6002:
5987:
5986:
5974:
5973:
5961:
5960:
5941:
5926:
5925:
5913:
5912:
5900:
5899:
5880:
5865:
5864:
5852:
5851:
5839:
5838:
5819:
5799:
5797:
5796:
5791:
5775:
5773:
5772:
5767:
5755:
5753:
5752:
5747:
5736:
5728:
5700:
5698:
5697:
5692:
5690:
5685:
5677:
5662:similarity class
5589:
5583:
5533:Klein four-group
5513:
5507:
5468:
5467:
5466:
5462:
5461:
5457:
5456:
5452:
5451:
5447:
5446:
5419:Klein four-group
5406:
5362:
5361:
5360:
5356:
5355:
5351:
5350:
5346:
5345:
5341:
5340:
5316:
5266:
5190:
5117:
5073:
5072:
5071:
5067:
5066:
5062:
5061:
5057:
5056:
5052:
5051:
5015:
4961:
4960:
4920:generating point
4826:
4824:
4823:
4818:
4816:
4806:
4804:
4796:
4793:
4791:
4790:
4785:
4783:
4773:
4771:
4763:
4760:
4758:
4757:
4752:
4750:
4740:
4738:
4730:
4727:
4725:
4724:
4719:
4717:
4707:
4705:
4697:
4694:
4692:
4691:
4686:
4684:
4674:
4672:
4664:
4661:
4659:
4658:
4653:
4651:
4641:
4639:
4631:
4628:
4626:
4625:
4620:
4618:
4608:
4606:
4598:
4595:
4593:
4592:
4587:
4585:
4575:
4573:
4565:
4562:
4560:
4559:
4554:
4543:
4536:
4534:
4533:
4528:
4526:
4516:
4514:
4506:
4503:
4501:
4500:
4495:
4493:
4483:
4481:
4473:
4470:
4468:
4467:
4462:
4451:
4444:
4442:
4441:
4436:
4434:
4424:
4422:
4414:
4411:
4409:
4408:
4403:
4401:
4391:
4389:
4381:
4378:
4376:
4375:
4370:
4368:
4358:
4356:
4348:
4341:
4339:
4338:
4333:
4331:
4321:
4319:
4311:
4308:
4306:
4305:
4300:
4289:
4286:
4284:
4283:
4278:
4276:
4266:
4264:
4256:
4244:
4242:
4241:
4236:
4234:
4228:
4224:
4221:
4218:
4209:
4204:
4198:
4196:
4195:
4190:
4188:
4185:
4177:
4151:
4149:
4148:
4143:
4135:
4125:
4123:
4115:
4111:
4109:
4108:
4103:
4098:
4090:
4089:
4075:
4061:
4059:
4058:
4053:
4045:
4035:
4033:
4025:
4021:
4019:
4018:
4013:
4008:
4000:
3999:
3985:
3971:
3969:
3968:
3963:
3955:
3945:
3943:
3935:
3931:
3929:
3928:
3923:
3918:
3910:
3909:
3895:
3877:
3875:
3874:
3869:
3867:
3858:
3849:
3844:
3840:
3838:
3837:
3832:
3830:
3827:
3822:
3813:
3804:
3799:
3795:
3793:
3792:
3787:
3779:
3769:
3767:
3759:
3750:
3748:
3747:
3742:
3740:
3731:
3722:
3717:
3713:
3711:
3710:
3705:
3703:
3700:
3695:
3686:
3677:
3672:
3668:
3666:
3665:
3660:
3649:
3640:
3638:
3637:
3632:
3630:
3621:
3612:
3607:
3603:
3601:
3600:
3595:
3593:
3590:
3579:
3574:
3570:
3568:
3567:
3562:
3554:
3544:
3542:
3534:
3513:
3511:
3510:
3505:
3503:
3500:
3483:
3478:
3474:
3472:
3471:
3466:
3464:
3461:
3444:
3439:
3435:
3433:
3432:
3427:
3416:
3392:
3391:
3388:
3387:
3386:
3382:
3381:
3377:
3376:
3372:
3371:
3367:
3366:
3360:
3359:
3358:
3354:
3353:
3349:
3348:
3344:
3343:
3339:
3338:
3317:
3316:
3310:
3309:
3308:
3304:
3303:
3299:
3298:
3294:
3293:
3289:
3288:
3282:
3281:
3280:
3276:
3275:
3271:
3270:
3266:
3265:
3261:
3260:
3254:
3253:
3247:
3246:
3212:is an irregular
3104:inscribed sphere
3053:
3044:
3037:
2974:Helical stacking
2955:
2948:
2941:
2940:
2928:spherical tiling
2922:Spherical tiling
2849:
2842:
2821:
2815:
2802:has two special
2772:
2758:
2751:
2749:
2748:
2745:
2742:
2724:
2706:
2704:
2703:
2698:
2696:
2695:
2676:
2674:
2673:
2668:
2666:
2650:
2648:
2647:
2642:
2640:
2639:
2616:
2614:
2613:
2608:
2606:
2605:
2604:
2598:
2539:
2537:
2536:
2531:
2514:
2499:
2498:
2497:
2493:
2492:
2488:
2487:
2483:
2482:
2478:
2477:
2456:
2454:
2453:
2448:
2446:
2441:
2429:
2427:
2426:
2421:
2419:
2265:
2263:
2262:
2257:
2255:
2250:
2249:
2244:
2238:
2229:
2227:
2226:
2221:
2219:
2186:
2182:
2181:
2173:
2165:
2156:
2155:
2147:
2138:
2137:
2120:
2116:
2115:
2107:
2099:
2090:
2089:
2084:
2075:
2074:
2058:
2054:
2053:
2045:
2031:
2022:
2021:
2003:
2001:
2000:
1995:
1974:
1972:
1971:
1966:
1964:
1960:
1959:
1953:
1949:
1926:
1922:
1918:
1914:
1913:
1907:
1903:
1847:
1845:
1844:
1839:
1837:
1827:
1823:
1815:
1794:
1790:
1786:
1778:
1760:
1752:
1743:
1739:
1731:
1668:
1667:
1661:
1660:
1648:
1646:
1645:
1640:
1638:
1631:
1630:
1625:
1621:
1619:
1614:
1601:
1596:
1583:
1578:
1565:
1560:
1548:
1547:
1525:
1521:
1519:
1514:
1501:
1496:
1483:
1478:
1465:
1460:
1448:
1447:
1423:
1422:
1417:
1413:
1412:
1407:
1406:
1405:
1392:
1387:
1382:
1380:
1375:
1362:
1357:
1344:
1339:
1326:
1321:
1311:
1296:
1291:
1290:
1289:
1276:
1271:
1266:
1264:
1259:
1246:
1241:
1228:
1223:
1210:
1205:
1195:
1182:
1180:
1179:
1174:
1172:
1171:
1156:, and distances
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1115:
1113:
1112:
1107:
1105:
1098:
1092:
1088:
1083:
1082:
1081:
1065:
1059:
1055:
1050:
1042:
1037:
1036:
1035:
1018:
1012:
1008:
1000:
992:
975:
966:
965:
946:
944:
943:
938:
936:
935:
934:
914:
912:
911:
906:
904:
903:
902:
882:
880:
879:
874:
859:
857:
856:
851:
826:
824:
823:
818:
816:
809:
808:
796:
792:
787:
765:
761:
760:
752:
729:
728:
716:
712:
711:
706:
683:
679:
671:
641:
639:
638:
633:
628:
627:
612:
610:
609:
604:
598:
597:
588:
580:
571:
570:
565:
561:
560:
559:
550:
541:
540:
530:
522:
507:
505:
504:
499:
494:
485:
484:
475:
473:
472:
467:
462:
461:
446:
441:
439:
438:
426:
422:
421:
420:
411:
402:
401:
375:
373:
372:
367:
355:
353:
352:
347:
330:
316:Hill tetrahedron
266:stella octangula
239:regular polygons
200:
188:stella octangula
183:
171:
134:convex polyhedra
94:convex polyhedra
63:
21:
17732:
17731:
17727:
17726:
17725:
17723:
17722:
17721:
17692:Platonic solids
17677:
17676:
17645:
17638:
17631:
17514:
17507:
17500:
17464:
17457:
17450:
17414:
17407:
17241:Regular polygon
17234:
17225:
17218:
17214:
17207:
17203:
17194:
17185:
17178:
17174:
17162:
17156:
17152:
17140:
17122:
17111:
17082:
17077:
17066:
17005:Dihedral others
16996:
16975:
16950:
16925:
16854:
16851:
16850:
16841:
16770:
16759:
16758:
16749:
16712:
16710:Platonic solids
16704:
16698:
16668:
16663:
16614:
16603:Star polyhedron
16585:
16550:
16498:
16475:Hexecontahedron
16457:Triacontahedron
16439:
16430:Enneadecahedron
16420:Heptadecahedron
16410:Pentadecahedron
16405:Tetradecahedron
16378:
16317:
16310:
16305:
16237:
16232:
16226:
16196:
16114:
16101:
16064:10.2307/3608204
16043:
16008:Coxeter, H.S.M.
15973:
15952:
15947:
15946:
15925:
15921:
15891:
15887:
15877:
15875:
15870:
15869:
15865:
15847:
15843:
15836:
15811:
15807:
15775:
15771:
15761:
15759:
15755:
15744:
15738:
15734:
15724:
15722:
15721:on 10 June 2007
15718:
15703:
15697:
15693:
15683:
15681:
15673:
15672:
15668:
15637:
15633:
15626:
15610:
15606:
15592:
15588:
15572:
15568:
15567:
15563:
15541:
15535:
15532:
15528:
15514:
15510:
15500:
15498:
15487:
15483:
15468:
15457:
15449:
15445:
15414:
15410:
15395:10.2307/2689983
15374:
15370:
15355:
15351:
15346:. Bulletin AMQ.
15343:
15337:
15333:
15317:
15316:
15311:
15303:
15299:
15294:
15290:
15275:10.2307/3647851
15246:
15240:
15236:
15228:
15224:
15216:
15212:
15204:
15197:
15193:, MathPages.com
15189:
15185:
15173:
15169:
15157:
15153:
15116:
15112:
15100:
15096:
15088:
15084:
15079:
15072:
15068:
15060:
15053:
15024:Coxeter, H.S.M.
15021:
15017:
15009:
15005:
14996:
14995:
14991:
14983:
14979:
14975:
14960:
14956:
14944:
14940:
14931:
14927:
14919:
14910:
14898:
14894:
14882:
14878:
14870:
14866:
14853:
14849:
14828:
14821:
14816:
14811:
14810:
14793:
14786:
14776:Plateau borders
14773:
14769:
14764:
14686:
14676:
14659:Platonic graphs
14656:
14648:
14502:
14467:Stanley Kubrick
14464:
14463:
14462:
14461:
14457:
14456:
14455:
14447:
14446:
14437:
14436:
14435:
14427:
14426:
14415:
14413:Popular culture
14399:
14393:
14373:polyhedral dice
14344:
14336:
14330:
14291:
14283:Main articles:
14281:
14241:
14238:
14237:
14236:
14234:
14221:
14218:
14215:
14214:
14212:
14197:
14194:
14193:
14192:
14190:
14181:
14178:
14177:
14176:
14174:
14161:molecules. All
14155:
14126:
14114:
14102:
14034:
14029:
13965:
13960:
13951:
13946:
13937:
13932:
13757:
13562:Compact hyper.
13551:
13343:
13338:
13333:
13328:
13323:
13321:
13314:
13309:
13304:
13299:
13294:
13292:
13285:
13280:
13275:
13270:
13265:
13263:
13256:
13251:
13246:
13241:
13236:
13234:
13227:
13222:
13217:
13212:
13207:
13205:
13198:
13193:
13188:
13183:
13178:
13176:
13169:
13164:
13159:
13154:
13149:
13147:
13140:
13135:
13130:
13125:
13120:
13118:
12854:
12655:
12648:
12639:Antiprism name
12634:
12423:
12410:
12407:
12405:
12400:
12398:
12393:
12391:
12380:
12374:
12366:
12354:arithmetic mean
12350:
12328:
12327:
12320:
12319:
12313:
12309:
12303:
12299:
12287:
12283:
12277:
12273:
12267:
12266:
12260:
12256:
12250:
12246:
12234:
12230:
12224:
12220:
12214:
12213:
12207:
12203:
12197:
12193:
12181:
12177:
12171:
12167:
12160:
12156:
12146:
12138:
12133:
12128:
12126:
12121:
12114:
12113:
12107:
12096:
12092:
12083:
12079:
12078:
12074:
12073:
12070:
12069:
12063:
12052:
12048:
12039:
12035:
12034:
12030:
12029:
12026:
12025:
12019:
12008:
12004:
11995:
11991:
11990:
11986:
11985:
11981:
11977:
11969:
11964:
11959:
11957:
11945:
11941:
11934:
11927:
11925:
11922:
11921:
11915:
11909:
11906:
11900:
11897:
11891:
11888:
11882:
11876:
11872:
11843:
11716:
11708:
11705:
11704:
11666:
11641:
11637:
11627:
11615:
11611:
11601:
11589:
11585:
11575:
11563:
11559:
11549:
11541:
11538:
11537:
11512:
11508:
11499:
11495:
11486:
11482:
11473:
11469:
11468:
11460:
11458:
11450:
11447:
11446:
11437:
11428:
11419:
11410:
11375:
11371:
11366:
11355:
11350:
11341:
11330:
11325:
11316:
11305:
11300:
11291:
11280:
11275:
11266:
11264:
11261:
11260:
11251:
11233:
11134:
11131:
11130:
11054:
11053:
11049:
11030:
11029:
11025:
11006:
11005:
11001:
10982:
10981:
10977:
10966:
10963:
10962:
10957:
10950:
10943:
10936:
10904:
10877:
10873:
10861:
10857:
10851:
10847:
10835:
10831:
10819:
10815:
10809:
10805:
10793:
10789:
10777:
10773:
10767:
10763:
10748:
10743:
10730:
10725:
10712:
10707:
10694:
10689:
10683:
10680:
10679:
10658:
10654:
10648:
10644:
10642:
10639:
10638:
10618:
10614:
10612:
10609:
10608:
10591:
10587:
10585:
10582:
10581:
10564:
10560:
10558:
10555:
10554:
10537:
10533:
10531:
10528:
10527:
10510:
10506:
10504:
10501:
10500:
10483:
10479:
10477:
10474:
10473:
10456:
10452:
10450:
10447:
10446:
10443:
10437:
10272:
10269:
10268:
10242:
10229:
10165:Thomson problem
10153:
10085:Lorenz Lindelöf
10066:
10034:
10033:
10025:
10015:
10011:
10007:
9999:
9989:
9985:
9981:
9973:
9963:
9959:
9955:
9946:
9945:
9935:
9931:
9927:
9919:
9911:
9901:
9897:
9893:
9885:
9875:
9871:
9867:
9858:
9857:
9847:
9843:
9839:
9831:
9821:
9817:
9813:
9805:
9797:
9787:
9783:
9779:
9770:
9769:
9759:
9755:
9751:
9743:
9733:
9729:
9725:
9717:
9707:
9703:
9699:
9691:
9678:
9677:
9675:
9672:
9671:
9630:of a triangle.
9570:
9541:
9521:
9513:
9505:
9501:
9499:
9491:
9488:
9487:
9471:
9468:
9467:
9447:
9439:
9437:
9434:
9433:
9417:
9414:
9413:
9397:
9394:
9393:
9379:dihedral angles
9347:
9346:
9291:
9237:
9236:
9182:
9128:
9127:
9073:
9018:
9016:
9013:
9012:
8998:
8997:
8981:
8972:
8956:
8947:
8946:
8930:
8921:
8905:
8895:
8893:
8890:
8889:
8859:
8740:
8732:
8729:
8728:
8712:
8709:
8708:
8692:
8689:
8688:
8672:
8669:
8668:
8652:
8649:
8648:
8632:
8629:
8628:
8612:
8609:
8608:
8592:
8589:
8588:
8572:
8569:
8568:
8552:
8549:
8548:
8532:
8529:
8528:
8512:
8509:
8508:
8492:
8489:
8488:
8460:
8451:
8447:
8439:
8430:
8426:
8418:
8409:
8405:
8397:
8386:
8375:
8358:
8342:
8340:
8332:
8329:
8328:
8314:
8313:
8304:
8300:
8291:
8287:
8278:
8274:
8267:
8261:
8260:
8251:
8247:
8238:
8234:
8225:
8221:
8214:
8208:
8207:
8198:
8194:
8185:
8181:
8172:
8168:
8161:
8154:
8152:
8149:
8148:
8113:
8109:
8103:
8099:
8090:
8086:
8080:
8076:
8067:
8063:
8057:
8053:
8044:
8040:
8034:
8030:
8024:
8020:
8014:
8006:
8003:
8002:
7986:
7983:
7982:
7966:
7963:
7962:
7946:
7943:
7942:
7926:
7923:
7922:
7906:
7903:
7902:
7886:
7883:
7882:
7866:
7863:
7862:
7856:Heron's formula
7843:
7819:
7808:
7787:
7786:
7781:
7775:
7770:
7764:
7758:
7753:
7747:
7741:
7736:
7730:
7724:
7723:
7717:
7712:
7706:
7701:
7695:
7690:
7684:
7678:
7673:
7667:
7661:
7660:
7654:
7649:
7643:
7637:
7632:
7626:
7621:
7615:
7610:
7604:
7598:
7597:
7591:
7586:
7580:
7574:
7569:
7563:
7557:
7552:
7546:
7541:
7535:
7534:
7529:
7524:
7519:
7514:
7504:
7503:
7494:
7490:
7482:
7479:
7478:
7441:
7440:
7435:
7430:
7425:
7419:
7418:
7412:
7408:
7406:
7400:
7396:
7394:
7388:
7384:
7382:
7376:
7372:
7369:
7368:
7362:
7358:
7356:
7350:
7346:
7344:
7338:
7334:
7332:
7326:
7322:
7319:
7318:
7312:
7308:
7306:
7300:
7296:
7294:
7288:
7284:
7282:
7276:
7272:
7268:
7264:
7260:
7256:
7242:
7239:
7238:
7147:
7138:
7134:
7126:
7117:
7113:
7105:
7096:
7092:
7084:
7073:
7062:
7045:
7029:
7027:
7019:
7016:
7015:
6994:
6993:
6988:
6981:
6980:
6972:
6969:
6968:
6950:
6949:
6944:
6937:
6936:
6928:
6925:
6924:
6906:
6905:
6900:
6893:
6892:
6884:
6881:
6880:
6860:
6859:
6851:
6831:
6823:
6820:
6819:
6811:
6791:
6783:
6780:
6779:
6771:
6751:
6743:
6736:
6735:
6733:
6730:
6729:
6726:
6722:
6705:
6704:
6697:
6693:
6692:
6690:
6685:
6677:
6675:
6670:
6662:
6659:
6658:
6653:
6645:
6643:
6636:
6632:
6631:
6629:
6624:
6616:
6613:
6612:
6607:
6599:
6597:
6592:
6584:
6582:
6575:
6571:
6570:
6563:
6562:
6553:
6549:
6541:
6538:
6537:
6526:
6510:
6509:
6497:
6493:
6484:
6480:
6471:
6467:
6456:
6453:
6452:
6440:
6436:
6427:
6423:
6414:
6410:
6399:
6396:
6395:
6383:
6379:
6370:
6366:
6357:
6353:
6342:
6335:
6334:
6332:
6329:
6328:
6325:
6321:
6304:
6303:
6298:
6295:
6294:
6289:
6286:
6285:
6280:
6273:
6272:
6258:
6255:
6254:
6251:
6247:
6230:
6229:
6224:
6222:
6217:
6215:
6210:
6203:
6202:
6188:
6185:
6184:
6171:
6168:
6165:
6164:
6162:
6135:
6127:
6119:
6111:
6103:
6095:
6079:
6077:
6074:
6073:
6056:
6055:
6043:
6039:
6030:
6026:
6017:
6013:
6003:
5998:
5995:
5994:
5982:
5978:
5969:
5965:
5956:
5952:
5942:
5937:
5934:
5933:
5921:
5917:
5908:
5904:
5895:
5891:
5881:
5876:
5873:
5872:
5860:
5856:
5847:
5843:
5834:
5830:
5820:
5815:
5811:
5809:
5806:
5805:
5785:
5782:
5781:
5761:
5758:
5757:
5727:
5719:
5716:
5715:
5712:
5707:
5681:
5676:
5674:
5671:
5670:
5650:
5633:
5621:
5610:
5602:
5584:
5570:
5565:
5561:
5558:
5552:
5551:
5540:
5530:
5522:
5508:
5480:
5464:
5459:
5454:
5449:
5444:
5442:
5440:
5433:
5426:
5415:
5391:
5386:
5382:
5379:
5375:
5374:
5358:
5353:
5348:
5343:
5338:
5336:
5334:
5325:
5286:
5271:
5258:
5244:
5237:
5236:
5229:
5228:
5217:
5206:
5199:
5177:
5172:
5168:
5165:
5161:
5160:
5149:
5141:symmetric group
5138:
5130:
5104:
5099:
5095:
5089:
5088:
5076:Schläfli symbol
5069:
5064:
5059:
5054:
5049:
5047:
5045:Coxeter diagram
5042:
5034:symmetric group
5031:
5024:
4970:
4968:
4957:
4953:
4944:
4877:
4846:Hill tetrahedra
4834:
4803:
4801:
4798:
4797:
4770:
4768:
4765:
4764:
4737:
4735:
4732:
4731:
4704:
4702:
4699:
4698:
4671:
4669:
4666:
4665:
4638:
4636:
4633:
4632:
4605:
4603:
4600:
4599:
4572:
4570:
4567:
4566:
4548:
4545:
4544:
4513:
4511:
4508:
4507:
4480:
4478:
4475:
4474:
4456:
4453:
4452:
4421:
4419:
4416:
4415:
4388:
4386:
4383:
4382:
4355:
4353:
4350:
4349:
4318:
4316:
4313:
4312:
4294:
4291:
4290:
4263:
4261:
4258:
4257:
4220:
4219:
4216:
4214:
4211:
4210:
4184:
4182:
4179:
4178:
4122:
4120:
4117:
4116:
4094:
4085:
4082:
4080:
4077:
4076:
4032:
4030:
4027:
4026:
4004:
3995:
3992:
3990:
3987:
3986:
3942:
3940:
3937:
3936:
3914:
3905:
3902:
3900:
3897:
3896:
3856:
3854:
3851:
3850:
3826:
3811:
3809:
3806:
3805:
3766:
3764:
3761:
3760:
3729:
3727:
3724:
3723:
3699:
3684:
3682:
3679:
3678:
3654:
3651:
3650:
3619:
3617:
3614:
3613:
3589:
3584:
3581:
3580:
3541:
3539:
3536:
3535:
3499:
3488:
3485:
3484:
3460:
3449:
3446:
3445:
3421:
3418:
3417:
3384:
3379:
3374:
3369:
3364:
3362:
3356:
3351:
3346:
3341:
3336:
3334:
3314:
3312:
3306:
3301:
3296:
3291:
3286:
3284:
3278:
3273:
3268:
3263:
3258:
3256:
3251:
3249:
3244:
3242:
3225:name for it is
3199:right triangles
3183:
3166:dihedral angles
3158:
3152:
3118:
3112:
3054:
3045:
3035:
3007:four dimensions
2976:
2924:
2869:
2855:
2809:
2796:
2760:
2756:
2746:
2743:
2729:
2728:
2726:
2722:
2691:
2687:
2685:
2682:
2681:
2662:
2660:
2657:
2656:
2635:
2631:
2629:
2626:
2625:
2623:symmetric group
2600:
2599:
2594:
2593:
2591:
2588:
2587:
2566:point inversion
2546:
2510:
2508:
2505:
2504:
2502:Schläfli symbol
2495:
2490:
2485:
2480:
2475:
2473:
2471:Coxeter diagram
2440:
2435:
2432:
2431:
2417:
2416:
2381:
2348:
2347:
2312:
2284:
2282:
2279:
2278:
2243:
2239:
2237:
2235:
2232:
2231:
2217:
2216:
2190:
2172:
2154:
2136:
2132:
2128:
2125:
2124:
2106:
2088:
2073:
2069:
2065:
2062:
2044:
2020:
2019:
2015:
2011:
2009:
2006:
2005:
1986:
1983:
1982:
1948:
1932:
1928:
1920:
1902:
1883:
1879:
1877:
1874:
1873:
1866:
1835:
1834:
1814:
1810:
1792:
1791:
1777:
1773:
1751:
1744:
1730:
1726:
1716:
1714:
1711:
1710:
1665:
1663:
1658:
1656:
1636:
1635:
1626:
1615:
1610:
1597:
1592:
1579:
1574:
1561:
1556:
1543:
1539:
1538:
1534:
1533:
1526:
1515:
1510:
1497:
1492:
1479:
1474:
1461:
1456:
1443:
1439:
1438:
1434:
1428:
1427:
1418:
1401:
1397:
1393:
1391:
1376:
1371:
1358:
1353:
1340:
1335:
1322:
1317:
1312:
1310:
1309:
1305:
1304:
1297:
1285:
1281:
1277:
1275:
1260:
1255:
1242:
1237:
1224:
1219:
1206:
1201:
1196:
1194:
1190:
1188:
1185:
1184:
1167:
1163:
1161:
1158:
1157:
1141:
1138:
1137:
1121:
1118:
1117:
1103:
1102:
1087:
1077:
1076:
1072:
1069:
1054:
1041:
1031:
1030:
1026:
1023:
1022:
1007:
991:
982:
964:
954:
952:
949:
948:
930:
929:
925:
923:
920:
919:
898:
897:
893:
891:
888:
887:
868:
865:
864:
845:
842:
841:
814:
813:
804:
800:
786:
782:
766:
751:
747:
743:
734:
733:
724:
720:
705:
701:
697:
684:
670:
666:
656:
654:
651:
650:
623:
619:
603:
599:
593:
589:
587:
569:
555:
551:
539:
538:
534:
521:
513:
510:
509:
483:
481:
478:
477:
457:
453:
440:
434:
430:
416:
412:
400:
399:
395:
381:
378:
377:
361:
358:
357:
341:
338:
337:
324:
235:Platonic solids
212:
211:
210:
209:
208:
201:
192:
191:
190:
184:
176:
175:
172:
161:
84:, six straight
48:
28:
23:
22:
15:
12:
11:
5:
17730:
17720:
17719:
17714:
17709:
17704:
17699:
17694:
17689:
17673:
17672:
17657:
17656:
17647:
17643:
17636:
17629:
17625:
17616:
17599:
17590:
17579:
17578:
17576:
17574:
17569:
17560:
17555:
17549:
17548:
17546:
17544:
17539:
17530:
17525:
17519:
17518:
17516:
17512:
17505:
17498:
17494:
17489:
17480:
17475:
17469:
17468:
17466:
17462:
17455:
17448:
17444:
17439:
17430:
17425:
17419:
17418:
17416:
17412:
17405:
17401:
17396:
17387:
17382:
17376:
17375:
17373:
17371:
17366:
17357:
17352:
17346:
17345:
17336:
17331:
17326:
17317:
17312:
17306:
17305:
17296:
17294:
17289:
17280:
17275:
17269:
17268:
17263:
17258:
17253:
17248:
17243:
17237:
17236:
17232:
17228:
17223:
17212:
17201:
17192:
17183:
17176:
17170:
17160:
17154:
17148:
17142:
17136:
17130:
17124:
17123:
17112:
17110:
17109:
17102:
17095:
17087:
17079:
17078:
17071:
17068:
17067:
17065:
17064:
17059:
17054:
17049:
17044:
17039:
17034:
17029:
17024:
17019:
17014:
17008:
17006:
17002:
17001:
16998:
16997:
16995:
16994:
16989:
16983:
16981:
16977:
16976:
16974:
16973:
16968:
16962:
16956:
16952:
16951:
16949:
16948:
16941:
16933:
16931:
16927:
16926:
16924:
16923:
16918:
16913:
16908:
16903:
16898:
16893:
16888:
16883:
16878:
16873:
16868:
16863:
16857:
16855:
16848:Catalan solids
16846:
16843:
16842:
16840:
16839:
16834:
16829:
16824:
16819:
16814:
16809:
16804:
16799:
16794:
16789:
16787:truncated cube
16784:
16779:
16773:
16771:
16754:
16751:
16750:
16748:
16747:
16742:
16737:
16732:
16727:
16721:
16719:
16706:
16705:
16697:
16696:
16689:
16682:
16674:
16665:
16664:
16662:
16661:
16659:parallelepiped
16656:
16651:
16646:
16641:
16636:
16631:
16625:
16623:
16616:
16615:
16613:
16612:
16606:
16600:
16593:
16591:
16587:
16586:
16584:
16583:
16577:
16571:
16565:
16562:Platonic solid
16558:
16556:
16552:
16551:
16549:
16548:
16547:
16546:
16540:
16534:
16522:
16517:
16512:
16506:
16504:
16500:
16499:
16497:
16496:
16490:
16484:
16478:
16472:
16466:
16460:
16454:
16447:
16445:
16441:
16440:
16438:
16437:
16432:
16427:
16425:Octadecahedron
16422:
16417:
16415:Hexadecahedron
16412:
16407:
16402:
16397:
16392:
16386:
16384:
16380:
16379:
16377:
16376:
16371:
16366:
16361:
16356:
16351:
16346:
16341:
16336:
16331:
16325:
16323:
16319:
16318:
16315:
16312:
16311:
16304:
16303:
16296:
16289:
16281:
16275:
16274:
16262:
16257:
16236:
16235:External links
16233:
16231:
16230:
16224:
16211:
16189:
16161:(2): 379–400.
16146:
16133:
16119:
16105:
16099:
16084:
16047:
16041:
16022:
16004:
15990:
15977:
15971:
15953:
15951:
15948:
15945:
15944:
15919:
15885:
15863:
15841:
15834:
15814:Holmes, Arthur
15805:
15769:
15758:on 7 June 2011
15732:
15691:
15666:
15631:
15624:
15604:
15586:
15561:
15539:
15526:
15508:
15481:
15455:
15443:
15408:
15368:
15349:
15331:
15297:
15288:
15259:(8): 679–693.
15234:
15222:
15210:
15195:
15183:
15167:
15151:
15110:
15094:
15082:
15077:
15066:
15051:
15038:(1–3): 59–71.
15015:
15013:, p. 181.
15003:
14989:
14977:
14974:
14973:
14968:
14961:
14954:
14938:
14925:
14908:
14892:
14884:Shavinina 2013
14876:
14864:
14847:
14818:
14817:
14815:
14812:
14809:
14808:
14784:
14766:
14765:
14763:
14760:
14759:
14758:
14753:
14748:
14742:
14737:
14732:
14727:
14722:
14716:
14702:
14697:
14692:
14685:
14682:
14679:
14678:
14663:Platonic solid
14654:
14646:
14643:complete graph
14629:
14628:
14622:
14621:
14592:
14588:
14587:
14584:
14578:
14577:
14574:
14568:
14567:
14564:
14558:
14557:
14554:
14548:
14547:
14544:
14538:
14537:
14534:
14528:
14527:
14524:
14518:
14517:
14509:
14508:
14501:
14498:
14459:
14458:
14449:
14448:
14440:
14439:
14438:
14429:
14428:
14420:
14419:
14418:
14417:
14416:
14414:
14411:
14395:Main article:
14392:
14389:
14363:Especially in
14343:
14340:
14332:Main article:
14329:
14326:
14280:
14277:
14251:phase diagrams
14239:
14195:
14179:
14151:Main article:
14125:
14122:
14113:
14110:
14101:
14098:
14054:polygonal mesh
14033:
14030:
14028:
14025:
13967:
13966:
13961:
13954:
13952:
13947:
13940:
13938:
13933:
13926:
13924:
13918:
13917:
13914:
13911:
13908:
13905:
13900:
13895:
13890:
13885:
13880:
13875:
13870:
13864:
13863:
13856:
13849:
13842:
13835:
13828:
13821:
13814:
13807:
13800:
13793:
13786:
13778:
13777:
13774:
13771:
13768:
13763:
13759:
13758:
13756:
13755:
13748:
13741:
13733:
13716:vertex figures
13710:
13709:
13706:
13703:
13700:
13697:
13692:
13687:
13682:
13677:
13672:
13667:
13662:
13656:
13655:
13648:
13641:
13634:
13627:
13620:
13613:
13606:
13599:
13592:
13585:
13578:
13570:
13569:
13566:
13563:
13560:
13557:
13553:
13552:
13550:
13549:
13542:
13535:
13527:
13496:
13495:
13490:
13485:
13480:
13475:
13470:
13465:
13460:
13454:
13453:
13446:
13439:
13432:
13425:
13418:
13411:
13404:
13396:
13395:
13391:
13390:
13385:
13380:
13375:
13370:
13365:
13360:
13355:
13349:
13348:
13319:
13290:
13261:
13232:
13203:
13174:
13145:
13115:
13114:
13107:
13100:
13093:
13086:
13079:
13072:
13065:
13057:
13056:
13053:
13043:
13042:
13021:
13020:
13017:
13014:
13011:
13008:
13005:
13002:
12999:
12993:
12992:
12985:
12979:
12972:
12965:
12958:
12951:
12944:
12937:
12936:
12934:
12931:
12924:
12917:
12910:
12903:
12896:
12889:
12888:
12883:
12880:
12875:
12870:
12865:
12860:
12851:
12823:
12822:
12819:
12816:
12813:
12810:
12807:
12804:
12801:
12798:
12796:Vertex config.
12792:
12791:
12784:
12781:
12774:
12767:
12760:
12753:
12746:
12739:
12735:
12734:
12732:
12729:
12722:
12715:
12708:
12701:
12694:
12687:
12683:
12682:
12677:
12674:
12669:
12664:
12659:
12652:
12645:
12640:
12633:
12632:
12625:
12618:
12610:
12580:
12579:
12576:
12569:
12562:
12555:
12548:
12541:
12533:
12532:
12529:
12522:
12515:
12508:
12501:
12494:
12486:
12485:
12482:
12479:
12476:
12472:
12471:
12468:
12465:
12460:
12455:
12450:
12445:
12439:
12438:
12422:
12419:
12390:with areas of
12376:Main article:
12373:
12370:
12365:
12362:
12349:
12346:
12342:
12341:
12325:
12316:
12312:
12306:
12302:
12298:
12295:
12290:
12286:
12280:
12276:
12272:
12269:
12268:
12263:
12259:
12253:
12249:
12245:
12242:
12237:
12233:
12227:
12223:
12219:
12216:
12215:
12210:
12206:
12200:
12196:
12192:
12189:
12184:
12180:
12174:
12170:
12166:
12163:
12162:
12159:
12153:
12150:
12145:
12142:
12139:
12134:
12127:
12122:
12119:
12110:
12105:
12099:
12095:
12091:
12086:
12082:
12077:
12072:
12071:
12066:
12061:
12055:
12051:
12047:
12042:
12038:
12033:
12028:
12027:
12022:
12017:
12011:
12007:
12003:
11998:
11994:
11989:
11984:
11983:
11980:
11976:
11973:
11970:
11965:
11958:
11956:
11951:
11948:
11944:
11940:
11937:
11935:
11933:
11930:
11929:
11913:
11904:
11895:
11886:
11871:
11868:
11867:
11866:
11855:
11849:
11846:
11841:
11838:
11835:
11832:
11829:
11826:
11823:
11820:
11817:
11814:
11811:
11808:
11805:
11802:
11799:
11796:
11793:
11790:
11787:
11784:
11781:
11778:
11775:
11772:
11769:
11766:
11763:
11760:
11757:
11754:
11751:
11748:
11745:
11742:
11739:
11736:
11733:
11730:
11727:
11724:
11721:
11715:
11712:
11665:
11662:
11649:
11644:
11640:
11634:
11631:
11626:
11623:
11618:
11614:
11608:
11605:
11600:
11597:
11592:
11588:
11582:
11579:
11574:
11571:
11566:
11562:
11556:
11553:
11548:
11545:
11533:
11532:
11515:
11511:
11507:
11502:
11498:
11494:
11489:
11485:
11481:
11476:
11472:
11466:
11463:
11457:
11454:
11433:
11424:
11415:
11406:
11397:
11396:
11385:
11378:
11374:
11370:
11365:
11358:
11353:
11349:
11345:
11340:
11333:
11328:
11324:
11320:
11315:
11308:
11303:
11299:
11295:
11290:
11283:
11278:
11274:
11270:
11247:
11232:
11229:
11228:
11227:
11216:
11213:
11210:
11207:
11204:
11201:
11198:
11195:
11192:
11189:
11186:
11183:
11180:
11177:
11174:
11171:
11168:
11165:
11162:
11159:
11156:
11153:
11150:
11147:
11144:
11141:
11138:
11084:
11083:
11072:
11069:
11066:
11063:
11057:
11052:
11048:
11045:
11042:
11039:
11033:
11028:
11024:
11021:
11018:
11015:
11009:
11004:
11000:
10997:
10994:
10991:
10985:
10980:
10976:
10973:
10970:
10955:
10948:
10941:
10934:
10903:
10902:Interior point
10900:
10888:
10883:
10880:
10876:
10872:
10869:
10864:
10860:
10854:
10850:
10846:
10841:
10838:
10834:
10830:
10827:
10822:
10818:
10812:
10808:
10804:
10799:
10796:
10792:
10788:
10785:
10780:
10776:
10770:
10766:
10762:
10759:
10756:
10751:
10746:
10742:
10738:
10733:
10728:
10724:
10720:
10715:
10710:
10706:
10702:
10697:
10692:
10688:
10676:law of cosines
10661:
10657:
10651:
10647:
10624:
10621:
10617:
10594:
10590:
10567:
10563:
10540:
10536:
10513:
10509:
10486:
10482:
10459:
10455:
10439:Main article:
10436:
10433:
10412:
10411:
10399:
10396:
10393:
10390:
10387:
10384:
10381:
10378:
10375:
10372:
10369:
10366:
10363:
10360:
10357:
10354:
10351:
10348:
10345:
10342:
10339:
10336:
10333:
10330:
10327:
10324:
10321:
10318:
10315:
10312:
10309:
10306:
10303:
10300:
10297:
10294:
10291:
10288:
10285:
10282:
10279:
10276:
10238:Main article:
10228:
10225:
10152:
10149:
10127:coincide with
10064:
10059:
10058:
10046:
10043:
10038:
10032:
10029:
10026:
10023:
10018:
10014:
10010:
10006:
10003:
10000:
9997:
9992:
9988:
9984:
9980:
9977:
9974:
9971:
9966:
9962:
9958:
9954:
9951:
9948:
9947:
9943:
9938:
9934:
9930:
9926:
9923:
9920:
9918:
9915:
9912:
9909:
9904:
9900:
9896:
9892:
9889:
9886:
9883:
9878:
9874:
9870:
9866:
9863:
9860:
9859:
9855:
9850:
9846:
9842:
9838:
9835:
9832:
9829:
9824:
9820:
9816:
9812:
9809:
9806:
9804:
9801:
9798:
9795:
9790:
9786:
9782:
9778:
9775:
9772:
9771:
9767:
9762:
9758:
9754:
9750:
9747:
9744:
9741:
9736:
9732:
9728:
9724:
9721:
9718:
9715:
9710:
9706:
9702:
9698:
9695:
9692:
9690:
9687:
9684:
9683:
9681:
9575:Spieker center
9569:
9566:
9554:
9549:
9544:
9540:
9536:
9533:
9530:
9527:
9524:
9520:
9516:
9512:
9508:
9504:
9498:
9495:
9475:
9462:as calculated
9450:
9446:
9442:
9421:
9401:
9345:
9342:
9339:
9336:
9333:
9330:
9327:
9324:
9320:
9317:
9314:
9311:
9308:
9305:
9302:
9299:
9296:
9292:
9290:
9287:
9284:
9281:
9278:
9275:
9272:
9269:
9266:
9263:
9260:
9257:
9254:
9251:
9248:
9245:
9242:
9239:
9238:
9235:
9232:
9229:
9226:
9223:
9220:
9217:
9214:
9211:
9208:
9205:
9202:
9199:
9196:
9193:
9190:
9187:
9183:
9181:
9178:
9175:
9172:
9169:
9166:
9163:
9160:
9157:
9154:
9151:
9148:
9145:
9142:
9139:
9136:
9133:
9130:
9129:
9126:
9123:
9120:
9117:
9114:
9111:
9108:
9105:
9102:
9099:
9096:
9093:
9090:
9087:
9084:
9081:
9078:
9074:
9072:
9069:
9066:
9063:
9060:
9057:
9054:
9051:
9048:
9045:
9042:
9039:
9036:
9033:
9030:
9027:
9024:
9021:
9020:
8996:
8991:
8988:
8985:
8980:
8977:
8973:
8971:
8966:
8963:
8960:
8955:
8952:
8949:
8948:
8945:
8940:
8937:
8934:
8929:
8926:
8922:
8920:
8915:
8912:
8909:
8904:
8901:
8898:
8897:
8874:
8870:
8866:
8862:
8857:
8854:
8851:
8848:
8845:
8842:
8839:
8836:
8833:
8829:
8826:
8823:
8820:
8817:
8814:
8811:
8808:
8805:
8801:
8798:
8795:
8792:
8789:
8786:
8783:
8780:
8777:
8773:
8770:
8767:
8764:
8761:
8758:
8755:
8752:
8749:
8746:
8739:
8736:
8716:
8696:
8676:
8656:
8636:
8616:
8596:
8576:
8556:
8536:
8516:
8496:
8463:
8459:
8454:
8450:
8446:
8442:
8438:
8433:
8429:
8425:
8421:
8417:
8412:
8408:
8404:
8400:
8396:
8393:
8389:
8385:
8382:
8378:
8374:
8371:
8368:
8365:
8362:
8355:
8351:
8348:
8345:
8339:
8336:
8312:
8307:
8303:
8299:
8294:
8290:
8286:
8281:
8277:
8273:
8270:
8268:
8266:
8263:
8262:
8259:
8254:
8250:
8246:
8241:
8237:
8233:
8228:
8224:
8220:
8217:
8215:
8213:
8210:
8209:
8206:
8201:
8197:
8193:
8188:
8184:
8180:
8175:
8171:
8167:
8164:
8162:
8160:
8157:
8156:
8134:
8130:
8127:
8124:
8121:
8116:
8112:
8106:
8102:
8098:
8093:
8089:
8083:
8079:
8075:
8070:
8066:
8060:
8056:
8052:
8047:
8043:
8037:
8033:
8027:
8023:
8019:
8013:
8010:
7990:
7970:
7950:
7930:
7910:
7890:
7870:
7841:
7816:∈ {1, 2, 3, 4}
7805:
7804:
7791:
7785:
7782:
7778:
7773:
7769:
7765:
7761:
7756:
7752:
7748:
7744:
7739:
7735:
7731:
7729:
7726:
7725:
7720:
7715:
7711:
7707:
7705:
7702:
7698:
7693:
7689:
7685:
7681:
7676:
7672:
7668:
7666:
7663:
7662:
7657:
7652:
7648:
7644:
7640:
7635:
7631:
7627:
7625:
7622:
7618:
7613:
7609:
7605:
7603:
7600:
7599:
7594:
7589:
7585:
7581:
7577:
7572:
7568:
7564:
7560:
7555:
7551:
7547:
7545:
7542:
7540:
7537:
7536:
7533:
7530:
7528:
7525:
7523:
7520:
7518:
7515:
7513:
7510:
7509:
7507:
7502:
7497:
7493:
7489:
7486:
7468:
7467:
7456:
7451:
7446:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7420:
7415:
7411:
7407:
7403:
7399:
7395:
7391:
7387:
7383:
7379:
7375:
7371:
7370:
7365:
7361:
7357:
7353:
7349:
7345:
7341:
7337:
7333:
7329:
7325:
7321:
7320:
7315:
7311:
7307:
7303:
7299:
7295:
7291:
7287:
7283:
7279:
7275:
7271:
7270:
7267:
7263:
7259:
7255:
7252:
7249:
7246:
7169:
7168:
7156:
7150:
7146:
7141:
7137:
7133:
7129:
7125:
7120:
7116:
7112:
7108:
7104:
7099:
7095:
7091:
7087:
7083:
7080:
7076:
7072:
7069:
7065:
7061:
7058:
7055:
7052:
7049:
7042:
7038:
7035:
7032:
7026:
7023:
7011:, which gives
6998:
6991:
6987:
6986:
6984:
6979:
6976:
6954:
6947:
6943:
6942:
6940:
6935:
6932:
6910:
6903:
6899:
6898:
6896:
6891:
6888:
6877:
6876:
6863:
6858:
6854:
6850:
6847:
6844:
6841:
6838:
6834:
6830:
6826:
6822:
6821:
6818:
6814:
6810:
6807:
6804:
6801:
6798:
6794:
6790:
6786:
6782:
6781:
6778:
6774:
6770:
6767:
6764:
6761:
6758:
6754:
6750:
6746:
6742:
6741:
6739:
6709:
6700:
6696:
6691:
6688:
6684:
6680:
6676:
6673:
6669:
6665:
6661:
6660:
6656:
6652:
6648:
6644:
6639:
6635:
6630:
6627:
6623:
6619:
6615:
6614:
6610:
6606:
6602:
6598:
6595:
6591:
6587:
6583:
6578:
6574:
6569:
6568:
6566:
6561:
6556:
6552:
6548:
6545:
6531:
6530:
6513:
6508:
6505:
6500:
6496:
6492:
6487:
6483:
6479:
6474:
6470:
6466:
6463:
6459:
6455:
6454:
6451:
6448:
6443:
6439:
6435:
6430:
6426:
6422:
6417:
6413:
6409:
6406:
6402:
6398:
6397:
6394:
6391:
6386:
6382:
6378:
6373:
6369:
6365:
6360:
6356:
6352:
6349:
6345:
6341:
6340:
6338:
6308:
6301:
6297:
6296:
6292:
6288:
6287:
6283:
6279:
6278:
6276:
6271:
6268:
6265:
6262:
6234:
6227:
6223:
6220:
6216:
6213:
6209:
6208:
6206:
6201:
6198:
6195:
6192:
6159:parallelepiped
6142:
6138:
6134:
6130:
6126:
6122:
6118:
6114:
6110:
6106:
6102:
6098:
6094:
6091:
6086:
6083:
6054:
6051:
6046:
6042:
6038:
6033:
6029:
6025:
6020:
6016:
6012:
6009:
6006:
6004:
6001:
5997:
5996:
5993:
5990:
5985:
5981:
5977:
5972:
5968:
5964:
5959:
5955:
5951:
5948:
5945:
5943:
5940:
5936:
5935:
5932:
5929:
5924:
5920:
5916:
5911:
5907:
5903:
5898:
5894:
5890:
5887:
5884:
5882:
5879:
5875:
5874:
5871:
5868:
5863:
5859:
5855:
5850:
5846:
5842:
5837:
5833:
5829:
5826:
5823:
5821:
5818:
5814:
5813:
5789:
5765:
5745:
5742:
5739:
5734:
5731:
5726:
5723:
5711:
5708:
5706:
5703:
5688:
5684:
5680:
5649:
5646:
5643:
5642:
5639:
5636:
5634:
5631:
5625:
5624:
5619:
5608:
5592:
5590:
5577:
5573:
5572:
5567:
5562:
5559:
5556:
5549:
5543:
5542:
5538:
5528:
5516:
5514:
5501:
5495:
5494:
5490:
5489:
5486:
5483:
5481:
5478:
5472:
5471:
5438:
5431:
5424:
5409:
5407:
5400:
5394:
5393:
5388:
5383:
5380:
5377:
5372:
5366:
5365:
5332:
5319:
5317:
5310:
5304:
5303:
5296:
5295:
5292:
5289:
5287:
5284:
5280:
5279:
5269:
5267:
5260:
5259:(No symmetry)
5254:
5253:
5250:
5247:
5245:
5242:
5234:
5226:
5220:
5219:
5215:
5204:
5193:
5191:
5184:
5180:
5179:
5174:
5169:
5166:
5163:
5158:
5152:
5151:
5147:
5136:
5129:triangle sides
5120:
5118:
5111:
5107:
5106:
5101:
5096:
5093:
5086:
5080:
5079:
5040:
5029:
5018:
5016:
5009:
5005:
5004:
4999:
4994:
4989:
4983:
4982:
4976:
4975:
4972:
4965:
4955:
4951:
4943:
4940:
4893:symmetry group
4876:
4873:
4833:
4830:
4812:
4809:
4779:
4776:
4746:
4743:
4713:
4710:
4680:
4677:
4647:
4644:
4614:
4611:
4581:
4578:
4552:
4522:
4519:
4489:
4486:
4460:
4430:
4427:
4397:
4394:
4364:
4361:
4327:
4324:
4298:
4272:
4269:
4251:
4250:
4248:
4246:
4231:
4227:
4207:
4202:
4200:
4174:
4173:
4171:
4169:
4167:
4165:
4162:
4161:
4159:
4157:
4155:
4153:
4141:
4138:
4131:
4128:
4113:
4101:
4097:
4093:
4088:
4084:
4072:
4071:
4069:
4067:
4065:
4063:
4051:
4048:
4041:
4038:
4023:
4011:
4007:
4003:
3998:
3994:
3982:
3981:
3979:
3977:
3975:
3973:
3961:
3958:
3951:
3948:
3933:
3921:
3917:
3913:
3908:
3904:
3892:
3891:
3889:
3887:
3885:
3883:
3880:
3879:
3864:
3861:
3847:
3842:
3825:
3819:
3816:
3802:
3797:
3785:
3782:
3775:
3772:
3757:
3753:
3752:
3737:
3734:
3720:
3715:
3698:
3692:
3689:
3675:
3670:
3658:
3647:
3643:
3642:
3627:
3624:
3610:
3605:
3588:
3577:
3572:
3560:
3557:
3550:
3547:
3532:
3528:
3527:
3525:
3523:
3521:
3519:
3516:
3515:
3498:
3495:
3492:
3481:
3476:
3459:
3456:
3453:
3442:
3437:
3425:
3414:
3410:
3409:
3406:
3403:
3400:
3397:
3396:
3182:
3179:
3154:Main article:
3151:
3148:
3114:Main article:
3111:
3108:
3057:
3056:
3047:
3034:
3031:
2975:
2972:
2969:
2968:
2963:
2957:
2956:
2949:
2923:
2920:
2868:
2865:
2862:
2861:
2859:
2857:
2851:
2850:
2843:
2836:
2832:
2831:
2828:
2825:
2807:
2795:
2792:
2791:
2790:
2785:is mapped to −
2779:
2776:
2775:
2774:
2753:
2719:
2694:
2690:
2665:
2638:
2634:
2603:
2597:
2581:symmetry group
2545:
2542:
2529:
2526:
2523:
2520:
2517:
2513:
2444:
2439:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2384:
2382:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2315:
2313:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2286:
2253:
2247:
2242:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2193:
2191:
2189:
2185:
2179:
2176:
2171:
2168:
2162:
2159:
2153:
2150:
2144:
2141:
2135:
2131:
2127:
2126:
2123:
2119:
2113:
2110:
2105:
2102:
2096:
2093:
2087:
2081:
2078:
2072:
2068:
2063:
2061:
2057:
2051:
2048:
2043:
2040:
2037:
2034:
2028:
2025:
2018:
2014:
2013:
1993:
1990:
1963:
1956:
1952:
1947:
1944:
1941:
1938:
1935:
1931:
1917:
1910:
1906:
1901:
1898:
1895:
1892:
1889:
1886:
1882:
1865:
1862:
1854:square degrees
1833:
1830:
1826:
1821:
1818:
1813:
1809:
1806:
1803:
1800:
1797:
1795:
1793:
1789:
1784:
1781:
1776:
1772:
1769:
1766:
1763:
1758:
1755:
1750:
1747:
1745:
1742:
1737:
1734:
1729:
1725:
1722:
1719:
1718:
1634:
1629:
1624:
1618:
1613:
1609:
1605:
1600:
1595:
1591:
1587:
1582:
1577:
1573:
1569:
1564:
1559:
1555:
1551:
1546:
1542:
1537:
1532:
1529:
1527:
1524:
1518:
1513:
1509:
1505:
1500:
1495:
1491:
1487:
1482:
1477:
1473:
1469:
1464:
1459:
1455:
1451:
1446:
1442:
1437:
1433:
1430:
1429:
1426:
1421:
1416:
1410:
1404:
1400:
1396:
1390:
1385:
1379:
1374:
1370:
1366:
1361:
1356:
1352:
1348:
1343:
1338:
1334:
1330:
1325:
1320:
1316:
1308:
1303:
1300:
1298:
1294:
1288:
1284:
1280:
1274:
1269:
1263:
1258:
1254:
1250:
1245:
1240:
1236:
1232:
1227:
1222:
1218:
1214:
1209:
1204:
1200:
1193:
1192:
1170:
1166:
1145:
1125:
1101:
1095:
1091:
1086:
1080:
1075:
1070:
1068:
1062:
1058:
1053:
1048:
1045:
1040:
1034:
1029:
1025:
1024:
1021:
1015:
1011:
1006:
1003:
998:
995:
990:
987:
983:
981:
978:
973:
969:
963:
960:
957:
956:
933:
928:
901:
896:
872:
849:
812:
807:
803:
799:
795:
790:
785:
781:
778:
775:
772:
769:
767:
764:
758:
755:
750:
746:
742:
739:
736:
735:
732:
727:
723:
719:
715:
709:
704:
700:
696:
693:
690:
687:
685:
682:
677:
674:
669:
665:
662:
659:
658:
647:dihedral angle
631:
626:
622:
618:
615:
607:
602:
596:
592:
586:
583:
578:
574:
568:
564:
558:
554:
548:
544:
537:
533:
528:
525:
520:
517:
497:
492:
488:
465:
460:
456:
452:
449:
444:
437:
433:
429:
425:
419:
415:
409:
405:
398:
394:
391:
388:
385:
365:
345:
323:
320:
202:
195:
194:
193:
185:
178:
177:
173:
166:
165:
164:
163:
162:
160:
157:
26:
9:
6:
4:
3:
2:
17729:
17718:
17715:
17713:
17710:
17708:
17705:
17703:
17700:
17698:
17695:
17693:
17690:
17688:
17685:
17684:
17682:
17671:
17667:
17663:
17658:
17655:
17651:
17648:
17646:
17639:
17632:
17626:
17624:
17620:
17617:
17615:
17611:
17607:
17603:
17600:
17598:
17594:
17591:
17589:
17585:
17581:
17580:
17577:
17575:
17573:
17570:
17568:
17564:
17561:
17559:
17556:
17554:
17551:
17550:
17547:
17545:
17543:
17540:
17538:
17534:
17531:
17529:
17526:
17524:
17521:
17520:
17517:
17515:
17508:
17501:
17495:
17493:
17490:
17488:
17484:
17481:
17479:
17476:
17474:
17471:
17470:
17467:
17465:
17458:
17451:
17445:
17443:
17440:
17438:
17434:
17431:
17429:
17426:
17424:
17421:
17420:
17417:
17415:
17408:
17402:
17400:
17397:
17395:
17391:
17388:
17386:
17383:
17381:
17378:
17377:
17374:
17372:
17370:
17367:
17365:
17361:
17358:
17356:
17353:
17351:
17348:
17347:
17344:
17340:
17337:
17335:
17332:
17330:
17329:Demitesseract
17327:
17325:
17321:
17318:
17316:
17313:
17311:
17308:
17307:
17304:
17300:
17297:
17295:
17293:
17290:
17288:
17284:
17281:
17279:
17276:
17274:
17271:
17270:
17267:
17264:
17262:
17259:
17257:
17254:
17252:
17249:
17247:
17244:
17242:
17239:
17238:
17235:
17229:
17226:
17222:
17215:
17211:
17204:
17200:
17195:
17191:
17186:
17182:
17177:
17175:
17173:
17169:
17159:
17155:
17153:
17151:
17147:
17143:
17141:
17139:
17135:
17131:
17129:
17126:
17125:
17120:
17116:
17108:
17103:
17101:
17096:
17094:
17089:
17088:
17085:
17075:
17069:
17063:
17060:
17058:
17055:
17053:
17050:
17048:
17045:
17043:
17040:
17038:
17035:
17033:
17030:
17028:
17025:
17023:
17020:
17018:
17015:
17013:
17010:
17009:
17007:
17003:
16993:
16990:
16988:
16985:
16984:
16982:
16978:
16972:
16969:
16967:
16964:
16963:
16960:
16957:
16953:
16947:
16946:
16942:
16940:
16939:
16935:
16934:
16932:
16928:
16922:
16919:
16917:
16914:
16912:
16909:
16907:
16904:
16902:
16899:
16897:
16894:
16892:
16889:
16887:
16884:
16882:
16879:
16877:
16874:
16872:
16869:
16867:
16864:
16862:
16859:
16858:
16856:
16849:
16844:
16838:
16835:
16833:
16830:
16828:
16825:
16823:
16820:
16818:
16815:
16813:
16810:
16808:
16805:
16803:
16800:
16798:
16795:
16793:
16790:
16788:
16785:
16783:
16782:cuboctahedron
16780:
16778:
16775:
16774:
16772:
16767:
16763:
16757:
16752:
16746:
16743:
16741:
16738:
16736:
16733:
16731:
16728:
16726:
16723:
16722:
16720:
16716:
16711:
16707:
16703:
16695:
16690:
16688:
16683:
16681:
16676:
16675:
16672:
16660:
16657:
16655:
16652:
16650:
16647:
16645:
16642:
16640:
16637:
16635:
16632:
16630:
16627:
16626:
16624:
16621:
16617:
16610:
16607:
16604:
16601:
16598:
16595:
16594:
16592:
16588:
16581:
16580:Johnson solid
16578:
16575:
16574:Catalan solid
16572:
16569:
16566:
16563:
16560:
16559:
16557:
16553:
16544:
16541:
16538:
16535:
16532:
16529:
16528:
16526:
16523:
16521:
16518:
16516:
16513:
16511:
16508:
16507:
16505:
16501:
16494:
16491:
16488:
16485:
16482:
16479:
16476:
16473:
16470:
16469:Hexoctahedron
16467:
16464:
16461:
16458:
16455:
16452:
16449:
16448:
16446:
16442:
16436:
16433:
16431:
16428:
16426:
16423:
16421:
16418:
16416:
16413:
16411:
16408:
16406:
16403:
16401:
16400:Tridecahedron
16398:
16396:
16393:
16391:
16390:Hendecahedron
16388:
16387:
16385:
16381:
16375:
16372:
16370:
16367:
16365:
16362:
16360:
16357:
16355:
16352:
16350:
16347:
16345:
16342:
16340:
16337:
16335:
16332:
16330:
16327:
16326:
16324:
16320:
16313:
16309:
16302:
16297:
16295:
16290:
16288:
16283:
16282:
16279:
16272:
16268:
16267:
16263:
16261:
16258:
16253:
16252:
16247:
16246:"Tetrahedron"
16244:
16239:
16238:
16227:
16221:
16218:. Routledge.
16217:
16212:
16208:
16204:
16203:
16195:
16190:
16186:
16182:
16178:
16174:
16169:
16164:
16160:
16156:
16152:
16147:
16143:
16139:
16134:
16130:
16129:
16124:
16120:
16113:
16112:
16106:
16102:
16096:
16092:
16091:
16085:
16081:
16077:
16073:
16069:
16065:
16061:
16057:
16053:
16048:
16044:
16038:
16034:
16030:
16029:
16023:
16019:
16015:
16014:
16009:
16005:
16001:
16000:
15995:
15991:
15987:
15983:
15978:
15974:
15968:
15964:
15960:
15955:
15954:
15939:
15938:
15933:
15930:
15923:
15915:
15911:
15907:
15903:
15899:
15896:(June 1903).
15895:
15889:
15873:
15867:
15859:
15855:
15851:
15845:
15837:
15835:9780177612992
15831:
15827:
15822:
15821:
15815:
15809:
15801:
15797:
15793:
15789:
15785:
15784:
15779:
15773:
15754:
15750:
15743:
15736:
15717:
15713:
15709:
15702:
15695:
15680:
15676:
15670:
15662:
15658:
15654:
15650:
15646:
15642:
15635:
15627:
15625:9780160876110
15621:
15617:
15616:
15608:
15601:
15600:
15595:
15590:
15582:
15578:
15575:, Solutions,
15571:
15570:"Problem 930"
15565:
15558:
15554:
15550:
15546:
15538:
15530:
15522:, p. 129
15521:
15520:
15512:
15496:
15492:
15485:
15478:
15475:
15473:
15466:
15464:
15462:
15460:
15452:
15447:
15439:
15435:
15431:
15427:
15423:
15419:
15412:
15404:
15400:
15396:
15392:
15388:
15384:
15383:
15378:
15372:
15364:
15360:
15353:
15342:
15335:
15327:
15321:
15310:
15309:
15301:
15292:
15284:
15280:
15276:
15272:
15267:
15262:
15258:
15254:
15253:
15245:
15238:
15231:
15226:
15219:
15214:
15208:, p. 11.
15207:
15202:
15200:
15192:
15187:
15180:
15176:
15171:
15164:
15160:
15155:
15147:
15143:
15138:
15133:
15129:
15125:
15121:
15114:
15107:
15103:
15098:
15091:
15086:
15075:
15070:
15063:
15058:
15056:
15046:
15041:
15037:
15033:
15029:
15025:
15019:
15012:
15007:
14999:
14993:
14986:
14981:
14972:
14969:
14966:
14963:
14962:
14958:
14951:
14947:
14942:
14935:
14934:"Tetrahedron"
14929:
14923:, Table I(i).
14922:
14917:
14915:
14913:
14905:
14901:
14900:Cromwell 1997
14896:
14889:
14885:
14880:
14873:
14868:
14860:
14859:
14851:
14842:
14841:
14836:
14835:"Tetrahedron"
14833:
14826:
14824:
14819:
14805:
14801:
14797:
14791:
14789:
14781:
14777:
14771:
14767:
14757:
14754:
14752:
14749:
14746:
14743:
14741:
14738:
14736:
14733:
14731:
14728:
14726:
14723:
14720:
14717:
14714:
14710:
14706:
14705:Demihypercube
14703:
14701:
14698:
14696:
14693:
14691:
14688:
14687:
14674:
14670:
14666:
14664:
14660:
14652:
14644:
14640:
14636:
14627:
14623:
14620:
14616:
14612:
14608:
14604:
14600:
14596:
14593:
14589:
14585:
14583:
14579:
14575:
14573:
14572:Automorphisms
14569:
14565:
14563:
14559:
14555:
14553:
14549:
14545:
14543:
14539:
14535:
14533:
14529:
14525:
14523:
14519:
14515:
14510:
14505:
14497:
14493:
14490:
14486:
14482:
14481:Marvin Minsky
14478:
14477:
14472:
14468:
14453:
14444:
14433:
14424:
14410:
14408:
14404:
14398:
14388:
14386:
14382:
14378:
14374:
14370:
14366:
14361:
14359:
14352:
14348:
14339:
14335:
14325:
14323:
14319:
14315:
14311:
14310:semiconductor
14307:
14302:
14300:
14296:
14293:If six equal
14290:
14286:
14276:
14274:
14270:
14266:
14261:
14259:
14254:
14252:
14247:
14232:
14228:
14210:
14209:central angle
14206:
14205:
14188:
14172:
14168:
14164:
14163:sp-hybridized
14160:
14154:
14147:
14142:
14135:
14130:
14121:
14119:
14109:
14107:
14097:
14095:
14091:
14087:
14083:
14079:
14075:
14071:
14067:
14063:
14059:
14056:of irregular
14055:
14051:
14047:
14038:
14024:
14022:
14018:
14013:
14011:
14006:
14004:
14000:
13996:
13995:mirror images
13992:
13988:
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12869:
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12861:
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12849:
12848:Trapezohedron
12846:
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12835:
12833:
12832:
12831:trapezohedron
12820:
12817:
12814:
12811:
12808:
12805:
12802:
12799:
12797:
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12793:
12789:
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12609:
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12516:
12513:
12509:
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12477:
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12469:
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12461:
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12449:
12446:
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12441:
12440:
12437:
12433:
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12389:
12385:
12379:
12369:
12361:
12359:
12355:
12345:
12323:
12314:
12304:
12300:
12293:
12288:
12278:
12274:
12261:
12251:
12247:
12240:
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12225:
12221:
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12198:
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12117:
12108:
12103:
12097:
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12089:
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12059:
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12049:
12045:
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12031:
12020:
12015:
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12001:
11996:
11992:
11987:
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11971:
11954:
11949:
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11938:
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11894:
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11675:
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11638:
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11606:
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11598:
11595:
11590:
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11577:
11572:
11569:
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11554:
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11513:
11509:
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11496:
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11487:
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11479:
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11470:
11464:
11461:
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11208:
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11199:
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11169:
11166:
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11160:
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11145:
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11139:
11136:
11129:
11128:
11127:
11125:
11121:
11117:
11113:
11109:
11105:
11101:
11097:
11093:
11089:
11086:For vertices
11070:
11067:
11064:
11061:
11050:
11046:
11043:
11040:
11037:
11026:
11022:
11019:
11016:
11013:
11002:
10998:
10995:
10992:
10989:
10978:
10974:
10971:
10968:
10961:
10960:
10959:
10954:
10947:
10940:
10933:
10929:
10925:
10921:
10917:
10913:
10909:
10899:
10881:
10878:
10874:
10870:
10867:
10862:
10852:
10844:
10839:
10836:
10832:
10828:
10825:
10820:
10810:
10802:
10797:
10794:
10790:
10786:
10783:
10778:
10768:
10757:
10754:
10749:
10744:
10736:
10731:
10726:
10718:
10713:
10708:
10700:
10695:
10690:
10677:
10659:
10655:
10649:
10645:
10622:
10619:
10615:
10592:
10588:
10565:
10538:
10534:
10511:
10507:
10484:
10480:
10457:
10453:
10442:
10432:
10429:
10423:
10420:
10415:
10397:
10394:
10391:
10388:
10382:
10379:
10376:
10373:
10370:
10367:
10361:
10358:
10355:
10352:
10349:
10346:
10340:
10337:
10334:
10331:
10328:
10325:
10319:
10316:
10313:
10310:
10307:
10304:
10298:
10295:
10292:
10289:
10286:
10283:
10277:
10274:
10267:
10266:
10265:
10263:
10259:
10255:
10251:
10247:
10241:
10236:
10234:
10224:
10222:
10217:
10213:
10211:
10207:
10203:
10198:
10196:
10192:
10188:
10184:
10179:
10177:
10172:
10168:
10166:
10162:
10158:
10148:
10146:
10142:
10138:
10134:
10130:
10126:
10122:
10118:
10114:
10110:
10106:
10102:
10098:
10094:
10090:
10086:
10082:
10077:
10075:
10071:
10067:
10044:
10041:
10036:
10030:
10027:
10016:
10012:
10004:
10001:
9990:
9986:
9978:
9975:
9964:
9960:
9952:
9949:
9936:
9932:
9924:
9921:
9916:
9913:
9902:
9898:
9890:
9887:
9876:
9872:
9864:
9861:
9848:
9844:
9836:
9833:
9822:
9818:
9810:
9807:
9802:
9799:
9788:
9784:
9776:
9773:
9760:
9756:
9748:
9745:
9734:
9730:
9722:
9719:
9708:
9704:
9696:
9693:
9688:
9685:
9679:
9670:
9669:
9668:
9665:
9662:
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9655:
9651:
9646:
9644:
9640:
9636:
9631:
9629:
9625:
9621:
9617:
9613:
9609:
9605:
9601:
9596:
9592:
9590:
9586:
9582:
9581:Gaspard Monge
9578:
9576:
9565:
9552:
9547:
9518:
9502:
9496:
9493:
9473:
9465:
9444:
9419:
9399:
9391:
9386:
9384:
9380:
9376:
9372:
9367:
9365:
9360:
9343:
9337:
9334:
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9325:
9315:
9312:
9309:
9306:
9303:
9297:
9294:
9288:
9282:
9279:
9276:
9273:
9270:
9261:
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9249:
9243:
9240:
9233:
9227:
9224:
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9218:
9215:
9206:
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9197:
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9179:
9173:
9170:
9167:
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9152:
9149:
9146:
9143:
9140:
9134:
9131:
9124:
9118:
9115:
9112:
9109:
9106:
9097:
9094:
9091:
9088:
9085:
9079:
9076:
9070:
9064:
9061:
9058:
9055:
9052:
9043:
9040:
9037:
9034:
9031:
9025:
9022:
8994:
8989:
8986:
8983:
8978:
8975:
8969:
8964:
8961:
8958:
8953:
8950:
8943:
8938:
8935:
8932:
8927:
8924:
8918:
8913:
8910:
8907:
8902:
8899:
8872:
8868:
8864:
8860:
8852:
8849:
8846:
8843:
8840:
8837:
8834:
8824:
8821:
8818:
8815:
8812:
8809:
8806:
8796:
8793:
8790:
8787:
8784:
8781:
8778:
8768:
8765:
8762:
8759:
8756:
8753:
8750:
8747:
8737:
8734:
8714:
8694:
8674:
8654:
8634:
8614:
8594:
8574:
8554:
8534:
8514:
8494:
8481:
8477:
8461:
8457:
8452:
8448:
8444:
8440:
8436:
8431:
8427:
8423:
8419:
8415:
8410:
8406:
8402:
8398:
8394:
8391:
8387:
8383:
8380:
8376:
8372:
8369:
8366:
8363:
8360:
8353:
8349:
8346:
8343:
8337:
8334:
8310:
8305:
8301:
8297:
8292:
8288:
8284:
8279:
8275:
8271:
8269:
8264:
8257:
8252:
8248:
8244:
8239:
8235:
8231:
8226:
8222:
8218:
8216:
8211:
8204:
8199:
8195:
8191:
8186:
8182:
8178:
8173:
8169:
8165:
8163:
8158:
8132:
8128:
8125:
8122:
8119:
8114:
8110:
8104:
8100:
8096:
8091:
8087:
8081:
8077:
8073:
8068:
8064:
8058:
8054:
8050:
8045:
8041:
8035:
8031:
8025:
8021:
8017:
8011:
8008:
7988:
7968:
7948:
7928:
7908:
7888:
7868:
7859:
7857:
7853:
7849:
7844:
7835:
7831:
7827:
7823:
7815:
7811:
7789:
7783:
7776:
7771:
7767:
7759:
7754:
7750:
7742:
7737:
7733:
7727:
7718:
7713:
7709:
7703:
7696:
7691:
7687:
7679:
7674:
7670:
7664:
7655:
7650:
7646:
7638:
7633:
7629:
7623:
7616:
7611:
7607:
7601:
7592:
7587:
7583:
7575:
7570:
7566:
7558:
7553:
7549:
7543:
7538:
7531:
7526:
7521:
7516:
7511:
7505:
7500:
7495:
7491:
7487:
7484:
7477:
7476:
7475:
7473:
7454:
7449:
7444:
7437:
7432:
7427:
7422:
7413:
7409:
7401:
7397:
7389:
7385:
7377:
7373:
7363:
7359:
7351:
7347:
7339:
7335:
7327:
7323:
7313:
7309:
7301:
7297:
7289:
7285:
7277:
7273:
7265:
7257:
7253:
7250:
7247:
7244:
7237:
7236:
7235:
7233:
7228:
7226:
7222:
7218:
7214:
7210:
7206:
7202:
7198:
7194:
7190:
7186:
7182:
7178:
7174:
7154:
7148:
7144:
7139:
7135:
7131:
7127:
7123:
7118:
7114:
7110:
7106:
7102:
7097:
7093:
7089:
7085:
7081:
7078:
7074:
7070:
7067:
7063:
7059:
7056:
7053:
7050:
7047:
7040:
7036:
7033:
7030:
7024:
7021:
7014:
7013:
7012:
6977:
6974:
6933:
6930:
6889:
6886:
6856:
6852:
6848:
6845:
6842:
6839:
6836:
6828:
6816:
6812:
6808:
6805:
6802:
6799:
6796:
6788:
6776:
6772:
6768:
6765:
6762:
6759:
6756:
6748:
6737:
6707:
6682:
6667:
6650:
6621:
6604:
6589:
6564:
6559:
6554:
6550:
6546:
6543:
6536:
6535:
6534:
6506:
6498:
6494:
6490:
6485:
6481:
6477:
6472:
6468:
6461:
6449:
6441:
6437:
6433:
6428:
6424:
6420:
6415:
6411:
6404:
6392:
6384:
6380:
6376:
6371:
6367:
6363:
6358:
6354:
6347:
6336:
6269:
6266:
6263:
6260:
6199:
6196:
6193:
6190:
6183:
6182:
6181:
6178:
6160:
6156:
6132:
6124:
6116:
6108:
6100:
6084:
6081:
6072:
6052:
6044:
6040:
6036:
6031:
6027:
6023:
6018:
6014:
6007:
6005:
5991:
5983:
5979:
5975:
5970:
5966:
5962:
5957:
5953:
5946:
5944:
5930:
5922:
5918:
5914:
5909:
5905:
5901:
5896:
5892:
5885:
5883:
5869:
5861:
5857:
5853:
5848:
5844:
5840:
5835:
5831:
5824:
5822:
5802:
5787:
5779:
5763:
5743:
5740:
5737:
5732:
5729:
5724:
5721:
5702:
5686:
5682:
5678:
5666:
5663:
5658:
5656:
5640:
5637:
5635:
5630:
5627:
5626:
5623:
5618:
5614:
5607:
5600:
5596:
5588:
5582:
5574:
5568:
5563:
5560:
5555:
5548:
5545:
5544:
5537:
5534:
5527:
5520:
5512:
5506:
5500:
5496:
5491:
5487:
5484:
5482:
5477:
5474:
5473:
5470:
5437:
5430:
5423:
5420:
5413:
5405:
5399:
5395:
5389:
5384:
5381:
5371:
5368:
5367:
5364:
5331:
5323:
5315:
5309:
5305:
5301:
5297:
5293:
5290:
5288:
5282:
5281:
5278:
5276:
5275:trivial group
5265:
5255:
5251:
5248:
5246:
5241:
5233:
5225:
5222:
5221:
5214:
5210:
5203:
5197:
5189:
5181:
5175:
5170:
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5153:
5146:
5142:
5135:
5128:
5124:
5116:
5108:
5102:
5097:
5094:
5092:
5085:
5082:
5081:
5077:
5046:
5039:
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5028:
5022:
5014:
5006:
5003:
5000:
4998:
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4993:
4990:
4988:
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4977:
4962:
4959:
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4937:
4933:
4928:
4923:
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4917:
4913:
4909:
4904:
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4894:
4890:
4881:
4872:
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4866:
4862:
4858:
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4851:
4847:
4843:
4839:
4829:
4810:
4807:
4777:
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4744:
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4711:
4708:
4678:
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4612:
4609:
4579:
4576:
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4487:
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4458:
4449:
4428:
4425:
4395:
4392:
4362:
4359:
4346:
4325:
4322:
4296:
4270:
4267:
4229:
4225:
4222:arc sec
4175:
4172:
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4168:
4166:
4164:
4163:
4139:
4136:
4129:
4126:
4099:
4095:
4091:
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4073:
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4036:
4009:
4005:
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3983:
3959:
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3893:
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3862:
3859:
3823:
3817:
3814:
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3770:
3754:
3735:
3732:
3696:
3690:
3687:
3656:
3644:
3625:
3622:
3586:
3558:
3555:
3548:
3545:
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3496:
3493:
3490:
3457:
3454:
3451:
3423:
3411:
3398:
3393:
3390:
3332:
3327:
3325:
3321:
3240:
3234:
3232:
3228:
3223:
3219:
3215:
3211:
3206:
3204:
3200:
3196:
3195:3-orthoscheme
3187:
3178:
3175:
3167:
3162:
3157:
3147:
3145:
3141:
3139:
3135:
3131:
3122:
3117:
3107:
3105:
3101:
3096:
3094:
3090:
3086:
3082:
3077:
3075:
3071:
3067:
3066:perpendicular
3062:
3052:
3048:
3043:
3039:
3038:
3030:
3028:
3024:
3020:
3016:
3012:
3008:
3003:
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2997:
2989:
2985:
2980:
2967:
2962:
2959:
2958:
2954:
2950:
2947:
2943:
2939:
2937:
2933:
2929:
2919:
2917:
2908:
2904:
2902:
2898:
2894:
2890:
2882:
2878:
2873:
2860:
2858:
2853:
2848:
2844:
2841:
2837:
2834:
2829:
2826:
2823:
2822:
2819:
2814:
2812:
2811:Coxeter plane
2805:
2801:
2788:
2784:
2780:
2777:
2771:
2767:
2763:
2757:3 ((1 2)(3 4)
2754:
2741:
2737:
2733:
2720:
2717:
2716:
2714:
2710:
2692:
2688:
2680:
2654:
2653:
2652:
2636:
2632:
2624:
2620:
2586:
2582:
2573:
2569:
2567:
2563:
2559:
2550:
2541:
2524:
2521:
2518:
2503:
2472:
2468:
2464:
2460:
2442:
2437:
2413:
2407:
2404:
2401:
2398:
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2389:
2383:
2378:
2372:
2369:
2366:
2363:
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2354:
2344:
2338:
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2329:
2326:
2323:
2320:
2314:
2309:
2303:
2300:
2297:
2294:
2291:
2276:
2272:
2267:
2251:
2245:
2240:
2210:
2207:
2204:
2201:
2198:
2192:
2187:
2183:
2177:
2174:
2169:
2166:
2160:
2157:
2151:
2148:
2142:
2139:
2133:
2129:
2121:
2117:
2111:
2108:
2103:
2100:
2094:
2091:
2085:
2079:
2076:
2070:
2066:
2059:
2055:
2049:
2046:
2041:
2038:
2035:
2032:
2026:
2023:
2016:
1991:
1988:
1980:
1975:
1961:
1954:
1950:
1945:
1942:
1939:
1936:
1933:
1929:
1915:
1908:
1904:
1899:
1896:
1893:
1890:
1887:
1884:
1880:
1871:
1861:
1859:
1856:, or 0.04387
1855:
1851:
1831:
1828:
1824:
1819:
1816:
1811:
1807:
1804:
1801:
1798:
1796:
1787:
1782:
1779:
1774:
1770:
1767:
1764:
1761:
1756:
1753:
1748:
1746:
1740:
1735:
1732:
1727:
1723:
1720:
1708:
1703:
1701:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1654:
1649:
1632:
1627:
1622:
1616:
1611:
1607:
1603:
1598:
1593:
1589:
1585:
1580:
1575:
1571:
1567:
1562:
1557:
1553:
1549:
1544:
1540:
1535:
1530:
1528:
1522:
1516:
1511:
1507:
1503:
1498:
1493:
1489:
1485:
1480:
1475:
1471:
1467:
1462:
1457:
1453:
1449:
1444:
1440:
1435:
1431:
1424:
1419:
1414:
1408:
1402:
1398:
1394:
1388:
1383:
1377:
1372:
1368:
1364:
1359:
1354:
1350:
1346:
1341:
1336:
1332:
1328:
1323:
1318:
1314:
1306:
1301:
1299:
1292:
1286:
1282:
1278:
1272:
1267:
1261:
1256:
1252:
1248:
1243:
1238:
1234:
1230:
1225:
1220:
1216:
1212:
1207:
1202:
1198:
1168:
1164:
1143:
1123:
1099:
1093:
1089:
1084:
1073:
1066:
1060:
1056:
1051:
1046:
1043:
1038:
1027:
1019:
1013:
1009:
1004:
1001:
996:
993:
988:
985:
979:
976:
971:
967:
961:
958:
926:
918:
894:
886:
870:
863:
847:
840:
831:
827:
810:
805:
801:
797:
793:
788:
783:
779:
776:
773:
770:
768:
762:
756:
753:
748:
744:
740:
737:
730:
725:
721:
717:
713:
707:
702:
698:
694:
691:
688:
686:
680:
675:
672:
667:
663:
660:
648:
643:
629:
624:
620:
616:
613:
605:
600:
594:
590:
584:
581:
576:
572:
566:
562:
556:
552:
546:
542:
535:
531:
526:
523:
518:
515:
495:
490:
486:
463:
458:
454:
450:
447:
442:
435:
431:
427:
423:
417:
413:
407:
403:
396:
392:
389:
386:
383:
363:
343:
329:
319:
317:
313:
309:
305:
301:
296:
294:
290:
286:
282:
277:
275:
271:
267:
263:
259:
255:
250:
248:
244:
240:
236:
231:
229:
225:
221:
217:
206:
199:
189:
182:
170:
156:
154:
150:
146:
141:
139:
135:
130:
128:
124:
120:
115:
113:
109:
106:
102:
97:
95:
91:
87:
83:
79:
75:
71:
67:
59:
55:
50:
46:
44:
39:
38:
33:
19:
17649:
17618:
17609:
17601:
17592:
17583:
17563:10-orthoplex
17299:Dodecahedron
17291:
17277:
17220:
17209:
17198:
17189:
17180:
17171:
17167:
17157:
17149:
17145:
17137:
17133:
17073:
16992:trapezohedra
16943:
16936:
16740:dodecahedron
16493:Apeirohedron
16444:>20 faces
16395:Dodecahedron
16343:
16271:kaleidocycle
16265:
16249:
16215:
16206:
16200:
16158:
16154:
16141:
16137:
16127:
16110:
16089:
16055:
16051:
16027:
16012:
15998:
15985:
15981:
15958:
15950:Bibliography
15935:
15922:
15905:
15901:
15888:
15876:. Retrieved
15866:
15857:
15844:
15819:
15808:
15782:
15772:
15760:. Retrieved
15753:the original
15748:
15735:
15725:15 September
15723:. Retrieved
15716:the original
15711:
15707:
15694:
15682:. Retrieved
15678:
15669:
15644:
15640:
15634:
15614:
15607:
15597:
15589:
15580:
15576:
15564:
15548:
15544:
15536:
15529:
15524:( Art. 163 )
15518:
15511:
15499:. Retrieved
15494:
15484:
15469:
15446:
15421:
15417:
15411:
15386:
15380:
15371:
15362:
15358:
15352:
15334:
15307:
15300:
15291:
15256:
15250:
15237:
15225:
15218:Bottema 1969
15213:
15186:
15170:
15154:
15127:
15123:
15113:
15105:
15102:Coxeter 1973
15097:
15090:Coxeter 1973
15085:
15074:Coxeter 1973
15069:
15062:Coxeter 1973
15035:
15031:
15018:
15006:
14992:
14980:
14971:Brittin 1945
14965:Coxeter 1948
14957:
14941:
14928:
14921:Coxeter 1948
14895:
14879:
14867:
14857:
14850:
14838:
14804:golden ratio
14799:
14796:Coxeter 1973
14770:
14712:
14632:
14619:planar graph
14494:
14474:
14465:
14400:
14377:Rubik's Cube
14362:
14355:
14351:4-sided dice
14337:
14303:
14292:
14273:tetrahedrane
14268:
14262:
14255:
14248:
14229:
14202:
14156:
14115:
14103:
14078:aerodynamics
14050:approximated
14043:
14027:Applications
14014:
14007:
13991:right-handed
13983:dodecahedron
13968:
13872:
13729:
13725:
13713:
13664:
13523:
13519:
13505:
13499:
13477:
13457:
13372:
13352:
13024:
12982:Plane tiling
12856:
12839:
12828:
12826:
12654:(Tetragonal)
12603:
12585:
12583:
12481:Equilateral
12447:
12424:
12416:
12381:
12367:
12351:
12343:
11910:
11901:
11892:
11883:
11877:
11873:
11870:Circumcenter
11697:
11693:
11689:
11685:
11681:
11677:
11673:
11669:
11667:
11664:Circumradius
11534:
11442:is given by
11439:
11434:
11430:
11425:
11421:
11416:
11412:
11407:
11403:
11401:
11398:
11253:
11248:
11244:
11236:
11234:
11123:
11119:
11115:
11111:
11107:
11103:
11099:
11095:
11091:
11087:
11085:
10952:
10945:
10938:
10931:
10927:
10923:
10919:
10915:
10911:
10907:
10905:
10444:
10424:
10418:
10416:
10413:
10261:
10257:
10253:
10249:
10246:law of sines
10243:
10230:
10218:
10214:
10199:
10191:rhombohedron
10180:
10173:
10169:
10154:
10144:
10140:
10136:
10132:
10128:
10124:
10120:
10116:
10112:
10104:
10100:
10096:
10088:
10078:
10073:
10069:
10062:
10060:
9666:
9663:
9657:
9653:
9649:
9647:
9643:Euler points
9642:
9638:
9632:
9623:
9615:
9607:
9603:
9597:
9593:
9584:
9579:
9571:
9486:is given by
9387:
9368:
9361:
8486:
7860:
7839:
7833:
7829:
7825:
7821:
7813:
7809:
7806:
7469:
7231:
7229:
7224:
7220:
7216:
7212:
7208:
7204:
7203:. The angle
7200:
7196:
7192:
7188:
7187:. The angle
7184:
7180:
7176:
7172:
7170:
6878:
6532:
6179:
5803:
5713:
5667:
5661:
5659:
5654:
5651:
5628:
5616:
5613:cyclic group
5605:
5603:
5598:
5594:
5553:
5546:
5535:
5525:
5518:
5475:
5435:
5428:
5421:
5416:
5411:
5369:
5329:
5326:
5321:
5272:
5239:
5231:
5223:
5212:
5209:cyclic group
5201:
5195:
5155:
5144:
5133:
5126:
5122:
5090:
5083:
5037:
5026:
5020:
4974:Description
4945:
4924:
4919:
4915:
4908:kaleidoscope
4905:
4886:
4854:
4852:to a cube.)
4842:mirror image
4837:
4835:
4447:
4344:
4254:
3328:
3319:
3235:
3230:
3226:
3216:that is the
3207:
3194:
3192:
3181:Orthoschemes
3171:
3142:
3138:right angles
3133:
3127:
3099:
3097:
3080:
3078:
3073:
3063:
3060:
3004:
2993:
2925:
2913:
2888:
2886:
2876:
2827:Face/vertex
2824:Centered by
2799:
2798:The regular
2797:
2786:
2782:
2769:
2765:
2761:
2739:
2735:
2731:
2578:
2555:
2268:
1976:
1867:
1704:
1694:
1690:
1682:
1670:
1655:of a face (2
1650:
839:circumsphere
836:
644:
335:
297:
278:
265:
251:
232:
215:
213:
145:circumsphere
142:
131:
116:
111:
98:
73:
70:tetrahedrons
69:
65:
57:
51:
49:
42:
36:
32:tetrahedroid
17572:10-demicube
17533:9-orthoplex
17483:8-orthoplex
17433:7-orthoplex
17390:6-orthoplex
17360:5-orthoplex
17315:Pentachoron
17303:Icosahedron
17278:Tetrahedron
16762:semiregular
16745:icosahedron
16725:tetrahedron
16435:Icosahedron
16383:11–20 faces
16369:Enneahedron
16359:Heptahedron
16349:Pentahedron
16344:Tetrahedron
15878:20 February
15762:11 November
15175:Fekete 1985
15011:Kepler 1619
14967:Table I(i).
14756:Orthoscheme
14719:Pentachoron
14651:wheel graph
14595:Hamiltonian
14385:Pyramorphix
14369:4-sided die
14365:roleplaying
14334:Color space
14328:Color space
14289:Electronics
14285:Electricity
14249:Quaternary
14204:Tetrahedron
14146:dot product
14106:spaceframes
13987:left-handed
12857:Tetrahedron
12467:Heptagonal
11106:, and feet
9648:The center
9585:Monge point
6071:determinant
5780:' area and
5410:Four equal
5320:Four equal
5300:Disphenoids
5123:equilateral
5021:equilateral
4969:equivalence
4954:, ), and (S
3218:convex hull
3210:orthoscheme
3136:vertex are
2986:within the
2800:tetrahedron
2467:alternating
1979:unit sphere
1707:solid angle
322:Measurement
228:deltahedron
88:, and four
58:tetrahedron
43:Tetrahedron
17717:Tetrahedra
17687:Deltahedra
17681:Categories
17558:10-simplex
17542:9-demicube
17492:8-demicube
17442:7-demicube
17399:6-demicube
17369:5-demicube
17283:Octahedron
17057:prismatoid
16987:bipyramids
16971:antiprisms
16945:hosohedron
16735:octahedron
16620:prismatoid
16605:(infinite)
16374:Decahedron
16364:Octahedron
16354:Hexahedron
16329:Monohedron
16322:1–10 faces
16209:: 227–232.
15647:(3): 145.
15206:Kahan 2012
15177:, p.
15161:, p.
15130:: 128631.
14948:, p.
14902:, p.
14886:, p.
14872:Cundy 1952
14814:References
14591:Properties
14173:molecule (
14058:tetrahedra
13762:Spherical
13556:Spherical
13493:V3.3.3.3.3
13031:octahedron
12893:Polyhedron
12838:Family of
12647:(Trigonal)
12608:antiprisms
12599:Family of
12484:Isosceles
12458:Pentagonal
12448:Triangular
10264:, we have
10206:disphenoid
9628:Euler line
9624:Euler line
9612:concurrent
9390:skew lines
5194:Two equal
4927:honeycombs
4869:octahedron
3440:109°28′16″
3203:disphenoid
3174:disphenoid
3156:Disphenoid
3150:Disphenoid
3093:concurrent
2854:Projective
2723:8 ((1 2 3)
2619:isomorphic
2562:symmetries
1850:steradians
1671:horizontal
300:tessellate
281:truncation
274:rectifying
270:octahedron
78:polyhedron
66:tetrahedra
37:Tetraedron
17606:orthoplex
17528:9-simplex
17478:8-simplex
17428:7-simplex
17385:6-simplex
17355:5-simplex
17324:Tesseract
17052:birotunda
17042:bifrustum
16807:snub cube
16702:polyhedra
16634:antiprism
16339:Trihedron
16308:Polyhedra
16251:MathWorld
16177:1019-8385
16144:(1): 1–6.
16080:250435684
16028:Polyhedra
15937:MathWorld
15320:cite book
15266:1304.0179
15146:0096-3003
14985:Park 2016
14840:MathWorld
14603:symmetric
14295:resistors
14124:Chemistry
14118:airfields
13766:Euclidean
13052:, (*332)
13019:V∞.3.3.3
13013:V6.3.3.3
13010:V5.3.3.3
13007:V4.3.3.3
13004:V3.3.3.3
13001:V2.3.3.3
12588:antiprism
12475:Improper
12463:Hexagonal
12311:‖
12297:‖
12294:−
12285:‖
12271:‖
12258:‖
12244:‖
12241:−
12232:‖
12218:‖
12205:‖
12191:‖
12188:−
12179:‖
12165:‖
12090:−
12046:−
12002:−
11947:−
11813:−
11789:−
11768:−
11364:≤
11170:≥
11062:≥
11047:⋅
11023:⋅
10999:⋅
10975:⋅
10875:θ
10871:
10859:Δ
10849:Δ
10833:θ
10829:
10817:Δ
10807:Δ
10791:θ
10787:
10775:Δ
10765:Δ
10755:−
10741:Δ
10723:Δ
10705:Δ
10687:Δ
10616:θ
10562:Δ
10386:∠
10383:
10377:⋅
10365:∠
10362:
10356:⋅
10344:∠
10341:
10323:∠
10320:
10314:⋅
10302:∠
10299:
10293:⋅
10281:∠
10278:
10187:Aristotle
10028:−
10013:α
10005:
9987:α
9979:
9961:α
9953:
9933:α
9925:
9914:−
9899:α
9891:
9873:α
9865:
9845:α
9837:
9819:α
9811:
9800:−
9785:α
9777:
9757:α
9749:
9731:α
9723:
9705:α
9697:
9686:−
9529:−
9519:×
9445:−
9329:−
9307:−
9253:−
9219:−
9198:−
9144:−
9110:−
9089:−
9035:−
8850:−
8816:−
8782:−
8748:−
8707:opposite
8667:opposite
8627:opposite
8462:γ
8458:
8445:−
8441:β
8437:
8424:−
8420:α
8416:
8403:−
8399:γ
8395:
8388:β
8384:
8377:α
8373:
8298:−
8245:−
8192:−
8097:−
8074:−
8051:−
7488:⋅
7248:⋅
7234:= 0 then
7149:γ
7145:
7132:−
7128:β
7124:
7111:−
7107:α
7103:
7090:−
7086:γ
7082:
7075:β
7071:
7064:α
7060:
6853:β
6849:
6829:⋅
6813:α
6809:
6789:⋅
6773:γ
6769:
6749:⋅
6683:⋅
6668:⋅
6651:⋅
6622:⋅
6605:⋅
6590:⋅
6547:⋅
6264:⋅
6194:⋅
6133:−
6117:−
6101:−
5601:triangles
5599:isosceles
5521:triangles
5519:isosceles
5414:triangles
5324:triangles
5322:isosceles
5127:isosceles
5023:triangles
4205:35°15′52″
4186:𝜿
4137:≈
4047:≈
3957:≈
3860:π
3828:𝜿
3824:−
3815:π
3781:≈
3733:π
3701:𝜿
3697:−
3688:π
3623:π
3591:𝜿
3575:70°31′44″
3556:≈
3501:𝟁
3494:−
3491:π
3479:70°31′44″
3462:𝜿
3455:−
3452:π
3408:dihedral
3089:incenters
2936:conformal
2893:rectangle
2856:symmetry
2583:known as
2399:−
2390:−
2370:−
2355:−
2336:−
2327:−
2170:−
2152:−
2134:−
2104:−
2071:−
2042:−
1940:±
1900:−
1885:±
1852:, 1809.8
1832:π
1829:−
1808:
1771:
1762:−
1754:π
1724:
885:midsphere
806:∘
798:≈
780:
749:−
741:
726:∘
718:≈
695:
664:
614:≈
567:⋅
532:⋅
448:≈
393:⋅
132:Like all
112:3-simplex
105:Euclidean
45:(journal)
17660:Topics:
17623:demicube
17588:polytope
17582:Uniform
17343:600-cell
17339:120-cell
17292:Demicube
17266:Pentagon
17246:Triangle
17032:bicupola
17012:pyramids
16938:dihedron
16334:Dihedron
16125:(1619).
16010:(1973).
15996:(1948).
15816:(1965).
15780:(1875).
15501:7 August
15451:Lee 1997
15438:15558830
15026:(1989).
14684:See also
14635:skeleton
14552:Diameter
14522:Vertices
14489:HAL 9000
14471:monolith
14381:Pyraminx
14322:crystals
14312:used in
14299:soldered
14187:ammonium
14185:) or an
14134:ammonium
14116:At some
14112:Aviation
13975:compound
13907:{12i,3}
13773:Paraco.
13565:Paraco.
13559:Euclid.
13483:V3.4.3.4
13468:V3.3.3.3
13055:, (332)
13047:Symmetry
12829:digonal
12821:∞.3.3.3
12815:7.3.3.3
12812:6.3.3.3
12809:5.3.3.3
12806:4.3.3.3
12803:3.3.3.3
12800:2.3.3.3
12586:digonal
12478:Regular
12358:Centroid
12348:Centroid
11239:and the
11231:Inradius
10607:and let
10204:and the
9616:centroid
9608:bimedian
9600:centroid
8727:. Then,
7215:, while
6997:‖
6983:‖
6953:‖
6939:‖
6909:‖
6895:‖
6307:‖
6275:‖
6233:‖
6205:‖
4980:Symmetry
4971:diagram
3800:54°44′8″
3673:54°44′8″
3027:3-sphere
3023:600-cell
2988:600-cell
2759:, etc.;
2725:, etc.;
2544:Symmetry
2463:demicube
1687:centroid
917:exsphere
862:insphere
314:and the
293:kleetope
258:compound
149:insphere
127:triangle
90:vertices
54:geometry
17597:simplex
17567:10-cube
17334:24-cell
17320:16-cell
17261:Hexagon
17115:regular
17074:italics
17062:scutoid
17047:rotunda
17037:frustum
16766:uniform
16715:regular
16700:Convex
16654:pyramid
16639:frustum
16185:2154824
16072:3608204
15988:: 6–10.
15800:3571917
15788:Bibcode
15649:Bibcode
15403:2689983
15283:3647851
14709:simplex
14700:Caltrop
14599:regular
14391:Geology
14318:valence
14306:silicon
14225:
14213:
14171:methane
13979:origami
13973:. This
13916:{3i,3}
13913:{6i,3}
13910:{9i,3}
13388:sr{3,3}
13383:tr{3,3}
13378:rr{3,3}
12606:-gonal
12601:uniform
12443:Digonal
12427:pyramid
11672:. Let
11241:inradii
10958:. Then
10208:of the
10157:simplex
9364:bisects
6175:
6163:
5776:is the
5595:scalene
5412:scalene
5196:scalene
5078:{3,3}.
4863:). The
3320:smaller
3313:√
3250:√
3243:√
3214:simplex
3085:cevians
3019:16-cell
2750:
2727:
2621:to the
1685:is the
1664:√
1657:√
802:109.471
153:tangent
123:polygon
119:pyramid
108:simplex
76:, is a
17537:9-cube
17487:8-cube
17437:7-cube
17394:6-cube
17364:5-cube
17251:Square
17128:Family
17027:cupola
16980:duals:
16966:prisms
16644:cupola
16520:vertex
16222:
16183:
16175:
16097:
16078:
16070:
16039:
15969:
15832:
15798:
15684:26 May
15622:
15436:
15401:
15281:
15144:
14649:, and
14542:Radius
14304:Since
14207:. The
14052:by, a
13488:V4.6.6
13478:V3.3.3
13473:V3.6.6
13463:V3.6.6
13458:V3.3.3
13368:t{3,3}
13363:r{3,3}
13358:t{3,3}
12984:image
12943:image
12895:image
12593:digons
12453:Square
12136:
12124:
11967:
11118:, and
11098:, and
10951:, and
10926:, and
10674:. The
10143:, but
10061:where
9604:median
9377:, the
8888:where
8687:, and
8587:, and
8147:where
7961:, and
7901:, and
7171:where
6967:, and
6879:where
6727:
6723:
6533:Hence
6527:
6326:
6322:
6252:
6248:
5756:where
5710:Volume
4987:Schön.
3144:Kepler
3015:5-cell
2996:chiral
2901:wedges
2897:square
2881:square
2835:Image
1805:arccos
1768:arcsin
1721:arccos
1679:median
915:, and
777:arctan
738:arccos
722:70.529
692:arctan
661:arccos
224:convex
17256:p-gon
16649:wedge
16629:prism
16489:(132)
16197:(PDF)
16115:(PDF)
16076:S2CID
16068:JSTOR
15756:(PDF)
15745:(PDF)
15719:(PDF)
15704:(PDF)
15573:(PDF)
15399:JSTOR
15344:(PDF)
15312:(PDF)
15279:JSTOR
15261:arXiv
15247:(PDF)
14762:Notes
14639:graph
14562:Girth
14532:Edges
14342:Games
14231:Water
14189:ion (
13903:{∞,3}
13898:{8,3}
13893:{7,3}
13888:{6,3}
13883:{5,3}
13878:{4,3}
13873:{3,3}
13868:{2,3}
13373:{3,3}
13353:{3,3}
12850:name
12364:Faces
11961:where
6725:where
6324:where
6155:graph
5019:Four
4932:above
4916:nodes
4891:of a
4140:0.408
4050:0.707
3960:1.225
3784:0.577
3559:1.155
3402:edge
3220:of a
3128:In a
2879:is a
2830:Edge
1858:spats
1700:proof
1653:slope
947:are:
617:0.118
451:1.732
243:Plato
86:edges
82:faces
40:, or
17614:cube
17287:Cube
17117:and
16730:cube
16611:(57)
16582:(92)
16576:(13)
16570:(13)
16539:(16)
16515:edge
16510:face
16483:(90)
16477:(60)
16471:(48)
16465:(32)
16459:(30)
16453:(24)
16220:ISBN
16173:ISSN
16095:ISBN
16037:ISBN
15967:ISBN
15880:2012
15830:ISBN
15796:OCLC
15764:2009
15727:2006
15686:2024
15620:ISBN
15503:2018
15434:PMID
15326:link
15142:ISSN
14707:and
14633:The
14401:The
14383:and
14356:The
14297:are
14287:and
14132:The
14015:The
14008:The
13989:and
13732:,3}
13016:...
12933:...
12882:...
12818:...
12731:...
12676:...
12578:...
12531:...
12470:...
11429:and
11252:for
10906:Let
10445:Let
10163:cf.
10123:and
10079:The
10072:and
9633:The
9464:here
9432:and
9011:and
8001:is:
7861:Let
7838:and
7223:and
7211:and
7199:and
5778:base
5098:*332
5074:and
5002:Ord.
4997:Orb.
4992:Cox.
4967:Edge
3405:arc
3222:tree
3021:and
2730:1 ±
2558:cube
2500:and
2459:dual
2271:cube
1705:Its
1675:apex
645:Its
254:dual
247:fire
186:The
138:nets
56:, a
17163:(p)
16764:or
16599:(4)
16564:(5)
16533:(9)
16495:(∞)
16163:doi
16060:doi
15910:doi
15657:doi
15553:doi
15426:doi
15391:doi
15271:doi
15257:110
15132:doi
15128:472
15040:doi
14888:333
14653:, W
14645:, K
14473:in
14068:of
14044:In
13660:3.3
13504:{3,
12411:600
12408:185
12406:124
12401:120
12394:800
12392:436
12130:and
11402:If
10868:cos
10826:cos
10784:cos
10380:sin
10359:sin
10338:sin
10317:sin
10296:sin
10275:sin
10167:).
10002:cos
9976:cos
9950:cos
9922:cos
9888:cos
9862:cos
9834:cos
9808:cos
9774:cos
9746:cos
9720:cos
9694:cos
8861:192
8449:cos
8428:cos
8407:cos
8392:cos
8381:cos
8370:cos
7485:288
7262:det
7136:cos
7115:cos
7094:cos
7079:cos
7068:cos
7057:cos
6846:cos
6806:cos
6766:cos
6090:det
5669:to
5597:or
5564:*22
5485:222
5427:or
5385:2*2
5171:*33
5121:An
5105:12
5100:332
3845:60°
3756:𝟁
3718:60°
3646:𝝉
3608:60°
3531:𝟀
3413:𝒍
3208:An
3134:one
3098:An
3079:An
3005:In
2715:):
1923:and
1702:).
264:or
68:or
62:pl.
52:In
17683::
17668:•
17664:•
17644:21
17640:•
17637:k1
17633:•
17630:k2
17608:•
17565:•
17535:•
17513:21
17509:•
17506:41
17502:•
17499:42
17485:•
17463:21
17459:•
17456:31
17452:•
17449:32
17435:•
17413:21
17409:•
17406:22
17392:•
17362:•
17341:•
17322:•
17301:•
17285:•
17217:/
17206:/
17196:/
17187:/
17165:/
16622:s
16248:.
16207:16
16205:.
16199:.
16181:MR
16179:.
16171:.
16159:13
16157:.
16153:.
16140:.
16074:.
16066:.
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16035:.
16031:.
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15984:.
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15747:.
15712:75
15710:.
15706:.
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15645:22
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15581:11
15579:,
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15458:^
15432:.
15422:10
15420:.
15397:.
15387:54
15385:.
15361:.
15322:}}
15318:{{
15277:.
15269:.
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15179:68
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15080:".
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15036:17
15034:.
15030:.
14950:68
14911:^
14904:55
14837:.
14822:^
14787:^
14711:–
14665:.
14617:,
14613:,
14609:,
14605:,
14601:,
14597:,
14576:24
14387:.
14275:.
14233:,
14191:NH
14175:CH
14108:.
14092:,
14088:,
14084:,
14080:,
14076:,
13718:.
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13705:3
13702:3
13699:3
13526:}
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12595:.
12429:.
12414:.
12399:47
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11845:24
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11688:,
11680:,
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11660:.
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11110:,
11094:,
11090:,
10944:,
10937:,
10922:,
10918:,
10526:,
10499:,
10472:,
10260:,
10256:,
10252:,
10197:.
10111:,
10093:sr
10076:.
10065:ij
10017:34
9991:24
9965:14
9937:34
9903:23
9877:13
9849:24
9823:23
9789:12
9761:14
9735:13
9709:12
9591:.
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8567:.
8547:,
8527:,
8507:,
8133:12
7941:,
7881:,
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7832:,
7828:,
7824:,
7812:,
7772:34
7755:24
7738:14
7714:34
7692:23
7675:13
7651:24
7634:23
7612:12
7588:14
7571:13
7554:12
7474::
7227:.
7179:,
7175:,
6923:,
6544:36
6250:or
5660:A
5641:2
5638:22
5622:.
5615:,
5571:2
5566:22
5550:2v
5529:2v
5488:4
5392:4
5387:2×
5373:2d
5333:2d
5294:1
5252:2
5243:1v
5235:1h
5218:.
5211:,
5178:3
5173:33
5159:3v
5143:,
5137:3v
5103:24
5036:,
4938:.
4903:.
4836:A
4828:.
4795:,
4762:,
4696:,
4663:,
4597:,
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4505:,
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4310:,
4288:,
3326:.
3193:A
3172:A
3095:.
3076:.
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2813:.
2768:,
2764:,
2738:±
2734:±
2568:.
2540:.
2266:.
1860:.
1736:27
1733:23
1279:16
1014:24
883:,
860:,
295:.
214:A
151:)
140:.
114:.
96:.
64::
34:,
17652:-
17650:n
17642:k
17635:2
17628:1
17621:-
17619:n
17612:-
17610:n
17604:-
17602:n
17595:-
17593:n
17586:-
17584:n
17511:4
17504:2
17497:1
17461:3
17454:2
17447:1
17411:2
17404:1
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17231:H
17224:2
17221:G
17213:4
17210:F
17202:8
17199:E
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17190:E
17184:6
17181:E
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17168:D
17161:2
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17150:n
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17138:n
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17106:e
17099:t
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14713:n
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13738:v
13730:n
13726:n
13724:*
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13539:t
13532:v
13524:n
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13518:*
13506:n
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12840:n
12629:e
12622:t
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12315:2
12305:0
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12141:B
12118:)
12109:T
12104:]
12098:0
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12085:3
12081:x
12076:[
12065:T
12060:]
12054:0
12050:x
12041:2
12037:x
12032:[
12021:T
12016:]
12010:0
12006:x
11997:1
11993:x
11988:[
11979:(
11975:=
11972:A
11955:B
11950:1
11943:A
11939:=
11932:C
11914:3
11911:x
11908:,
11905:2
11902:x
11899:,
11896:1
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11890:,
11887:0
11884:x
11878:C
11854:.
11848:V
11840:)
11837:C
11834:c
11831:+
11828:B
11825:b
11822:+
11819:A
11816:a
11810:(
11807:)
11804:C
11801:c
11798:+
11795:B
11792:b
11786:A
11783:a
11780:(
11777:)
11774:C
11771:c
11765:B
11762:b
11759:+
11756:A
11753:a
11750:(
11747:)
11744:C
11741:c
11738:+
11735:B
11732:b
11729:+
11726:A
11723:a
11720:(
11714:=
11711:R
11698:V
11694:C
11690:B
11686:A
11682:c
11678:b
11674:a
11670:R
11648:r
11643:4
11639:A
11633:3
11630:1
11625:+
11622:r
11617:3
11613:A
11607:3
11604:1
11599:+
11596:r
11591:2
11587:A
11581:3
11578:1
11573:+
11570:r
11565:1
11561:A
11555:3
11552:1
11547:=
11544:V
11531:.
11514:4
11510:A
11506:+
11501:3
11497:A
11493:+
11488:2
11484:A
11480:+
11475:1
11471:A
11465:V
11462:3
11456:=
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11440:r
11435:4
11431:A
11426:3
11422:A
11417:2
11413:A
11408:1
11404:A
11384:,
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11357:2
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11332:2
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11314:+
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11282:2
11277:1
11273:r
11269:1
11254:i
11249:i
11245:r
11237:r
11215:.
11212:)
11209:M
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11203:+
11200:L
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11176:(
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11161:+
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11112:K
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11088:A
11071:.
11068:V
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11041:P
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11020:C
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10972:A
10969:P
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10942:b
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10928:D
10924:C
10920:B
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10912:V
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10761:(
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10750:2
10745:l
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10732:2
10727:k
10719:+
10714:2
10709:j
10701:=
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10660:j
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10650:i
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10623:j
10620:i
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10589:P
10566:i
10539:4
10535:P
10512:3
10508:P
10485:2
10481:P
10458:1
10454:P
10419:O
10398:.
10395:A
10392:B
10389:O
10374:B
10371:C
10368:O
10353:C
10350:A
10347:O
10335:=
10332:A
10329:C
10326:O
10311:C
10308:B
10305:O
10290:B
10287:A
10284:O
10262:C
10258:B
10254:A
10250:O
10145:O
10141:v
10137:M
10133:v
10129:v
10125:M
10121:O
10117:v
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10105:O
10101:O
10097:O
10089:O
10074:j
10070:i
10063:α
10045:0
10042:=
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10031:1
10022:)
10009:(
9996:)
9983:(
9970:)
9957:(
9942:)
9929:(
9917:1
9908:)
9895:(
9882:)
9869:(
9854:)
9841:(
9828:)
9815:(
9803:1
9794:)
9781:(
9766:)
9753:(
9740:)
9727:(
9714:)
9701:(
9689:1
9680:|
9658:T
9654:M
9650:T
9553:.
9548:6
9543:|
9539:)
9535:)
9532:c
9526:b
9523:(
9515:a
9511:(
9507:|
9503:d
9497:=
9494:V
9474:V
9449:c
9441:b
9420:a
9400:d
9344:.
9341:)
9338:W
9335:+
9332:v
9326:u
9323:(
9319:)
9316:v
9313:+
9310:u
9304:W
9301:(
9298:=
9295:z
9289:,
9286:)
9283:v
9280:+
9277:u
9274:+
9271:W
9268:(
9265:)
9262:u
9259:+
9256:W
9250:v
9247:(
9244:=
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9234:,
9231:)
9228:V
9225:+
9222:u
9216:w
9213:(
9210:)
9207:u
9204:+
9201:w
9195:V
9192:(
9189:=
9186:y
9180:,
9177:)
9174:u
9171:+
9168:w
9165:+
9162:V
9159:(
9156:)
9153:w
9150:+
9147:V
9141:u
9138:(
9135:=
9132:Y
9125:,
9122:)
9119:U
9116:+
9113:w
9107:v
9104:(
9101:)
9098:w
9095:+
9092:v
9086:U
9083:(
9080:=
9077:x
9071:,
9068:)
9065:w
9062:+
9059:v
9056:+
9053:U
9050:(
9047:)
9044:v
9041:+
9038:U
9032:w
9029:(
9026:=
9023:X
8995:,
8990:z
8987:y
8984:x
8979:=
8976:s
8970:,
8965:Y
8962:X
8959:z
8954:=
8951:r
8944:,
8939:X
8936:Z
8933:y
8928:=
8925:q
8919:,
8914:Z
8911:Y
8908:x
8903:=
8900:p
8873:w
8869:v
8865:u
8856:)
8853:s
8847:r
8844:+
8841:q
8838:+
8835:p
8832:(
8828:)
8825:s
8822:+
8819:r
8813:q
8810:+
8807:p
8804:(
8800:)
8797:s
8794:+
8791:r
8788:+
8785:q
8779:p
8776:(
8772:)
8769:s
8766:+
8763:r
8760:+
8757:q
8754:+
8751:p
8745:(
8738:=
8735:V
8715:W
8695:w
8675:V
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8635:U
8615:u
8595:w
8575:v
8555:u
8535:W
8515:V
8495:U
8453:2
8432:2
8411:2
8367:2
8364:+
8361:1
8354:6
8350:c
8347:b
8344:a
8338:=
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8311:.
8306:2
8302:z
8293:2
8289:b
8285:+
8280:2
8276:a
8272:=
8265:Z
8258:,
8253:2
8249:y
8240:2
8236:c
8232:+
8227:2
8223:a
8219:=
8212:Y
8205:,
8200:2
8196:x
8187:2
8183:c
8179:+
8174:2
8170:b
8166:=
8159:X
8129:Z
8126:Y
8123:X
8120:+
8115:2
8111:Z
8105:2
8101:c
8092:2
8088:Y
8082:2
8078:b
8069:2
8065:X
8059:2
8055:a
8046:2
8042:c
8036:2
8032:b
8026:2
8022:a
8018:4
8012:=
8009:V
7989:V
7969:z
7949:y
7929:x
7909:c
7889:b
7869:a
7840:d
7836:}
7834:d
7830:c
7826:b
7822:a
7820:{
7814:j
7810:i
7790:|
7784:0
7777:2
7768:d
7760:2
7751:d
7743:2
7734:d
7728:1
7719:2
7710:d
7704:0
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7688:d
7680:2
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7665:1
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7630:d
7624:0
7617:2
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7593:2
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7567:d
7559:2
7550:d
7544:0
7539:1
7532:1
7527:1
7522:1
7517:1
7512:0
7506:|
7501:=
7496:2
7492:V
7455:.
7450:|
7445:)
7438:1
7433:1
7428:1
7423:1
7414:3
7410:d
7402:3
7398:c
7390:3
7386:b
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7374:a
7364:2
7360:d
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7340:2
7336:b
7328:2
7324:a
7314:1
7310:d
7302:1
7298:c
7290:1
7286:b
7278:1
7274:a
7266:(
7258:|
7254:=
7251:V
7245:6
7232:d
7225:b
7221:a
7217:γ
7213:c
7209:a
7205:β
7201:c
7197:b
7193:d
7189:α
7185:d
7181:γ
7177:β
7173:α
7155:,
7140:2
7119:2
7098:2
7054:2
7051:+
7048:1
7041:6
7037:c
7034:b
7031:a
7025:=
7022:V
6990:c
6978:=
6975:c
6946:b
6934:=
6931:b
6902:a
6890:=
6887:a
6857:.
6843:c
6840:a
6837:=
6833:c
6825:a
6817:,
6803:c
6800:b
6797:=
6793:c
6785:b
6777:,
6763:b
6760:a
6757:=
6753:b
6745:a
6738:{
6708:|
6699:2
6695:c
6687:c
6679:b
6672:c
6664:a
6655:c
6647:b
6638:2
6634:b
6626:b
6618:a
6609:c
6601:a
6594:b
6586:a
6577:2
6573:a
6565:|
6560:=
6555:2
6551:V
6507:,
6504:)
6499:3
6495:c
6491:,
6486:2
6482:c
6478:,
6473:1
6469:c
6465:(
6462:=
6458:c
6450:,
6447:)
6442:3
6438:b
6434:,
6429:2
6425:b
6421:,
6416:1
6412:b
6408:(
6405:=
6401:b
6393:,
6390:)
6385:3
6381:a
6377:,
6372:2
6368:a
6364:,
6359:1
6355:a
6351:(
6348:=
6344:a
6337:{
6300:c
6291:b
6282:a
6270:=
6267:V
6261:6
6226:c
6219:b
6212:a
6200:=
6197:V
6191:6
6172:6
6169:/
6166:1
6141:)
6137:d
6129:c
6125:,
6121:d
6113:b
6109:,
6105:d
6097:a
6093:(
6085:6
6082:1
6053:.
6050:)
6045:3
6041:d
6037:,
6032:2
6028:d
6024:,
6019:1
6015:d
6011:(
6008:=
6000:d
5992:,
5989:)
5984:3
5980:c
5976:,
5971:2
5967:c
5963:,
5958:1
5954:c
5950:(
5947:=
5939:c
5931:,
5928:)
5923:3
5919:b
5915:,
5910:2
5906:b
5902:,
5897:1
5893:b
5889:(
5886:=
5878:b
5870:,
5867:)
5862:3
5858:a
5854:,
5849:2
5845:a
5841:,
5836:1
5832:a
5828:(
5825:=
5817:a
5788:h
5764:A
5744:.
5741:h
5738:A
5733:3
5730:1
5725:=
5722:V
5687:2
5683:/
5679:3
5632:2
5629:C
5620:2
5617:Z
5609:2
5606:C
5569:4
5557:2
5554:C
5547:C
5539:4
5536:V
5526:C
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5436:D
5432:2
5429:Z
5425:4
5422:V
5390:8
5378:4
5376:S
5370:D
5330:D
5291:1
5285:1
5283:C
5249:*
5240:C
5238:=
5232:C
5230:=
5227:s
5224:C
5216:2
5213:Z
5205:s
5202:C
5176:6
5164:3
5162:C
5156:C
5148:3
5145:S
5134:C
5091:T
5087:d
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5041:4
5038:S
5030:d
5027:T
4956:4
4952:3
4811:6
4808:1
4778:2
4775:3
4745:3
4742:4
4712:6
4709:1
4679:2
4676:1
4646:3
4643:1
4613:2
4610:1
4580:2
4577:3
4551:1
4521:6
4518:1
4488:3
4485:1
4459:1
4429:6
4426:1
4396:2
4393:1
4363:2
4360:3
4326:3
4323:1
4297:1
4271:3
4268:4
4230:2
4226:3
4130:6
4127:1
4100:l
4096:/
4092:R
4087:2
4040:2
4037:1
4010:l
4006:/
4002:R
3997:1
3950:2
3947:3
3920:l
3916:/
3912:R
3907:0
3863:3
3818:2
3774:3
3771:1
3736:3
3691:2
3657:1
3626:3
3587:2
3549:3
3546:4
3497:2
3458:2
3424:2
3315:3
3252:3
3245:2
2883:.
2808:2
2787:x
2783:x
2773:)
2770:k
2766:j
2762:i
2752:)
2747:2
2744:/
2740:k
2736:j
2732:i
2693:4
2689:A
2664:T
2637:4
2633:S
2602:d
2596:T
2528:}
2525:3
2522:,
2519:4
2516:{
2512:h
2443:2
2438:2
2414:.
2411:)
2408:1
2405:,
2402:1
2396:,
2393:1
2387:(
2379:,
2376:)
2373:1
2367:,
2364:1
2361:,
2358:1
2352:(
2345:,
2342:)
2339:1
2333:,
2330:1
2324:,
2321:1
2318:(
2310:,
2307:)
2304:1
2301:,
2298:1
2295:,
2292:1
2289:(
2252:3
2246:6
2241:2
2214:)
2211:1
2208:,
2205:0
2202:,
2199:0
2196:(
2188:,
2184:)
2178:3
2175:1
2167:,
2161:3
2158:2
2149:,
2143:9
2140:2
2130:(
2122:,
2118:)
2112:3
2109:1
2101:,
2095:3
2092:2
2086:,
2080:9
2077:2
2067:(
2060:,
2056:)
2050:3
2047:1
2039:,
2036:0
2033:,
2027:9
2024:8
2017:(
1992:y
1989:x
1962:)
1955:2
1951:1
1946:,
1943:1
1937:,
1934:0
1930:(
1916:)
1909:2
1905:1
1897:,
1894:0
1891:,
1888:1
1881:(
1825:)
1820:3
1817:1
1812:(
1802:3
1799:=
1788:)
1783:3
1780:1
1775:(
1765:3
1757:2
1749:=
1741:)
1728:(
1695:C
1691:C
1683:C
1666:2
1659:2
1633:.
1628:2
1623:)
1617:2
1612:4
1608:d
1604:+
1599:2
1594:3
1590:d
1586:+
1581:2
1576:2
1572:d
1568:+
1563:2
1558:1
1554:d
1550:+
1545:2
1541:a
1536:(
1531:=
1523:)
1517:4
1512:4
1508:d
1504:+
1499:4
1494:3
1490:d
1486:+
1481:4
1476:2
1472:d
1468:+
1463:4
1458:1
1454:d
1450:+
1445:4
1441:a
1436:(
1432:4
1425:,
1420:2
1415:)
1409:3
1403:2
1399:R
1395:2
1389:+
1384:4
1378:2
1373:4
1369:d
1365:+
1360:2
1355:3
1351:d
1347:+
1342:2
1337:2
1333:d
1329:+
1324:2
1319:1
1315:d
1307:(
1302:=
1293:9
1287:4
1283:R
1273:+
1268:4
1262:4
1257:4
1253:d
1249:+
1244:4
1239:3
1235:d
1231:+
1226:4
1221:2
1217:d
1213:+
1208:4
1203:1
1199:d
1169:i
1165:d
1144:R
1124:a
1100:.
1094:6
1090:a
1085:=
1079:E
1074:r
1067:,
1061:8
1057:a
1052:=
1047:R
1044:r
1039:=
1033:M
1028:r
1020:,
1010:a
1005:=
1002:R
997:3
994:1
989:=
986:r
980:,
977:a
972:4
968:6
962:=
959:R
932:E
927:r
900:M
895:r
871:r
848:R
811:.
794:)
789:2
784:(
774:2
771:=
763:)
757:3
754:1
745:(
731:,
714:)
708:2
703:2
699:(
689:=
681:)
676:3
673:1
668:(
630:.
625:3
621:a
606:2
601:6
595:3
591:a
585:=
582:a
577:3
573:6
563:)
557:2
553:a
547:4
543:3
536:(
527:3
524:1
519:=
516:V
496:a
491:3
487:6
464:.
459:2
455:a
443:3
436:2
432:a
428:=
424:)
418:2
414:a
408:4
404:3
397:(
390:4
387:=
384:A
364:A
344:a
60:(
47:.
20:)
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