1961:
5520:
5527:
5513:
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2249:
5541:
5455:
1541:
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5448:
1548:
1534:
5534:
5483:
1527:
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1506:
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1391:
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1520:
1513:
5427:
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1404:
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1425:
1418:
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1432:
348:
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1084:
2027:
582:
2045:
1567:
1245:
1585:
218:
29:
1973:
1069:
pentagonal antiprism between pentagon-to-triangle is 100.8°, and the dihedral angle of a pentagonal pyramid between the same faces is 37.4°. Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is 37.4° + 100.8° = 138.2°.
867:
1351:
can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges. Because there are five times as many icosahedron edges as cube faces, there are five
1989:
Dice are the common objects with the different polyhedron, one of them is the regular icosahedron. The twenty-sided dice was found in many ancient times. One example is the dice from the
Ptolemaic of Egypt, which was later the Greek letters inscribed on the faces in the period of Greece and Roman.
915:
is obtained by calculating the volume of all pyramids with the base of triangular faces and the height with the distance from a triangular face's centroid to the center inside the regular icosahedron, the circumradius of a regular icosahedron; alternatively, it can be ascertained by slicing it off
359:
Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular
1068:
of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces is approximately 38.2°. The dihedral angle of a
2331:
based on the placement of
Platonic solids in a concentric sequence of increasing radius of the inscribed and circumscribed spheres whose radii gave the distance of the six known planets from the common center. The ordering of the solids, from innermost to outermost, consisted of:
706:
511:
1038:
2776:, which are always in the golden ratio to the regular pentagon's edge. When a cube is inscribed in a dodecahedron and an icosahedron is inscribed in the cube, the dodecahedron and icosahedron that do not share any vertices have the same edge length.
1841:
Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive
2154:, by subdividing the net into triangles, followed by calculating the grid on the Earth's surface, and transferring the results from the sphere to the polyhedron. This projection was created during the time that Fuller realized that the
3894:
1040:
A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by
397:
1788:(another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron, and in a cube).
923:
2772:
Reciprocally, the edge length of a cube inscribed in a dodecahedron is in the golden ratio to the dodecahedron's edge length. The cube's edges lie in pentagonal face planes of the dodecahedron as
2018:
is another board game, where the player names the categories in the card with given the set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.
2311:
defined the
Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length. Following their identification with the elements by Plato,
1349:
1194:
is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the
862:{\displaystyle r_{I}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}\approx 0.756a,\qquad r_{C}={\frac {\sqrt {\varphi ^{2}+1}}{2}}a\approx 0.951a,\qquad r_{M}={\frac {\varphi }{2}}a\approx 0.809a.}
123:
2515:
561:
2547:
1149:
1774:
1743:
3392:
1837:
1155:. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points.
1057:
discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the
2570:
1943:
1234:
701:
674:
647:
379:. All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles. This process is known as
161:
2221:
1315:. Because the golden sections are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.
2008:) is commonly used in determining success or failure of an action. It may be numbered from "0" to "9" twice, in which form it usually serves as a ten-sided die (
2187:
913:
893:
620:
1874:-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the
1352:
ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the
4349:
2662:
308:. Other applications of the regular icosahedron are the usage of its net in cartography, twenty-sided dice that may have been found in ancient times and
1683:
has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces.
1854:
because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex
1304:
to each other. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.
285:
can be constructed by removing the pentagonal pyramids. The regular icosahedron has many relations with other
Platonic solids, one of them is the
1061:, but taken to different powers. As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).
3507:
2124:
1960:
2924:
1882:). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the
2227:. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.
2880:
3912:
2357:
1635:
but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles).
4342:
1202:
wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.
5327:
2392:
2940:
2811:
1372:
stated 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular
3750:
3636:
3588:
3569:
3517:
3492:
3324:
3259:
3201:
3929:
2791:
4335:
340:, such that the resulting polyhedron has 20 equilateral triangles as its faces. This process construction is known as the
5418:
4762:
3956:
2408:
3087:
2601:
2444:
506:{\displaystyle \left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right),}
3839:
3827:
3806:
3777:
3685:
3664:
3469:
3363:
3284:
3146:
3063:
129:
2852:
2726:
1321:
1692:
1668:
1700:
3420:
1033:{\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.}
87:
2480:
1887:
516:
3533:
3415:
3164:
3128:
1395:
The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this
3272:
Selfish, Scared and Stupid: Stop
Fighting Human Nature and Increase Your Performance, Engagement and Influence
2980:
2692:
2428:
2402:
2136:
2132:
1663:. Some of them are constructed involving the removal of the part of a regular icosahedron, a process known as
360:
planes, with edges drawn between each of them. Because of the constructions above, the regular icosahedron is
5363:
4254:
3099:
3055:
2950:
2894:
2821:
2698:
1680:
1373:
274:
2617:
2633:
1672:
1612:
1598:
1377:
2520:
1311:
by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two
1098:
5398:
5383:
4548:
4489:
1300:
Aside from comparing the mensuration between the regular icosahedron and regular dodecahedron, they are
300:
The appearance of regular icosahedron can be found in nature, such as the virus with icosahedral-shaped
5403:
5393:
5388:
5378:
4578:
4538:
4144:
3457:
3344:
3193:
1752:
1721:
1679:, which remove one, two, and three pentagonal pyramids from the icosahedron, respectively. The similar
293:
and has the historical background on the comparison mensuration. It also has many relations with other
5340:
4573:
4568:
3177:
2298:
2294:
1815:
1676:
254:
to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20
2754:
5592:
4785:
4138:
3137:
2323:
2286:
1647:
are the polyhedra whose faces are all regular, but not uniform. This means they do not include the
1269:
1195:
5353:
4755:
4679:
4674:
4553:
4459:
4266:
4194:
4114:
3949:
3918:
3902:
K.J.M. MacLean, A Geometric
Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
3338:
2290:
1914:
1898:
1287:, meaning that it contains a Hamiltonian cycle, or a cycle that visits each vertex exactly once.
321:
39:
5519:
1272:, meaning that the removal of any two of its vertices leaves a connected subgraph. According to
602:
of a convex polyhedron is a sphere tangent to every edge. Therefore, given that the edge length
5587:
5526:
5512:
5505:
5498:
5413:
4543:
4484:
4474:
4419:
4200:
1206:
2934:
2801:
2736:
2708:
2611:
2454:
2438:
2418:
2367:
2248:
598:
of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The
5299:
5292:
5285:
4563:
4479:
4434:
4382:
3695:
3299:
3132:
2862:
2676:
2643:
2627:
2588:
2470:
2147:
2000:
1879:
875:
of polyhedra is the sum of its every face. Therefore, the surface area of regular icosahedra
372:. There are only eight different convex deltahedra, one of which is the regular icosahedron.
5454:
4824:
4802:
4790:
2552:
1540:
5564:
5540:
5461:
5447:
4956:
4903:
4523:
4449:
4397:
3868:
3441:
3316:
Number, Shape, & Symmetry: An
Introduction to Number Theory, Geometry, and Group Theory
3276:
3172:
3156:
3076:(2006). "Coxeter Theory: The Cognitive Aspects". In Davis, Chandler; Ellers, Erich (eds.).
3040:
2337:
2306:
2224:
1921:
1883:
1749:
with an icosahedron as its base). Thus the 600-cell contains 120 icosahedra of edge length
1547:
1533:
1212:
1088:
1078:
1046:
917:
679:
652:
625:
391:
337:
325:
286:
255:
243:
202:
187:
139:
134:
80:
3449:
2200:
336:
to each of its faces. These pyramids cover the pentagonal faces, replacing them with five
8:
5582:
5560:
5358:
5311:
5210:
4689:
4558:
4533:
4518:
4454:
3885:
3334:
3097:
Buker, W. E.; Eggleton, R. B. (1969). "The
Platonic Solids (Solution to problem E2053)".
2341:
2128:
1893:
There are distortions of the icosahedron that, while no longer regular, are nevertheless
1863:
1708:
1277:
1236:
of size two, which is generated by the reflection through the center of the icosahedron.
1054:
267:
5482:
3541:
2990:
1631:. They all have 30 edges. The regular icosahedron and great dodecahedron share the same
1526:
273:
Many polyhedrons are constructed from the regular icosahedron. For example, most of the
5180:
5130:
5080:
5037:
5007:
4967:
4930:
4748:
4704:
4669:
4528:
4423:
4372:
4311:
4188:
4182:
3942:
3856:
3783:
3769:
3724:
3699:
3615:
3527:
3236:
3228:
3116:
2333:
2276:
2172:
1704:
1628:
1616:
1594:
1505:
1496:
1396:
1152:
1042:
898:
878:
605:
376:
329:
309:
247:
5533:
5468:
3906:
3901:
3306:
1776:. The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as
1519:
1512:
5408:
5319:
4684:
4494:
4469:
4413:
4301:
4225:
4177:
4150:
4120:
3823:
3802:
3773:
3746:
3729:
3681:
3660:
3632:
3584:
3565:
3513:
3488:
3465:
3359:
3320:
3280:
3255:
3240:
3197:
3142:
3083:
3073:
3059:
3032:
2834:
2773:
2116:
2108:
2092:, described its shells as the like-shaped various regular polyhedra; one of which is
1966:
Twenty-sided dice from
Ptolemaic of Egypt, inscribed with Greek letters at the faces.
1910:
1843:
1648:
1620:
1602:
1482:
1284:
1168:
70:
5491:
3787:
3598:
Minas-Nerpel, Martina (2007). "A Demotic
Inscribed Icosahedron from Dakhleh Oasis".
2779:
1878:), just as hexagons can be used as faces in semi-regular polyhedra (for example the
1489:
1454:
1390:
5426:
5323:
4888:
4877:
4866:
4855:
4846:
4837:
4776:
4772:
4623:
4306:
4286:
4108:
3852:
3848:
3765:
3719:
3711:
3607:
3484:
3445:
3429:
3401:
3220:
3108:
3028:
2317:
2112:
1995:
1656:
1632:
1461:
1431:
1417:
1403:
1273:
1261:
1205:
The full symmetry group of the icosahedron (including reflections) is known as the
1191:
594:
of a convex polyhedron is a sphere inside the polyhedron, touching every face. The
209:
5547:
5475:
5440:
5433:
3543:
Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade
2655:
Numerical values for the volumes of the inscribed Platonic solids may be found in
2197:
maximizing the smallest distance among the points, the minimum solution known for
1475:
1468:
1445:
1438:
1424:
1410:
4913:
4898:
4260:
4172:
4167:
4132:
4087:
4077:
4067:
4062:
3864:
3817:
3796:
3700:"New Ligand Platforms Featuring Boron-Rich Clusters as Organomimetic Sbstituents"
3675:
3555:
3437:
3353:
3314:
3270:
3249:
3152:
3049:
3036:
2312:
2254:
2166:
1994:, which was made out of golden and with numbers written on each face. In several
1867:
1847:
1804:
1781:
1777:
1301:
1268:, graphs that can be drawn in the plane without crossing its edges; and they are
1209:. It is isomorphic to the product of the rotational symmetry group and the group
1183:
1179:
895:
equals the area of 20 equilateral triangles. The volume of a regular icosahedron
380:
365:
333:
290:
251:
182:
60:
50:
3187:
1276:, the icosahedral graph endowed with these heretofore properties represents the
347:
5348:
5263:
4444:
4367:
4316:
4219:
4082:
4072:
3611:
3168:
2268:
2194:
2190:
1894:
1808:
1575:
1312:
1253:
1160:
1065:
361:
341:
259:
171:
167:
3924:
3406:
3387:
2271:. The regular polyhedra have been known since antiquity, but are named after
2060:
The regular icosahedron may also appear in many fields of science as follows:
5576:
5280:
5168:
5161:
5154:
5118:
5111:
5104:
5068:
5061:
4649:
4505:
4439:
4237:
4231:
4126:
4057:
4047:
3819:
Problems And Solutions For Groups, Lie Groups, Lie Algebras With Applications
3715:
3509:
Lectures on the ikosahedron and the solution of equations of the fifth degree
3294:
2159:
2100:
2089:
2069:
2014:
1712:
1652:
1644:
1172:
282:
3581:
A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra
2009:
355:
rectangles, with edges connecting their corners, form a regular icosahedron.
5220:
4052:
3733:
3561:
3433:
3082:. Providence, Rhode Island: American Mathematical Society. pp. 17–43.
2328:
2258:
2236:
2151:
2120:
2050:
2005:
1875:
1792:
1353:
1265:
1187:
1175:
1092:
1083:
1058:
872:
595:
564:
352:
5229:
5190:
5140:
5090:
5047:
5017:
4949:
4935:
4092:
4026:
4016:
4006:
4001:
3760:
Strauss, James H.; Strauss, Ellen G. (2008). "The Structure of Viruses".
3551:
3503:
2868:
2085:
2033:
1991:
1199:
394:
for the vertices of a regular icosahedron, giving the edge length 2, is:
369:
305:
263:
43:
3619:
2695:
for further history, and related symmetries on seven and eleven letters.
2026:
5556:
5215:
5199:
5149:
5099:
5056:
5026:
4940:
4714:
4602:
4392:
4359:
4277:
4031:
4021:
4011:
3996:
3986:
3965:
3860:
3557:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
3232:
3120:
2742:
2549:
is Coxeter's notation for the midradius, also noting that Coxeter uses
2104:
1947:
symmetry, i.e. have different planes of symmetry from the tetrahedron.
1851:
1365:
1308:
1164:
239:
197:
5555:
The stellation process on the icosahedron creates a number of related
2321:
sketched each of them, in particular, the regular icosahedron. In his
2044:
581:
5271:
5185:
5135:
5085:
5042:
5012:
4981:
4709:
4699:
4644:
4628:
4464:
4291:
2155:
1906:
1746:
1660:
1566:
1050:
599:
3224:
3112:
3077:
1745:, 20 of which meet at each vertex to form an icosahedral pyramid (a
1584:
1244:
368:. A polyhedron with only equilateral triangles as faces is called a
5245:
5000:
4996:
4923:
4595:
4327:
3991:
3181:. Vol. 6. University of Toronto Studies (Mathematical Series).
2081:
2065:
1902:
1855:
1785:
1696:
1624:
591:
294:
278:
227:
217:
3837:
Whyte, L. L. (1952). "Unique arrangements of points on a sphere".
2080:
is enclosed in a regular icosahedron, as is the head of a typical
1846:
for folding in three dimensions, icosahedra cannot be used as the
1686:
5254:
5224:
4991:
4986:
4977:
4918:
4719:
4694:
4296:
1982:
twenty-sided die, excluding the six letters Q, U, V, X, Y, and Z.
3248:
Dronskowski, Richard; Kikkawa, Shinichi; Stein, Andreas (2017).
2301:(cube) and the shape of the universe as a whole (dodecahedron).
1803:
As mentioned above, the regular icosahedron is unique among the
375:
The regular icosahedron can also be constructed starting from a
5194:
5144:
5094:
5051:
5021:
4972:
4908:
3743:
Viral Nanoparticles: Tools for Material Science and Biomedicine
2763:, pp. 8–9, §5. How to draw an icosahedron on a blackboard.
2302:
2267:
As mentioned above, the regular icosahedron is one of the five
2150:
used the net of a regular icosahedron to create a map known as
2073:
301:
258:
as its faces, 30 edges, and 12 vertices. It is an example of a
3393:
International Journal of Mathematics and Mathematical Sciences
2968:
1718:
The unit-radius 600-cell has tetrahedral cells of edge length
28:
3460:(2003) . Roach, Peter; Hartmann, James; Setter, Jane (eds.).
2272:
3934:
3629:
The Routledge International Handbook of Innovation Education
2223:
places the points at the vertices of a regular icosahedron,
622:
of a regular icosahedron, the radius of insphere (inradius)
4944:
4387:
2345:
1972:
920:, and adding up their volume. The expressions of both are:
3163:
2785:
2748:
1369:
3481:
Connections: The Geometric Bridge Between Art and Science
3373:
Hofmeister, H. (2004). "Fivefold Twinned Nanoparticles".
3019:
Berman, Martin (1971). "Regular-faced convex polyhedra".
2077:
3816:
Steeb, Willi-hans; Hardy, Yorick; Tanski, Igor (2012).
3247:
2930:
2285:, whose elementary units were attributed these shapes:
1798:
2912:
2096:, whose skeleton is shaped like a regular icosahedron.
2012:); most modern versions are labeled from "1" to "20".
1905:
as the tetrahedron, and are somewhat analogous to the
1755:
1724:
1324:
320:
The regular icosahedron can be constructed like other
2555:
2523:
2483:
2203:
2175:
2111:
with a shape resembling the regular icosahedron. The
1924:
1818:
1215:
1101:
926:
901:
881:
709:
682:
655:
628:
608:
519:
400:
142:
90:
3657:
Calculus Gems: Brief Lives and Memorable Mathematics
3141:(3rd ed.). Dover Publications. pp. 16–17.
2996:
2900:
2382:
2380:
3915:
A discussion of viral structure and the icosahedron
3741:Steinmitz, Nicole F.; Manchester, Marianne (2011).
3740:
2714:
2564:
2541:
2509:
2242:Sketch of a regular icosahedron by Johannes Kepler
2215:
2181:
1937:
1831:
1768:
1737:
1343:
1228:
1143:
1032:
907:
887:
861:
695:
668:
641:
614:
555:
505:
155:
117:
2956:
2533:
2493:
2377:
2169:, concerning the minimum-energy configuration of
1707:of two sizes, and each of its 120 vertices is an
5574:
3815:
2450:
1627:of the regular icosahedron. They share the same
3560:(1st trade paperback ed.). New York City:
3418:(1966). "Convex polyhedra with regular faces".
3268:
3251:Handbook of Solid State Chemistry, 6 Volume Set
2858:
1687:Relations to the 600-cell and other 4-polytopes
1344:{\textstyle {\frac {1}{\varphi }}\approx 0.618}
383:, and the regular icosahedron is also known as
3759:
3375:Encyclopedia of Nanoscience and Nanotechnology
2874:
4756:
4343:
3950:
3351:
3312:
3096:
2732:
2656:
2607:
2053:, created by the net of a regular icosahedron
1990:Another example was found in the treasure of
3597:
2805:
1151:angles, dividing a sphere into 120 triangle
649:, the radius of circumsphere (circumradius)
3801:. Wooden Books. Bloomsbury Publishing USA.
3355:A Mathematical History of the Golden Number
3313:Herrmann, Diane L.; Sally, Paul J. (2013).
2189:charged particles on a sphere, and for the
1870:and leave a positive defect for folding in
1295:
916:into two regular pentagonal pyramids and a
118:{\displaystyle 12\times \left(3^{5}\right)}
4763:
4749:
4350:
4336:
3957:
3943:
3372:
3133:"2.1 Regular polyhedra; 2.2 Reciprocation"
2918:
2510:{\displaystyle {}_{1}\!\mathrm {R} /\ell }
676:, and the radius of midsphere (midradius)
556:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
216:
27:
16:Convex polyhedron with 20 triangular faces
3886:"3D convex uniform polyhedra x3o5o – ike"
3723:
3645:
3626:
3464:. Cambridge: Cambridge University Press.
3405:
2414:
2363:
2281:dialogue, identified these with the five
3694:
3478:
3185:
2986:
2946:
2906:
2890:
2817:
2693:icosahedral symmetry: related geometries
2434:
2398:
1399:of the intersections in a single plane.
1243:
1082:
579:
346:
5328:List of regular polytopes and compounds
3822:. World Scientific Publishing Company.
3659:. Mathematical Association of America.
3654:
3578:
3414:
3385:
3340:The Thirteen Books of Euclid's Elements
3293:
3269:Flanagan, Kieran; Gregory, Dan (2015).
3211:Cundy, H. Martyn (1952). "Deltahedra".
3127:
3072:
2886:
2760:
2720:
2672:
2668:
2623:
2584:
2477:, Table I(i), pp. 292–293. See column "
2474:
2466:
266:. The icosahedral graph represents the
5575:
3794:
3388:"Hamiltonian paths on Platonic graphs"
3047:
3018:
2704:
2639:
2593:
2386:
2143:icosahedron as a basic structure unit.
1364:The icosahedron has a large number of
1307:An icosahedron can be inscribed in an
4331:
3938:
3925:Video of icosahedral mirror sculpture
3836:
3673:
3550:
3539:
3502:
3456:
3333:
3210:
3002:
2974:
2962:
2931:Dronskowski, Kikkawa & Stein 2017
2797:
2688:
2422:
2371:
1388:
4357:
3883:
2542:{\displaystyle {}_{1}\!\mathrm {R} }
2119:also occurs in crystals, especially
1799:Relations to other uniform polytopes
1290:
1239:
1144:{\displaystyle \pi /5,\pi /3,\pi /2}
3892:
3680:. Vol. 2. Dover Publications.
3600:The Journal of Egyptian Archaeology
2004:, the twenty-sided die (labeled as
1091:has 15 mirror planes (seen as cyan
13:
3974:Listed by number of faces and type
3921:– Models made with Modular Origami
3770:10.1016/b978-0-12-373741-0.50005-2
2535:
2495:
2327:, he also proposed a model of the
1769:{\textstyle {\frac {1}{\varphi }}}
1738:{\textstyle {\frac {1}{\varphi }}}
14:
5604:
3877:
3840:The American Mathematical Monthly
3798:Platonic & Archimedean Solids
3021:Journal of the Franklin Institute
1913:, including some forms which are
1095:on this sphere) meeting at order
586:3D model of a regular icosahedron
5546:
5539:
5532:
5525:
5518:
5511:
5504:
5497:
5490:
5481:
5474:
5467:
5460:
5453:
5446:
5439:
5432:
5425:
2247:
2235:
2043:
2025:
1971:
1959:
1669:gyroelongated pentagonal pyramid
1583:
1574:
1565:
1546:
1539:
1532:
1525:
1518:
1511:
1504:
1495:
1488:
1481:
1474:
1467:
1460:
1453:
1444:
1437:
1430:
1423:
1416:
1409:
1402:
1389:
3931:Principle of virus architecture
3895:"Dr Mike's Math Games for Kids"
3579:MacLean, Kenneth J. M. (2007).
3421:Canadian Journal of Mathematics
2827:
2766:
2682:
2649:
1832:{\displaystyle 138.19^{\circ }}
1703:. The 600-cell has icosahedral
1638:
972:
820:
766:
580:
315:
275:Kepler–Poinsot polyhedron
5341:stellations of the icosahedron
3853:10.1080/00029890.1952.11988207
3650:. Oxford University Publisher.
3462:English Pronouncing Dictionary
3358:. Courier Dover Publications.
2575:
2572:as the edge length (see p. 2).
2460:
2451:Steeb, Hardy & Tanski 2012
1950:
1791:A semiregular 4-polytope, the
1701:regular 4-dimensional polytope
1318:An icosahedron of edge length
1163:of the regular icosahedron is
575:
542:
526:
1:
4185:(two infinite groups and 75)
3964:
3909:The Encyclopedia of Polyhedra
3627:Shavinina, Larisa V. (2013).
3352:Herz-Fischler, Roger (2013).
3100:American Mathematical Monthly
3056:American Mathematical Society
3011:
2835:"Dungeons & Dragons Dice"
2076:. The outer protein shell of
1681:dissected regular icosahedron
1359:
570:
364:, a family of polyhedra with
351:Three mutually perpendicular
242:that can be constructed from
5334:
4730:Degenerate polyhedra are in
4203:(two infinite groups and 50)
3648:Geometry: Ancient and Modern
3051:Fundamentals of Graph Theory
3033:10.1016/0016-0032(71)90071-8
2257:Platonic solid model of the
1888:non-convex regular polychora
1673:metabidiminished icosahedron
1613:small stellated dodecahedron
1599:small stellated dodecahedron
1556:
7:
5399:Compound of five tetrahedra
5384:Medial triambic icosahedron
4549:pentagonal icositetrahedron
4490:truncated icosidodecahedron
3655:Simmons, George F. (2007).
3646:Silvester, John R. (2001).
3186:Cromwell, Peter R. (1997).
2859:Flanagan & Gregory 2015
1862:dimensions, at least three
1264:. This means that they are
1198:uses this simple fact, and
1186:on five letters. Since the
1072:
10:
5611:
5554:
5404:Compound of ten tetrahedra
5394:Compound of five octahedra
5389:Great triambic icosahedron
5379:Small triambic icosahedron
5337:
5317:
4744:
4579:pentagonal hexecontahedron
4539:deltoidal icositetrahedron
3745:. Pan Stanford Publisher.
3704:Pure and Applied Chemistry
3612:10.1177/030751330709300107
3532:: CS1 maint: postscript (
3345:Cambridge University Press
3194:Cambridge University Press
2875:Strauss & Strauss 2008
2774:regular pentagon diagonals
1780:formed by its unit-length
1378:regular compound polyhedra
1280:of a regular icosahedron.
1171:on five letters. This non-
1076:
270:of a regular icosahedron.
5503:
5438:
5367:
5357:
5352:
4728:
4662:
4637:
4619:
4612:
4587:
4574:disdyakis triacontahedron
4569:deltoidal hexecontahedron
4503:
4411:
4366:
4276:
4255:Kepler–Poinsot polyhedron
4247:
4212:
4160:
4101:
4040:
3979:
3972:
3907:Virtual Reality Polyhedra
3762:Viruses and Human Disease
3407:10.1155/S0161171204307118
3254:. John Sons & Wiley.
3178:The Fifty-Nine Icosahedra
2733:Herrmann & Sally 2013
2671:, See table II, line 4.;
2657:Buker & Eggleton 1969
1795:, has icosahedral cells.
1711:; the icosahedron is the
1677:tridiminished icosahedron
1374:Kepler–Poinsot polyhedron
1283:The icosahedral graph is
215:
208:
193:
181:
166:
128:
79:
69:
59:
49:
35:
26:
21:
3716:10.1351/PAC-CON-13-01-13
3674:Smith, David E. (1958).
3583:. Loving Healing Press.
3213:The Mathematical Gazette
2977:, p. 262, 478, 480.
2351:
2324:Mysterium Cosmographicum
1296:In other Platonic solids
1178:is the only non-trivial
322:gyroelongated bipyramids
4680:gyroelongated bipyramid
4554:rhombic triacontahedron
4460:truncated cuboctahedron
4267:Uniform star polyhedron
4195:quasiregular polyhedron
3512:. Courier Corporation.
3386:Hopkins, Brian (2004).
2336:, regular icosahedron,
1691:The icosahedron is the
390:One possible system of
177:138.190 (approximately)
40:Gyroelongated bipyramid
5414:Excavated dodecahedron
4675:truncated trapezohedra
4544:disdyakis dodecahedron
4510:(duals of Archimedean)
4485:rhombicosidodecahedron
4475:truncated dodecahedron
4201:semiregular polyhedron
3677:History of Mathematics
3479:Kappraff, Jay (1991).
3434:10.4153/cjm-1966-021-8
3048:Bickle, Allan (2020).
2877:, p. 35–62.
2566:
2565:{\displaystyle 2\ell }
2543:
2511:
2217:
2183:
2001:Dungeons & Dragons
1939:
1833:
1770:
1739:
1345:
1249:
1230:
1207:full icosahedral group
1156:
1145:
1034:
909:
889:
863:
697:
670:
643:
616:
587:
557:
507:
356:
157:
119:
4564:pentakis dodecahedron
4480:truncated icosahedron
4435:truncated tetrahedron
4248:non-convex polyhedron
3795:Sutton, Daud (2002).
3540:Klein, Felix (1884).
3300:Kunstformen der Natur
3277:John Wiley & Sons
2788:, p. 8–26.
2567:
2544:
2512:
2225:inscribed in a sphere
2218:
2184:
2148:R. Buckminster Fuller
2094:Circogonia icosahedra
2084:. Several species of
2037:Circogonia icosahedra
1940:
1938:{\displaystyle T_{h}}
1880:truncated icosahedron
1834:
1784:. In the unit-radius
1771:
1740:
1384:21 of 59 stellations
1370:Coxeter et al. (1938)
1346:
1247:
1231:
1229:{\displaystyle C_{2}}
1146:
1086:
1035:
910:
890:
864:
698:
696:{\displaystyle r_{M}}
671:
669:{\displaystyle r_{C}}
644:
642:{\displaystyle r_{I}}
617:
585:
558:
508:
350:
338:equilateral triangles
256:equilateral triangles
158:
156:{\displaystyle I_{h}}
120:
5565:icosahedral symmetry
5374:(Convex) icosahedron
4524:rhombic dodecahedron
4450:truncated octahedron
2553:
2521:
2481:
2338:regular dodecahedron
2216:{\displaystyle n=12}
2201:
2173:
1922:
1884:icosahedral 120-cell
1850:of a convex regular
1816:
1753:
1722:
1693:dimensional analogue
1322:
1213:
1196:Abel–Ruffini theorem
1099:
1089:Icosahedral symmetry
1079:Icosahedral symmetry
924:
918:pentagonal antiprism
899:
879:
707:
680:
653:
626:
606:
517:
398:
392:Cartesian coordinate
326:pentagonal antiprism
287:regular dodecahedron
244:pentagonal antiprism
188:Regular dodecahedron
140:
135:Icosahedral symmetry
88:
81:Vertex configuration
5312:pentagonal polytope
5211:Uniform 10-polytope
4771:Fundamental convex
4559:triakis icosahedron
4534:tetrakis hexahedron
4519:triakis tetrahedron
4455:rhombicuboctahedron
3884:Klitzing, Richard.
3309:for an online book.
2786:Coxeter et al. 1938
2749:Coxeter et al. 1938
2342:regular tetrahedron
2129:allotropes of boron
1709:icosahedral pyramid
1385:
1153:fundamental domains
1055:Apollonius of Perga
703:are, respectively:
330:pentagonal pyramids
248:pentagonal pyramids
232:regular icosahedron
22:Regular icosahedron
5354:Uniform duals
5181:Uniform 9-polytope
5131:Uniform 8-polytope
5081:Uniform 7-polytope
5038:Uniform 6-polytope
5008:Uniform 5-polytope
4968:Uniform polychoron
4931:Uniform polyhedron
4779:in dimensions 2–10
4529:triakis octahedron
4414:Archimedean solids
4189:regular polyhedron
4183:uniform polyhedron
4145:Hectotriadiohedron
3893:Hartley, Michael.
3538:, translated from
3416:Johnson, Norman W.
3171:; Flather, H. T.;
3079:The Coxeter Legacy
3074:Borovik, Alexandre
2608:Herz-Fischler 2013
2562:
2539:
2507:
2334:regular octahedron
2213:
2193:of constructing a
2179:
2117:icosahedral shapes
2099:In chemistry, the
1935:
1829:
1766:
1735:
1649:Archimedean solids
1629:vertex arrangement
1617:great dodecahedron
1595:great dodecahedron
1397:stellation diagram
1383:
1341:
1270:3-vertex-connected
1250:
1226:
1157:
1141:
1030:
905:
885:
859:
693:
666:
639:
612:
588:
553:
503:
377:regular octahedron
357:
310:role-playing games
277:is constructed by
153:
115:
5571:
5570:
5409:Great icosahedron
5359:Regular compounds
5333:
5332:
5320:Polytope families
4777:uniform polytopes
4739:
4738:
4658:
4657:
4495:snub dodecahedron
4470:icosidodecahedron
4325:
4324:
4226:Archimedean solid
4213:convex polyhedron
4121:Icosidodecahedron
3919:Origami Polyhedra
3752:978-981-4267-94-6
3638:978-0-203-38714-6
3590:978-1-932690-99-6
3571:978-0-7679-0816-0
3519:978-0-486-49528-6
3494:978-981-281-139-4
3400:(30): 1613–1616.
3326:978-1-4665-5464-1
3261:978-3-527-69103-6
3203:978-0-521-55432-9
3138:Regular Polytopes
3129:Coxeter, H. S. M.
2806:Minas-Nerpel 2007
2182:{\displaystyle n}
2072:have icosahedral
1996:roleplaying games
1911:snub dodecahedron
1886:, one of the ten
1811:is approximately
1778:interior features
1764:
1733:
1715:of the 600-cell.
1621:great icosahedron
1603:great icosahedron
1554:
1553:
1333:
1291:Related polyhedra
1258:icosahedral graph
1248:Icosahedral graph
1240:Icosahedral graph
1169:alternating group
999:
941:
908:{\displaystyle V}
888:{\displaystyle A}
842:
803:
799:
752:
749:
615:{\displaystyle a}
540:
328:by attaching two
324:, started from a
246:by attaching two
224:
223:
5600:
5550:
5543:
5536:
5529:
5522:
5515:
5508:
5501:
5494:
5485:
5478:
5471:
5464:
5457:
5450:
5443:
5436:
5429:
5419:Final stellation
5335:
5324:Regular polytope
4885:
4874:
4863:
4822:
4765:
4758:
4751:
4742:
4741:
4617:
4616:
4613:Dihedral uniform
4588:Dihedral regular
4511:
4427:
4376:
4352:
4345:
4338:
4329:
4328:
4161:elemental things
4139:Enneacontahedron
4109:Icositetrahedron
3959:
3952:
3945:
3936:
3935:
3898:
3889:
3872:
3833:
3812:
3791:
3756:
3737:
3727:
3691:
3670:
3651:
3642:
3623:
3594:
3575:
3547:
3537:
3531:
3523:
3498:
3485:World Scientific
3483:(2nd ed.).
3475:
3453:
3411:
3409:
3382:
3369:
3348:
3343:(3rd ed.).
3335:Heath, Thomas L.
3330:
3304:
3290:
3265:
3244:
3219:(318): 263–266.
3207:
3182:
3160:
3124:
3093:
3069:
3044:
3006:
3000:
2994:
2984:
2978:
2972:
2966:
2960:
2954:
2944:
2938:
2928:
2922:
2916:
2910:
2904:
2898:
2884:
2878:
2872:
2866:
2856:
2850:
2849:
2847:
2845:
2831:
2825:
2815:
2809:
2795:
2789:
2783:
2777:
2770:
2764:
2758:
2752:
2746:
2740:
2730:
2724:
2718:
2712:
2702:
2696:
2686:
2680:
2666:
2660:
2653:
2647:
2637:
2631:
2621:
2615:
2605:
2599:
2579:
2573:
2571:
2569:
2568:
2563:
2548:
2546:
2545:
2540:
2538:
2532:
2531:
2526:
2516:
2514:
2513:
2508:
2503:
2498:
2492:
2491:
2486:
2464:
2458:
2448:
2442:
2432:
2426:
2412:
2406:
2396:
2390:
2384:
2375:
2361:
2318:Harmonices Mundi
2251:
2239:
2222:
2220:
2219:
2214:
2188:
2186:
2185:
2180:
2158:is smaller than
2146:In cartography,
2113:crystal twinning
2047:
2029:
1975:
1963:
1946:
1944:
1942:
1941:
1936:
1934:
1933:
1840:
1838:
1836:
1835:
1830:
1828:
1827:
1807:in possessing a
1775:
1773:
1772:
1767:
1765:
1757:
1744:
1742:
1741:
1736:
1734:
1726:
1633:edge arrangement
1587:
1578:
1569:
1550:
1543:
1536:
1529:
1522:
1515:
1508:
1499:
1492:
1485:
1478:
1471:
1464:
1457:
1448:
1441:
1434:
1427:
1420:
1413:
1406:
1393:
1386:
1382:
1350:
1348:
1347:
1342:
1334:
1326:
1274:Steinitz theorem
1262:polyhedral graph
1256:, including the
1235:
1233:
1232:
1227:
1225:
1224:
1192:quintic equation
1150:
1148:
1147:
1142:
1137:
1123:
1109:
1053:, among others.
1039:
1037:
1036:
1031:
1026:
1025:
1010:
1009:
1000:
995:
994:
993:
980:
968:
967:
952:
951:
942:
937:
914:
912:
911:
906:
894:
892:
891:
886:
868:
866:
865:
860:
843:
835:
830:
829:
804:
792:
791:
782:
781:
776:
775:
753:
751:
750:
745:
739:
735:
734:
724:
719:
718:
702:
700:
699:
694:
692:
691:
675:
673:
672:
667:
665:
664:
648:
646:
645:
640:
638:
637:
621:
619:
618:
613:
584:
562:
560:
559:
554:
549:
541:
536:
512:
510:
509:
504:
499:
495:
465:
461:
431:
427:
220:
162:
160:
159:
154:
152:
151:
124:
122:
121:
116:
114:
110:
109:
31:
19:
18:
5610:
5609:
5603:
5602:
5601:
5599:
5598:
5597:
5593:Platonic solids
5573:
5572:
5303:
5296:
5289:
5172:
5165:
5158:
5122:
5115:
5108:
5072:
5065:
4899:Regular polygon
4892:
4883:
4876:
4872:
4865:
4861:
4852:
4843:
4836:
4832:
4820:
4814:
4810:
4798:
4780:
4769:
4740:
4735:
4724:
4663:Dihedral others
4654:
4633:
4608:
4583:
4512:
4509:
4508:
4499:
4428:
4417:
4416:
4407:
4370:
4368:Platonic solids
4362:
4356:
4326:
4321:
4272:
4261:Star polyhedron
4243:
4208:
4156:
4133:Hexecontahedron
4115:Triacontahedron
4097:
4088:Enneadecahedron
4078:Heptadecahedron
4068:Pentadecahedron
4063:Tetradecahedron
4036:
3975:
3968:
3963:
3880:
3875:
3830:
3809:
3780:
3753:
3696:Spokoyny, A. M.
3688:
3667:
3639:
3591:
3572:
3525:
3524:
3522:, Dover edition
3520:
3495:
3472:
3366:
3327:
3287:
3262:
3225:10.2307/3608204
3204:
3169:du Val, Patrick
3165:Coxeter, H.S.M.
3149:
3113:10.2307/2317282
3090:
3066:
3014:
3009:
3001:
2997:
2985:
2981:
2973:
2969:
2961:
2957:
2945:
2941:
2929:
2925:
2919:Hofmeister 2004
2917:
2913:
2905:
2901:
2885:
2881:
2873:
2869:
2857:
2853:
2843:
2841:
2833:
2832:
2828:
2816:
2812:
2796:
2792:
2784:
2780:
2771:
2767:
2759:
2755:
2747:
2743:
2731:
2727:
2719:
2715:
2703:
2699:
2687:
2683:
2667:
2663:
2654:
2650:
2638:
2634:
2622:
2618:
2606:
2602:
2598:
2580:
2576:
2554:
2551:
2550:
2534:
2527:
2525:
2524:
2522:
2519:
2518:
2499:
2494:
2487:
2485:
2484:
2482:
2479:
2478:
2465:
2461:
2449:
2445:
2433:
2429:
2413:
2409:
2397:
2393:
2385:
2378:
2362:
2358:
2354:
2313:Johannes Kepler
2297:(icosahedron),
2289:(tetrahedron),
2269:Platonic solids
2265:
2264:
2263:
2262:
2261:
2252:
2244:
2243:
2240:
2202:
2199:
2198:
2174:
2171:
2170:
2167:Thomson problem
2142:
2139:contain boron B
2058:
2057:
2056:
2055:
2054:
2048:
2040:
2039:
2030:
1987:
1986:
1985:
1984:
1983:
1976:
1968:
1967:
1964:
1953:
1929:
1925:
1923:
1920:
1919:
1918:
1901:under the same
1866:must meet at a
1823:
1819:
1817:
1814:
1813:
1812:
1805:Platonic solids
1801:
1756:
1754:
1751:
1750:
1725:
1723:
1720:
1719:
1689:
1641:
1609:
1608:
1607:
1606:
1590:
1589:
1588:
1580:
1579:
1571:
1570:
1559:
1394:
1362:
1325:
1323:
1320:
1319:
1313:golden sections
1298:
1293:
1242:
1220:
1216:
1214:
1211:
1210:
1190:of the general
1184:symmetric group
1180:normal subgroup
1159:The rotational
1133:
1119:
1105:
1100:
1097:
1096:
1081:
1075:
1021:
1017:
1005:
1001:
989:
985:
981:
979:
963:
959:
947:
943:
936:
925:
922:
921:
900:
897:
896:
880:
877:
876:
834:
825:
821:
787:
783:
780:
771:
767:
744:
740:
730:
726:
725:
723:
714:
710:
708:
705:
704:
687:
683:
681:
678:
677:
660:
656:
654:
651:
650:
633:
629:
627:
624:
623:
607:
604:
603:
578:
573:
545:
535:
518:
515:
514:
473:
469:
439:
435:
405:
401:
399:
396:
395:
385:snub octahedron
318:
291:dual polyhedron
201:
183:Dual polyhedron
147:
143:
141:
138:
137:
105:
101:
97:
89:
86:
85:
42:
17:
12:
11:
5:
5608:
5607:
5596:
5595:
5590:
5585:
5569:
5568:
5552:
5551:
5544:
5537:
5530:
5523:
5516:
5509:
5502:
5495:
5487:
5486:
5479:
5472:
5465:
5458:
5451:
5444:
5437:
5430:
5422:
5421:
5416:
5411:
5406:
5401:
5396:
5391:
5386:
5381:
5376:
5370:
5369:
5366:
5361:
5356:
5351:
5345:
5344:
5331:
5330:
5315:
5314:
5305:
5301:
5294:
5287:
5283:
5274:
5257:
5248:
5237:
5236:
5234:
5232:
5227:
5218:
5213:
5207:
5206:
5204:
5202:
5197:
5188:
5183:
5177:
5176:
5174:
5170:
5163:
5156:
5152:
5147:
5138:
5133:
5127:
5126:
5124:
5120:
5113:
5106:
5102:
5097:
5088:
5083:
5077:
5076:
5074:
5070:
5063:
5059:
5054:
5045:
5040:
5034:
5033:
5031:
5029:
5024:
5015:
5010:
5004:
5003:
4994:
4989:
4984:
4975:
4970:
4964:
4963:
4954:
4952:
4947:
4938:
4933:
4927:
4926:
4921:
4916:
4911:
4906:
4901:
4895:
4894:
4890:
4886:
4881:
4870:
4859:
4850:
4841:
4834:
4828:
4818:
4812:
4806:
4800:
4794:
4788:
4782:
4781:
4770:
4768:
4767:
4760:
4753:
4745:
4737:
4736:
4729:
4726:
4725:
4723:
4722:
4717:
4712:
4707:
4702:
4697:
4692:
4687:
4682:
4677:
4672:
4666:
4664:
4660:
4659:
4656:
4655:
4653:
4652:
4647:
4641:
4639:
4635:
4634:
4632:
4631:
4626:
4620:
4614:
4610:
4609:
4607:
4606:
4599:
4591:
4589:
4585:
4584:
4582:
4581:
4576:
4571:
4566:
4561:
4556:
4551:
4546:
4541:
4536:
4531:
4526:
4521:
4515:
4513:
4506:Catalan solids
4504:
4501:
4500:
4498:
4497:
4492:
4487:
4482:
4477:
4472:
4467:
4462:
4457:
4452:
4447:
4445:truncated cube
4442:
4437:
4431:
4429:
4412:
4409:
4408:
4406:
4405:
4400:
4395:
4390:
4385:
4379:
4377:
4364:
4363:
4355:
4354:
4347:
4340:
4332:
4323:
4322:
4320:
4319:
4317:parallelepiped
4314:
4309:
4304:
4299:
4294:
4289:
4283:
4281:
4274:
4273:
4271:
4270:
4264:
4258:
4251:
4249:
4245:
4244:
4242:
4241:
4235:
4229:
4223:
4220:Platonic solid
4216:
4214:
4210:
4209:
4207:
4206:
4205:
4204:
4198:
4192:
4180:
4175:
4170:
4164:
4162:
4158:
4157:
4155:
4154:
4148:
4142:
4136:
4130:
4124:
4118:
4112:
4105:
4103:
4099:
4098:
4096:
4095:
4090:
4085:
4083:Octadecahedron
4080:
4075:
4073:Hexadecahedron
4070:
4065:
4060:
4055:
4050:
4044:
4042:
4038:
4037:
4035:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3983:
3981:
3977:
3976:
3973:
3970:
3969:
3962:
3961:
3954:
3947:
3939:
3933:
3932:
3927:
3922:
3916:
3910:
3904:
3899:
3890:
3879:
3878:External links
3876:
3874:
3873:
3847:(9): 606–611.
3834:
3828:
3813:
3807:
3792:
3778:
3757:
3751:
3738:
3710:(5): 903–919.
3692:
3686:
3671:
3665:
3652:
3643:
3637:
3624:
3606:(1): 137–148.
3595:
3589:
3576:
3570:
3562:Broadway Books
3548:
3518:
3500:
3493:
3476:
3470:
3454:
3412:
3383:
3370:
3364:
3349:
3331:
3325:
3310:
3291:
3285:
3266:
3260:
3245:
3208:
3202:
3183:
3161:
3147:
3125:
3094:
3089:978-0821837221
3088:
3070:
3064:
3045:
3027:(5): 329–352.
3015:
3013:
3010:
3008:
3007:
3005:, p. 147.
2995:
2979:
2967:
2955:
2939:
2923:
2911:
2899:
2879:
2867:
2851:
2826:
2810:
2790:
2778:
2765:
2753:
2741:
2725:
2713:
2697:
2681:
2661:
2648:
2632:
2616:
2600:
2597:
2596:
2591:
2581:
2574:
2561:
2558:
2537:
2530:
2506:
2502:
2497:
2490:
2459:
2443:
2427:
2415:Shavinina 2013
2407:
2391:
2376:
2364:Silvester 2001
2355:
2353:
2350:
2293:(octahedron),
2253:
2246:
2245:
2241:
2234:
2233:
2232:
2231:
2230:
2229:
2228:
2212:
2209:
2206:
2195:spherical code
2191:Tammes problem
2178:
2163:
2144:
2140:
2137:β-rhombohedral
2097:
2088:discovered by
2049:
2042:
2041:
2031:
2024:
2023:
2022:
2021:
2020:
1977:
1970:
1969:
1965:
1958:
1957:
1956:
1955:
1954:
1952:
1949:
1932:
1928:
1917:and some with
1895:vertex-uniform
1826:
1822:
1809:dihedral angle
1800:
1797:
1763:
1760:
1732:
1729:
1705:cross sections
1688:
1685:
1653:Catalan solids
1645:Johnson solids
1640:
1637:
1592:
1591:
1582:
1581:
1573:
1572:
1564:
1563:
1562:
1561:
1560:
1558:
1555:
1552:
1551:
1544:
1537:
1530:
1523:
1516:
1509:
1501:
1500:
1493:
1486:
1479:
1472:
1465:
1458:
1450:
1449:
1442:
1435:
1428:
1421:
1414:
1407:
1400:
1361:
1358:
1340:
1337:
1332:
1329:
1297:
1294:
1292:
1289:
1254:Platonic graph
1241:
1238:
1223:
1219:
1161:symmetry group
1140:
1136:
1132:
1129:
1126:
1122:
1118:
1115:
1112:
1108:
1104:
1077:Main article:
1074:
1071:
1066:dihedral angle
1029:
1024:
1020:
1016:
1013:
1008:
1004:
998:
992:
988:
984:
978:
975:
971:
966:
962:
958:
955:
950:
946:
940:
935:
932:
929:
904:
884:
858:
855:
852:
849:
846:
841:
838:
833:
828:
824:
819:
816:
813:
810:
807:
802:
798:
795:
790:
786:
779:
774:
770:
765:
762:
759:
756:
748:
743:
738:
733:
729:
722:
717:
713:
690:
686:
663:
659:
636:
632:
611:
577:
574:
572:
569:
552:
548:
544:
539:
534:
531:
528:
525:
522:
502:
498:
494:
491:
488:
485:
482:
479:
476:
472:
468:
464:
460:
457:
454:
451:
448:
445:
442:
438:
434:
430:
426:
423:
420:
417:
414:
411:
408:
404:
362:Platonic solid
342:gyroelongation
317:
314:
283:Johnson solids
281:. Some of the
260:Platonic solid
238:) is a convex
222:
221:
213:
212:
206:
205:
195:
191:
190:
185:
179:
178:
175:
168:Dihedral angle
164:
163:
150:
146:
132:
130:Symmetry group
126:
125:
113:
108:
104:
100:
96:
93:
83:
77:
76:
73:
67:
66:
63:
57:
56:
53:
47:
46:
37:
33:
32:
24:
23:
15:
9:
6:
4:
3:
2:
5606:
5605:
5594:
5591:
5589:
5588:Planar graphs
5586:
5584:
5581:
5580:
5578:
5566:
5562:
5558:
5553:
5549:
5545:
5542:
5538:
5535:
5531:
5528:
5524:
5521:
5517:
5514:
5510:
5507:
5500:
5496:
5493:
5489:
5488:
5484:
5480:
5477:
5473:
5470:
5466:
5463:
5459:
5456:
5452:
5449:
5445:
5442:
5435:
5431:
5428:
5424:
5423:
5420:
5417:
5415:
5412:
5410:
5407:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5371:
5365:
5362:
5360:
5355:
5350:
5347:
5346:
5343:
5342:
5336:
5329:
5325:
5321:
5316:
5313:
5309:
5306:
5304:
5297:
5290:
5284:
5282:
5278:
5275:
5273:
5269:
5265:
5261:
5258:
5256:
5252:
5249:
5247:
5243:
5239:
5238:
5235:
5233:
5231:
5228:
5226:
5222:
5219:
5217:
5214:
5212:
5209:
5208:
5205:
5203:
5201:
5198:
5196:
5192:
5189:
5187:
5184:
5182:
5179:
5178:
5175:
5173:
5166:
5159:
5153:
5151:
5148:
5146:
5142:
5139:
5137:
5134:
5132:
5129:
5128:
5125:
5123:
5116:
5109:
5103:
5101:
5098:
5096:
5092:
5089:
5087:
5084:
5082:
5079:
5078:
5075:
5073:
5066:
5060:
5058:
5055:
5053:
5049:
5046:
5044:
5041:
5039:
5036:
5035:
5032:
5030:
5028:
5025:
5023:
5019:
5016:
5014:
5011:
5009:
5006:
5005:
5002:
4998:
4995:
4993:
4990:
4988:
4987:Demitesseract
4985:
4983:
4979:
4976:
4974:
4971:
4969:
4966:
4965:
4962:
4958:
4955:
4953:
4951:
4948:
4946:
4942:
4939:
4937:
4934:
4932:
4929:
4928:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4896:
4893:
4887:
4884:
4880:
4873:
4869:
4862:
4858:
4853:
4849:
4844:
4840:
4835:
4833:
4831:
4827:
4817:
4813:
4811:
4809:
4805:
4801:
4799:
4797:
4793:
4789:
4787:
4784:
4783:
4778:
4774:
4766:
4761:
4759:
4754:
4752:
4747:
4746:
4743:
4733:
4727:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4667:
4665:
4661:
4651:
4648:
4646:
4643:
4642:
4640:
4636:
4630:
4627:
4625:
4622:
4621:
4618:
4615:
4611:
4605:
4604:
4600:
4598:
4597:
4593:
4592:
4590:
4586:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4516:
4514:
4507:
4502:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4440:cuboctahedron
4438:
4436:
4433:
4432:
4430:
4425:
4421:
4415:
4410:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4380:
4378:
4374:
4369:
4365:
4361:
4353:
4348:
4346:
4341:
4339:
4334:
4333:
4330:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4284:
4282:
4279:
4275:
4268:
4265:
4262:
4259:
4256:
4253:
4252:
4250:
4246:
4239:
4238:Johnson solid
4236:
4233:
4232:Catalan solid
4230:
4227:
4224:
4221:
4218:
4217:
4215:
4211:
4202:
4199:
4196:
4193:
4190:
4187:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4165:
4163:
4159:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4127:Hexoctahedron
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4106:
4104:
4100:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4058:Tridecahedron
4056:
4054:
4051:
4049:
4048:Hendecahedron
4046:
4045:
4043:
4039:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3984:
3982:
3978:
3971:
3967:
3960:
3955:
3953:
3948:
3946:
3941:
3940:
3937:
3930:
3928:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3903:
3900:
3896:
3891:
3887:
3882:
3881:
3870:
3866:
3862:
3858:
3854:
3850:
3846:
3842:
3841:
3835:
3831:
3829:9789813104112
3825:
3821:
3820:
3814:
3810:
3808:9780802713865
3804:
3800:
3799:
3793:
3789:
3785:
3781:
3779:9780123737410
3775:
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3687:0-486-20430-8
3683:
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3672:
3668:
3666:9780883855614
3662:
3658:
3653:
3649:
3644:
3640:
3634:
3631:. Routledge.
3630:
3625:
3621:
3617:
3613:
3609:
3605:
3601:
3596:
3592:
3586:
3582:
3577:
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3471:3-12-539683-2
3467:
3463:
3459:
3458:Jones, Daniel
3455:
3451:
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3365:9780486152325
3361:
3357:
3356:
3350:
3346:
3342:
3341:
3336:
3332:
3328:
3322:
3319:. CRC Press.
3318:
3317:
3311:
3308:
3302:
3301:
3296:
3292:
3288:
3286:9780730312796
3282:
3278:
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3174:
3173:Petrie, J. F.
3170:
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3150:
3148:0-486-61480-8
3144:
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3071:
3067:
3065:9781470455491
3061:
3057:
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3042:
3038:
3034:
3030:
3026:
3022:
3017:
3016:
3004:
2999:
2992:
2988:
2987:Cromwell 1997
2983:
2976:
2971:
2964:
2959:
2952:
2948:
2947:Cromwell 1997
2943:
2936:
2935:435–436
2932:
2927:
2920:
2915:
2908:
2907:Spokoyny 2013
2903:
2896:
2892:
2891:Cromwell 1997
2888:
2883:
2876:
2871:
2864:
2860:
2855:
2840:
2836:
2830:
2823:
2819:
2818:Cromwell 1997
2814:
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2500:
2488:
2476:
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2468:
2463:
2456:
2452:
2447:
2440:
2436:
2435:Kappraff 1991
2431:
2424:
2420:
2416:
2411:
2404:
2400:
2399:Cromwell 1997
2395:
2388:
2383:
2381:
2373:
2369:
2368:140–141
2365:
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2160:South America
2157:
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2130:
2126:
2122:
2121:nanoparticles
2118:
2114:
2110:
2106:
2102:
2098:
2095:
2091:
2090:Ernst Haeckel
2087:
2083:
2079:
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2071:
2067:
2063:
2062:
2061:
2052:
2046:
2038:
2035:
2028:
2019:
2017:
2016:
2015:Scattergories
2011:
2007:
2003:
2002:
1997:
1993:
1981:
1980:Scattergories
1974:
1962:
1948:
1930:
1926:
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1713:vertex figure
1710:
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1266:planar graphs
1263:
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1093:great circles
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393:
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367:
366:regular faces
363:
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334:regular faces
331:
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252:regular faces
249:
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38:
34:
30:
25:
20:
5373:
5364:Regular star
5338:
5307:
5276:
5267:
5259:
5250:
5241:
5221:10-orthoplex
4960:
4957:Dodecahedron
4878:
4867:
4856:
4847:
4838:
4829:
4825:
4815:
4807:
4803:
4795:
4791:
4731:
4650:trapezohedra
4601:
4594:
4402:
4398:dodecahedron
4151:Apeirohedron
4102:>20 faces
4053:Dodecahedron
3844:
3838:
3818:
3797:
3764:. Elsevier.
3761:
3742:
3707:
3703:
3676:
3656:
3647:
3628:
3603:
3599:
3580:
3556:
3552:Livio, Mario
3542:
3508:
3504:Klein, Felix
3499:</ref>
3480:
3461:
3425:
3419:
3397:
3391:
3378:
3374:
3354:
3339:
3315:
3303:(in German).
3298:
3271:
3250:
3216:
3212:
3188:
3176:
3136:
3104:
3098:
3078:
3050:
3024:
3020:
2998:
2982:
2970:
2958:
2942:
2926:
2914:
2902:
2887:Haeckel 1904
2882:
2870:
2854:
2842:. Retrieved
2838:
2829:
2813:
2793:
2781:
2768:
2761:Borovik 2006
2756:
2751:, p. 4.
2744:
2728:
2721:Hopkins 2004
2716:
2700:
2684:
2673:MacLean 2007
2669:Johnson 1966
2664:
2651:
2635:
2624:Simmons 2007
2619:
2603:
2585:MacLean 2007
2577:
2475:Coxeter 1973
2467:MacLean 2007
2462:
2446:
2430:
2410:
2394:
2359:
2329:Solar System
2322:
2316:
2307:
2282:
2277:
2275:who, in his
2266:
2259:Solar System
2152:Dymaxion map
2093:
2086:radiolarians
2070:herpes virus
2059:
2051:Dymaxion map
2036:
2013:
1999:
1988:
1979:
1897:. These are
1892:
1876:snub 24-cell
1871:
1859:
1802:
1793:snub 24-cell
1790:
1717:
1690:
1665:diminishment
1664:
1642:
1639:Diminishment
1610:
1376:. Three are
1363:
1354:golden ratio
1317:
1306:
1299:
1282:
1257:
1251:
1204:
1188:Galois group
1176:simple group
1158:
1063:
1059:golden ratio
873:surface area
870:
596:circumsphere
589:
565:golden ratio
563:denotes the
389:
384:
374:
358:
353:golden ratio
319:
316:Construction
306:radiolarians
299:
272:
235:
231:
225:
5230:10-demicube
5191:9-orthoplex
5141:8-orthoplex
5091:7-orthoplex
5048:6-orthoplex
5018:5-orthoplex
4973:Pentachoron
4961:Icosahedron
4936:Tetrahedron
4420:semiregular
4403:icosahedron
4383:tetrahedron
4093:Icosahedron
4041:11–20 faces
4027:Enneahedron
4017:Heptahedron
4007:Pentahedron
4002:Tetrahedron
3428:: 169–200.
3295:Haeckel, E.
2705:Bickle 2020
2677:43–44
2640:Sutton 2002
2594:Berman 1971
2589:43–44
2471:43–44
2387:Berman 1971
2034:radiolarian
1992:Tipu Sultan
1951:Appearances
1667:. They are
1366:stellations
1285:Hamiltonian
1200:Felix Klein
576:Mensuration
370:deltahedron
264:deltahedron
236:icosahedron
234:(or simply
44:Deltahedron
5583:Deltahedra
5577:Categories
5216:10-simplex
5200:9-demicube
5150:8-demicube
5100:7-demicube
5057:6-demicube
5027:5-demicube
4941:Octahedron
4715:prismatoid
4645:bipyramids
4629:antiprisms
4603:hosohedron
4393:octahedron
4278:prismatoid
4263:(infinite)
4032:Decahedron
4022:Octahedron
4012:Hexahedron
3987:Monohedron
3980:1–10 faces
3913:Tulane.edu
3546:. Teubner.
3450:0132.14603
3381:: 431–452.
3107:(2): 192.
3012:References
3003:Livio 2003
2989:, p.
2975:Heath 1908
2963:Whyte 1952
2949:, p.
2933:, p.
2893:, p.
2861:, p.
2844:August 20,
2839:gmdice.com
2820:, p.
2800:, p.
2798:Smith 1958
2735:, p.
2707:, p.
2689:Klein 1884
2675:, p.
2642:, p.
2626:, p.
2610:, p.
2587:, p.
2469:, p.
2453:, p.
2437:, p.
2423:Cundy 1952
2417:, p.
2401:, p.
2372:Cundy 1952
2366:, p.
2105:carboranes
1998:, such as
1852:polychoron
1661:antiprisms
1623:are three
1601:, and the
1360:Stellation
1309:octahedron
1165:isomorphic
571:Properties
240:polyhedron
194:Properties
5561:compounds
5557:polyhedra
5264:orthoplex
5186:9-simplex
5136:8-simplex
5086:7-simplex
5043:6-simplex
5013:5-simplex
4982:Tesseract
4710:birotunda
4700:bifrustum
4465:snub cube
4360:polyhedra
4292:antiprism
3997:Trihedron
3966:Polyhedra
3554:(2003) .
3528:cite book
3241:250435684
3189:Polyhedra
2560:ℓ
2517:", where
2505:ℓ
2156:Greenland
2109:compounds
1907:snub cube
1903:rotations
1899:invariant
1825:∘
1762:φ
1747:4-pyramid
1731:φ
1625:facetings
1557:Facetings
1336:≈
1331:φ
1131:π
1117:π
1103:π
1051:Fibonacci
1012:≈
987:φ
954:≈
848:≈
837:φ
809:≈
785:φ
755:≈
728:φ
600:midsphere
521:φ
490:±
478:φ
475:±
453:φ
450:±
441:±
425:φ
422:±
413:±
295:polytopes
262:and of a
203:composite
95:×
5339:Notable
5318:Topics:
5281:demicube
5246:polytope
5240:Uniform
5001:600-cell
4997:120-cell
4950:Demicube
4924:Pentagon
4904:Triangle
4690:bicupola
4670:pyramids
4596:dihedron
3992:Dihedron
3788:80803624
3734:24311823
3698:(2013).
3620:40345834
3506:(1888).
3337:(1908).
3297:(1904).
3175:(1938).
3131:(1973).
2308:Elements
2283:elements
2255:Kepler's
2131:such as
2082:myovirus
2066:virology
1856:polytope
1786:120-cell
1697:600-cell
1278:skeleton
1073:Symmetry
592:insphere
279:faceting
268:skeleton
228:geometry
71:Vertices
5368:Others
5349:Regular
5255:simplex
5225:10-cube
4992:24-cell
4978:16-cell
4919:Hexagon
4773:regular
4732:italics
4720:scutoid
4705:rotunda
4695:frustum
4424:uniform
4373:regular
4358:Convex
4312:pyramid
4297:frustum
3869:0050303
3861:2306764
3725:3845684
3442:0185507
3233:3608204
3157:0370327
3121:2317282
3041:0290245
2612:138–140
2315:in his
2278:Timaeus
2165:In the
2125:borides
2123:. Many
1695:of the
1260:, is a
1182:of the
1173:abelian
1167:to the
289:as its
172:degrees
5195:9-cube
5145:8-cube
5095:7-cube
5052:6-cube
5022:5-cube
4909:Square
4786:Family
4685:cupola
4638:duals:
4624:prisms
4302:cupola
4178:vertex
3867:
3859:
3826:
3805:
3786:
3776:
3749:
3732:
3722:
3684:
3663:
3635:
3618:
3587:
3568:
3516:
3491:
3468:
3448:
3440:
3362:
3323:
3283:
3258:
3239:
3231:
3200:
3155:
3145:
3119:
3086:
3062:
3039:
2691:. See
2344:, and
2303:Euclid
2074:shells
1915:chiral
1864:facets
1844:defect
1821:138.19
1782:chords
1675:, and
1659:, and
1657:prisms
1619:, and
1252:Every
1049:, and
1047:Pappus
513:where
302:shells
230:, the
198:convex
5563:with
4914:p-gon
4307:wedge
4287:prism
4147:(132)
3857:JSTOR
3784:S2CID
3616:JSTOR
3237:S2CID
3229:JSTOR
3117:JSTOR
2352:Notes
2299:earth
2295:water
2273:Plato
2115:with
2101:closo
1848:cells
1339:0.618
1087:Full
1015:2.182
957:8.660
851:0.809
812:0.951
758:0.756
332:with
250:with
61:Edges
51:Faces
5559:and
5272:cube
4945:Cube
4775:and
4388:cube
4269:(57)
4240:(92)
4234:(13)
4228:(13)
4197:(16)
4173:edge
4168:face
4141:(90)
4135:(60)
4129:(48)
4123:(32)
4117:(30)
4111:(24)
3824:ISBN
3803:ISBN
3774:ISBN
3747:ISBN
3730:PMID
3682:ISBN
3661:ISBN
3633:ISBN
3585:ISBN
3566:ISBN
3534:link
3514:ISBN
3489:ISBN
3466:ISBN
3398:2004
3360:ISBN
3321:ISBN
3307:here
3305:See
3281:ISBN
3256:ISBN
3198:ISBN
3143:ISBN
3084:ISBN
3060:ISBN
2846:2019
2346:cube
2287:fire
2135:and
2127:and
2107:are
2032:The
1978:The
1909:and
1868:peak
1699:, a
1643:The
1611:The
1593:The
1302:dual
1064:The
1043:Hero
871:The
590:The
381:snub
304:and
36:Type
4821:(p)
4422:or
4257:(4)
4222:(5)
4191:(9)
4153:(∞)
3849:doi
3766:doi
3720:PMC
3712:doi
3608:doi
3446:Zbl
3430:doi
3402:doi
3221:doi
3109:doi
3029:doi
3025:291
2802:295
2737:257
2709:147
2455:211
2439:475
2419:333
2305:'s
2291:air
2078:HIV
2064:In
2010:d10
2006:d20
1858:in
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