2514:
1896:
1728:
2488:
1019:
1942:
1903:
2870:
1452:
1750:
1761:
2859:
2247:
2261:
2228:
375:
2501:
2235:
2199:
396:
382:
368:
2527:
2221:
267:
281:
1873:
2365:
1772:
1188:
2254:
389:
1040:
1179:
2469:
2456:
2430:
2417:
1990:
2443:
2012:
1917:
2881:
2308:
2268:
2214:
2001:
1979:
2023:
2848:
2783:
2315:
2287:
2275:
2797:
2349:
1739:
2811:
2301:
1590:
1953:
1928:
1882:
1859:
2769:
274:
1967:
1311:
1304:
1958:
1933:
1887:
1864:
2837:
2755:
1603:
288:
1297:
1290:
1170:
998:
2294:
165:
172:
144:
2329:
1564:
158:
151:
1577:
1318:
709:. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, and perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.
1094:
Around the same time as the
Pythagoreans, Plato described a theory of matter in which the five elements (earth, air, fire, water and spirit) each comprised tiny copies of one of the five regular solids. Matter was built up from a mixture of these polyhedra, with each substance having different
700:
and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five
Platonic solids, but also some of the relations of duality amongst them (that is, that the centres of the faces of the cube gives the vertices of an octahedron). Examples of these stones are on
1078:
and for a decade tried to establish the
Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the
835:. Some of these star polyhedra may have been discovered by others before Kepler's time, but Kepler was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Two hundred years later
1669:
if its combinatorial symmetries are transitive on its flags â that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
878:, or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the Platonic solids. This was first done by Bertrand around the same time that Cayley named them.
1511:
just as polyhedra are inscribed in a sphere (which contains zero ideal points). The sequence extends when hyperbolic tilings are themselves used as facets of noncompact hyperbolic tessellations, as in the
843:(circuits around each corner), enabling him to discover two new regular star polyhedra along with rediscovering Kepler's. These four are the only regular star polyhedra, and have come to be known as the
1083:
for which he is now famous. In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of
Platonic solids. Kepler's work, and the discovery since that time of
1665:
is a connected set of elements of each dimension â for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be
800:(since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of that
1039:
2633:
590:
of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of any polyhedron which is star-shaped with respect to some interior point.
1121:
In the first decades, Coxeter and Petrie allowed "saddle" vertices with alternating ridges and valleys, enabling them to construct three infinite folded surfaces which they called
768:, used all five in a correspondence between the polyhedra and the nature of the universe as it was then perceived â this correspondence is recorded in Plato's dialogue
997:
3164:(Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
1556:. The tetrahedron does not have a projective counterpart as it does not have pairs of parallel faces which can be identified, as the other four Platonic solids do.
598:
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point (this is an extension of
1018:
483:
The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:
792:
A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
847:. It was not until the mid-19th century, several decades after Poinsot published, that Cayley gave them their modern English names: (Kepler's)
1684:
744:
records of the regular convex solids originated from
Classical Greece. When these solids were all discovered and by whom is not known, but
1209:
Finite regular skew polyhedra exist in 4-space. These finite regular skew polyhedra in 4-space can be seen as a subset of the faces of
3012:
Hagino, K., Onuma, R., Kawachi, M. and
Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in
1649:(poset) of elements. The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the
896:. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the
525:
A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
2723:
connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with two
1080:
2915:= 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron. All of these have Euler characteristic 2.
1107:
The 20th century saw a succession of generalisations of the idea of a regular polyhedron, leading to several new classes.
2078:
whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set of
1712:
3206:
1340:
1335:
908:, which is visually almost indistinguishable from a regular dodecahedron. Truly icosahedral crystals may be formed by
3322:
3304:
3288:
3230:
3161:
3126:
3093:
3059:
2970:
1617:
These occur as dual pairs in the same way as the original
Platonic solids do. Their Euler characteristics are all 1.
827:
as their secret sign, but they did not use them to construct polyhedra. It was not until the early 17th century that
716:
preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near
3259:
1637:
in any number of dimensions. The second half of the century saw the development of abstract algebraic ideas such as
1517:
804:
would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).
2570:
1715:. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
677:
The SchlÀfli symbol of the dual is just the original written backwards, for example the dual of {5, 3} is {3, 5}.
2987:
97:), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.
1732:
1765:
1480:
1325:
752:) was the first to give a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII).
51:. In classical contexts, many different equivalent definitions are used; a common one is that the faces are
3292:
3225:, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge University Press, p. 192,
2518:
2492:
1330:
927:, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C
852:
848:
844:
666:
655:
309:
295:
257:
121:
94:
1754:
1513:
410:
881:
By the end of the 19th century there were therefore nine regular polyhedra â five convex and four star.
686:
618:
pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa.
3368:
2075:
1456:
745:
431:
417:
358:
2016:
1994:
1626:
1116:
1099:
would show this idea to be along the right lines, though not related directly to the regular solids.
1096:
1008:
947:
2709:
2027:
2005:
1638:
1267:
1248:
2924:
1204:
1122:
32:
939:) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.
2929:
1776:
1673:
Five such regular abstract polyhedra, which can not be realised faithfully, were identified by
979:; the shapes of these creatures are indicated by their names. The outer protein shells of many
424:
3220:
2405:
The usual five regular polyhedra can also be represented as spherical tilings (tilings of the
1495:
that can be closed by bending one way or the other. If the tiling is properly scaled, it will
1468:
3311:
E. P. Dutton, New York. (Dover
Publications edition, 1958). Chapter X: The Regular Polytopes.
3275:
3046:
1983:
1646:
1529:
866:
The KeplerâPoinsot polyhedra may be constructed from the
Platonic solids by a process called
713:
85:
the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the
52:
3173:
1439:, discovered over the preceding century, led to the discovery of more new polyhedra such as
3000:
2400:
2338:
2148:
2034:
1549:
1533:
1436:
1239:
generate regular skew polyhedra as {4, 4 | n}. In the infinite limit these approach a
1218:
1154:
901:
801:
599:
587:
578:
Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.
403:
216:
200:
8:
2473:
1568:
897:
644:
207:
1895:
3339:
3187:
3132:
3118:
3085:
2934:
2505:
1743:
1687:
in his paper "The combinatorially regular polyhedra of index 2" (1987). All five have C
1631:
By now, polyhedra were firmly understood as three-dimensional examples of more general
1210:
860:
769:
659:
302:
3063:
2513:
2487:
1727:
1126:
1066:
believed that there was a harmony between the regular polyhedra and the orbits of the
807:
This concept of a regular polyhedron would remain unchallenged for almost 2000 years.
66:
3336:
3318:
3284:
3226:
3157:
3122:
3089:
3055:
2966:
2531:
1679:
1642:
1594:
1541:
1345:
856:
706:
670:
316:
59:
44:
3136:
1451:
3114:
3081:
3025:
2960:
2939:
2728:
2712:
2658:
1941:
1902:
1749:
1674:
1607:
1545:
1484:
1476:
1460:
1440:
1428:
963:, some of whose shells are shaped like various regular polyhedra. Examples include
765:
702:
696:
Stones carved in shapes resembling clusters of spheres or knobs have been found in
2869:
3029:
2705:
2666:
1581:
1537:
1214:
1071:
1004:
943:
875:
832:
828:
615:
506:
116:
55:
48:
40:
36:
2246:
1760:
3314:
3209:
Beitrage zur
Algebra und Geometrie 52(2):357â387 · November 2010, Table 3, p.27
3186:
Cutler, Anthony M.; Schulte, Egon (2010). "Regular Polyhedra of Index Two, I".
2639:
2541:
2260:
2227:
2082:
1553:
1432:
1063:
797:
786:
760:(400 BC) with having made models of them, and mentions that one of the earlier
753:
561:
495:
134:
111:
106:
90:
86:
28:
3353:
2858:
2234:
2198:
374:
3362:
2220:
1872:
1850:
1488:
1424:
1261:
1221:
1130:
956:
840:
836:
502:
395:
381:
367:
280:
266:
2500:
2364:
2740:
2724:
2526:
2460:
2079:
1653:
or empty set. These abstract elements can be mapped into ordinary space or
1492:
1029:
960:
909:
905:
824:
761:
725:
648:
544:
3109:
STRAUSS, JAMES H.; STRAUSS, ELLEN G. (2008). "The Structure of Viruses".
2421:
2253:
2071:
2065:
1771:
1711:
faces around each vertex, can be repeated indefinitely as tilings of the
1504:
1240:
1187:
1075:
1025:
626:
513:
388:
179:
3293:
http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html
2468:
2455:
2442:
2429:
2416:
2011:
1989:
1178:
2880:
2674:
2547:
2447:
2307:
2267:
2213:
2022:
2000:
1978:
1508:
1464:
952:
867:
637:
603:
193:
24:
2847:
2796:
2782:
2654:â„ 3 enforces that the polygonal faces must have at least three sides.
2348:
2314:
2286:
2274:
1916:
1418:
3344:
2810:
2300:
1738:
1589:
1500:
1054:
920:
889:
Each of the Platonic solids occurs naturally in one form or another.
871:
820:
816:
774:. Euclid's reference to Plato led to their common description as the
729:
537:
2768:
2673:= 2 admits a new infinite class of regular polyhedra, which are the
1966:
1657:
as geometrical figures. Some abstract polyhedra have well-formed or
273:
2985:
Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem",
2836:
2801:
2773:
2754:
2697:
2551:
1952:
1927:
1881:
1858:
1695:
symmetry but can only be realised with half the symmetry, that is C
1633:
1602:
1236:
1046:
988:
912:
which are very rare in nature but can be produced in a laboratory.
697:
557:
530:
287:
3192:
1957:
1932:
1886:
1863:
1703:
or icosahedral symmetry. They are all topologically equivalent to
1169:
987:
is enclosed in a regular icosahedron, as is the head of a typical
3149:
2815:
2720:
1576:
1382:
1364:
1317:
1310:
1303:
1232:
1088:
893:
2665:(2-gons) can be represented as spherical lunes, having non-zero
1296:
1289:
2787:
2736:
2406:
2293:
1704:
1523:
1353:
1228:
1084:
1067:
1050:
916:
831:
realised that pentagrams could be used as the faces of regular
749:
488:
164:
1443:
which could only take regular geometric form in those spaces.
171:
143:
3317:; Regular Polytopes (third edition). Dover Publications Inc.
2759:
2662:
1244:
1163:
Infinite regular skew polyhedra in 3-space (partially drawn)
980:
757:
721:
717:
157:
150:
3354:
YouTube video 'there are 48 regular polyhedra' by jan Misali
2542:
Regular polyhedra that can only exist as spherical polyhedra
1079:
realisation that the orbits of planets are not circles, and
819:(star pentagon) were also known to the ancient Greeks â the
3334:
2434:
2328:
1563:
1528:
Another group of regular polyhedra comprise tilings of the
1516:{7,3,3}; they are inscribed in an equidistant surface (a 2-
789:) planar figure with all edges equal and all corners equal.
732:, and dating back more than 2,500 years (Lindemann, 1987).
633:
186:
39:. A regular polyhedron is highly symmetrical, being all of
2677:. On a spherical surface, the regular polyhedron {2,
984:
487:
The vertices of a convex regular polyhedron all lie on a
1446:
781:
One might characterise the Greek definition as follows:
2731:, a dihedron can exist as nondegenerate form, with two
915:
A more recent discovery is of a series of new types of
904:
are amongst them, but crystals can have the shape of a
2642:
known to antiquity are the only integer solutions for
3264:
Comptes rendus des séances de l'Académie des Sciences
3262:(1858). Note sur la théorie des polyÚdres réguliers,
2573:
1491:
that act like finite polyhedra. Such tilings have an
874:(or faceting). Every stellation of one polyhedron is
602:.) However, the converse does not hold, not even for
2735:-sided faces covering the sphere, each face being a
621:
The regular polyhedra show this duality as follows:
2556:
For a regular polyhedron whose SchlÀfli symbol is {
1419:
Regular polyhedra in non-Euclidean and other spaces
1129:{l,m|n} for these figures, with {l,m} implying the
892:The tetrahedron, cube, and octahedron all occur as
609:
125:; and five regular compounds of regular polyhedra:
16:
Polyhedron with regular congruent polygons as faces
2627:
2564:}, the number of polygonal faces may be found by:
1095:proportions in the mix. Two thousand years later
942:Regular polyhedra appear in biology as well. The
870:. The reciprocal process to stellation is called
3360:
3309:An Introduction to the Geometry of n Dimensions.
58:which are assembled in the same way around each
3218:
951:has a regular dodecahedral structure, about 10
3108:
884:
3185:
3040:
3038:
2693:. All these lunes share two common vertices.
1620:
1552:, and are the projective counterparts of the
1507:. These Euclidean tilings are inscribed in a
1467:. One such facet is shown in as seen in this
1074:studied data on planetary motion compiled by
564:, which are named after the Platonic solids:
560:of all the polyhedra. They lie in just three
2833:
2751:
2628:{\displaystyle N_{2}={\frac {4n}{2m+2n-mn}}}
2320:
1524:Regular tilings of the real projective plane
796:This definition rules out, for example, the
687:Regular polytope § History of discovery
1157:, vertices zig-zagging between two planes.
1110:
251:
3299:Geometrical And Structural Crystallography
3035:
2965:. Cambridge University Press. p. 77.
2704:, 2} (2-hedron) in three-dimensional
1102:
359:Polytope compound § Regular compounds
65:A regular polyhedron is identified by its
3191:
3006:
2685:abutting lunes, with interior angles of 2
2661:, this restriction may be relaxed, since
1908:
1255:Finite regular skew polyhedra in 4-space
1198:
1091:, have invalidated the Pythagorean idea.
810:
3075:
2958:
1450:
1231:, two dual solutions are related to the
629:is self-dual, i.e. it pairs with itself.
478:
100:
81:is the number of sides of each face and
3219:McMullen, Peter; Schulte, Egon (2002),
581:
3361:
3243:
3241:
2394:
1227:Two dual solutions are related to the
701:display in the John Evans room of the
3335:
3154:The Beauty of Geometry: Twelve Essays
1447:Regular polyhedra in hyperbolic space
983:form regular polyhedra. For example,
756:(Coxeter, 1948, Section 1.9) credits
520:
3174:The Regular Polyhedra (of index two)
2747:if the vertices are equally spaced.
1049:typically has a regular icosahedral
1012:has a regular dodecahedral structure
352:
3238:
1707:. Their construction, by arranging
1463:, {6,3}, facets with vertices on a
1235:, and an infinite set of self-dual
1033:has a regular icosahedral structure
955:across. In the early 20th century,
556:The regular polyhedra are the most
13:
3207:Regular Polyhedra of Index Two, II
3119:10.1016/b978-0-12-373741-0.50005-2
3086:10.1016/b978-0-12-384684-6.00002-1
3080:. Elsevier. 2012. pp. 46â62.
815:Regular star polygons such as the
593:
473:
128:
14:
3380:
3328:
959:described a number of species of
586:The five Platonic solids have an
2879:
2868:
2857:
2846:
2835:
2809:
2795:
2781:
2767:
2753:
2657:When considering polyhedra as a
2525:
2512:
2499:
2486:
2467:
2454:
2441:
2428:
2415:
2363:
2347:
2327:
2313:
2306:
2299:
2292:
2285:
2273:
2266:
2259:
2252:
2245:
2233:
2226:
2219:
2212:
2197:
2021:
2010:
1999:
1988:
1977:
1965:
1956:
1951:
1940:
1931:
1926:
1915:
1901:
1894:
1885:
1880:
1871:
1862:
1857:
1770:
1759:
1748:
1737:
1726:
1641:, culminating in the idea of an
1601:
1588:
1575:
1562:
1316:
1309:
1302:
1295:
1288:
1186:
1177:
1168:
1038:
1017:
996:
724:) in the late 19th century of a
610:Duality of the regular polyhedra
516:of the polyhedron are congruent.
394:
387:
380:
373:
366:
286:
279:
272:
265:
170:
163:
156:
149:
142:
109:regular polyhedra, known as the
3212:
3200:
3179:
2988:The College Mathematics Journal
1661:realisations, others do not. A
1520:), which has two ideal points.
3167:
3143:
3102:
3069:
2994:
2979:
2952:
2059:
1733:Medial rhombic triacontahedron
1481:paracompact regular honeycombs
1:
2945:
2074:of a regular polyhedron is a
1766:Ditrigonal dodecadodecahedron
1125:. Coxeter offered a modified
712:It is also possible that the
691:
574:Icosahedral (or dodecahedral)
3156:, Dover Publications, 1999,
3113:. Elsevier. pp. 35â62.
3030:10.1371/journal.pone.0081749
2519:Great stellated dodecahedron
2493:Small stellated dodecahedron
2484:
2482:
2413:
2411:
2280:
2240:
2210:
2190:
2133:
2113:
2032:
1972:
1848:
1816:
1781:
1721:
1719:
1560:
1166:
1081:the laws of planetary motion
923:(see Curl, 1991). Although C
853:great stellated dodecahedron
849:small stellated dodecahedron
667:great stellated dodecahedron
656:small stellated dodecahedron
323:
310:Great stellated dodecahedron
296:Small stellated dodecahedron
293:
263:
261:
214:
177:
140:
138:
7:
3279:. Available as Haeckel, E.
3050:. Available as Haeckel, E.
2959:Cromwell, Peter R. (1997).
2918:
2907:} is dual to the dihedron {
2715:consisting of two (planar)
1755:Medial triambic icosahedron
1514:heptagonal tiling honeycomb
1435:) and other spaces such as
1153:. Their vertex figures are
1141:-gons around a vertex. The
885:Regular polyhedra in nature
551:
498:of the polyhedron are equal
10:
3385:
3222:Abstract Regular Polytopes
2545:
2398:
2063:
1624:
1621:Abstract regular polyhedra
1457:hexagonal tiling honeycomb
1202:
1114:
910:quasicrystalline materials
684:
680:
547:, tangent to all vertices.
356:
255:
132:
3111:Viruses and Human Disease
3014:Braarudosphaera bigelowii
2991:37(5), 2006, pp. 390â391.
2206:
2203:
2194:
2191:
1627:Abstract regular polytope
1279:
1273:
1266:
1259:
1249:stereographic projections
1162:
1117:Regular skew apeirohedron
1009:Braarudosphaera bigelowii
948:Braarudosphaera bigelowii
735:
3001:The Scottish Solids Hoax
2739:, and vertices around a
1639:Polyhedral combinatorics
1268:Stereographic projection
1111:Regular skew apeirohedra
977:Circorrhegma dodecahedra
845:KeplerâPoinsot polyhedra
785:A regular polygon is a (
258:KeplerâPoinsot polyhedra
252:KeplerâPoinsot polyhedra
122:KeplerâPoinsot polyhedra
95:KeplerâPoinsot polyhedra
2925:Quasiregular polyhedron
1205:Regular skew polyhedron
1103:Further generalisations
1070:. In the 17th century,
919:molecule, known as the
673:are dual to each other.
662:are dual to each other.
651:are dual to each other.
640:are dual to each other.
540:, tangent to all edges.
533:, tangent to all faces.
3301:. John Wiley and Sons.
3283:, Prestel USA (1998),
3054:, Prestel USA (1998),
2930:Semiregular polyhedron
2629:
1777:Excavated dodecahedron
1548:. They are (globally)
1483:have Euclidean tiling
1472:
1199:Regular skew polyhedra
1123:regular skew polyhedra
1097:Dalton's atomic theory
973:Lithocubus geometricus
811:Regular star polyhedra
505:of the polyhedron are
3305:Sommerville, D. M. Y.
3297:Smith, J. V. (1982).
3276:Kunstformen der Natur
3047:Kunstformen der Natur
3016:(Prymnesiophyceae)".
2874:Pentagonal hosohedron
2650:â„ 3. The restriction
2630:
2107:Petrial dodecahedron
1647:partially ordered set
1625:Further information:
1530:real projective plane
1454:
1155:regular skew polygons
1062:In ancient times the
1030:Circogonia icosahedra
969:Circogonia icosahedra
965:Circoporus octahedrus
571:Octahedral (or cubic)
479:Equivalent properties
101:The regular polyhedra
3340:"Regular Polyhedron"
3273:Haeckel, E. (1904).
3064:Kurt StĂŒber's Biolib
3062:. Online version at
3044:Haeckel, E. (1904).
2911:,2}. Note that when
2885:Hexagonal hosohedron
2708:can be considered a
2681:} is represented as
2571:
2401:Spherical polyhedron
2110:Petrial icosahedron
1683:(1977) and again by
1550:projective polyhedra
1532:. These include the
1219:regular skew polygon
902:regular dodecahedron
802:triangular bipyramid
588:Euler characteristic
582:Euler characteristic
89:), and four regular
3281:Art forms in nature
3052:Art forms in nature
2852:Trigonal hosohedron
2395:Spherical polyhedra
2104:Petrial octahedron
2097:Petrial tetrahedron
2090:
1469:Poincaré disk model
1256:
1213:. They have planar
1211:uniform 4-polytopes
898:regular icosahedron
740:The earliest known
3337:Weisstein, Eric W.
3176:, David A. Richter
2935:Uniform polyhedron
2903:The hosohedron {2,
2841:Digonal hosohedron
2625:
2506:Great dodecahedron
2088:
1744:Dodecadodecahedron
1473:
1254:
861:great dodecahedron
855:, and (Poinsot's)
839:also allowed star
660:great dodecahedron
536:An intersphere or
521:Concentric spheres
303:Great dodecahedron
21:regular polyhedron
3369:Regular polyhedra
3270:, pp. 79â82.
3249:Regular polytopes
2901:
2900:
2863:Square hosohedron
2623:
2539:
2538:
2532:Great icosahedron
2481:
2480:
2392:
2391:
2089:Regular petrials
2055:
2054:
1680:Regular Polytopes
1643:abstract polytope
1615:
1614:
1595:Hemi-dodecahedron
1542:hemi-dodecahedron
1441:complex polyhedra
1416:
1415:
1196:
1195:
1053:(head) about 100
857:great icosahedron
707:Oxford University
671:great icosahedron
600:Viviani's theorem
471:
470:
353:Regular compounds
350:
349:
317:Great icosahedron
249:
248:
45:vertex-transitive
3376:
3350:
3349:
3252:
3245:
3236:
3235:
3216:
3210:
3204:
3198:
3197:
3195:
3183:
3177:
3171:
3165:
3147:
3141:
3140:
3106:
3100:
3099:
3073:
3067:
3042:
3033:
3010:
3004:
2998:
2992:
2983:
2977:
2976:
2956:
2940:Regular polytope
2883:
2872:
2861:
2850:
2839:
2813:
2799:
2785:
2771:
2757:
2750:
2749:
2729:spherical tiling
2727:. However, as a
2688:
2659:spherical tiling
2634:
2632:
2631:
2626:
2624:
2622:
2596:
2588:
2583:
2582:
2529:
2516:
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2412:
2367:
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2296:
2289:
2277:
2270:
2263:
2256:
2249:
2237:
2230:
2223:
2216:
2207:6 skew decagons
2204:4 skew hexagons
2201:
2091:
2087:
2025:
2014:
2003:
1992:
1981:
1969:
1960:
1955:
1944:
1935:
1930:
1919:
1905:
1898:
1889:
1884:
1875:
1866:
1861:
1774:
1763:
1752:
1741:
1730:
1720:
1713:hyperbolic plane
1675:H. S. M. Coxeter
1608:Hemi-icosahedron
1605:
1592:
1579:
1566:
1559:
1558:
1546:hemi-icosahedron
1501:asymptotic limit
1477:hyperbolic space
1461:hexagonal tiling
1411:
1404:
1396:
1320:
1313:
1306:
1299:
1292:
1257:
1253:
1243:and look like a
1190:
1181:
1172:
1160:
1159:
1042:
1021:
1000:
823:was used by the
766:Timaeus of Locri
703:Ashmolean Museum
507:regular polygons
398:
391:
384:
377:
370:
363:
362:
290:
283:
276:
269:
262:
174:
167:
160:
153:
146:
139:
56:regular polygons
3384:
3383:
3379:
3378:
3377:
3375:
3374:
3373:
3359:
3358:
3331:
3315:Coxeter, H.S.M.
3291:, or online at
3256:
3255:
3246:
3239:
3233:
3217:
3213:
3205:
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3184:
3180:
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2842:
2840:
2819:
2814:
2805:
2800:
2791:
2786:
2777:
2772:
2763:
2758:
2706:Euclidean space
2686:
2640:Platonic solids
2597:
2589:
2587:
2578:
2574:
2572:
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2098:
2083:Petrie polygons
2068:
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2026:
2015:
2004:
1993:
1982:
1964:
1950:
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1939:
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1893:
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1698:
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1629:
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1597:
1593:
1584:
1582:Hemi-octahedron
1580:
1571:
1567:
1554:Platonic solids
1538:hemi-octahedron
1526:
1459:, {6,3,3}, has
1449:
1421:
1407:
1406:
1399:
1398:
1392:
1388:
1386:
1377:
1375:
1370:
1368:
1359:
1357:
1346:{4, 4 | n}
1341:{8, 4 | 3}
1336:{4, 8 | 3}
1331:{6, 4 | 3}
1326:{4, 6 | 3}
1283:
1277:
1215:regular polygon
1207:
1201:
1191:
1182:
1173:
1127:SchlÀfli symbol
1119:
1113:
1105:
1072:Johannes Kepler
1058:
1043:
1034:
1022:
1013:
1005:coccolithophore
1001:
944:coccolithophore
938:
934:
930:
926:
887:
829:Johannes Kepler
813:
776:Platonic solids
738:
694:
689:
683:
612:
596:
594:Interior points
584:
562:symmetry groups
554:
523:
496:dihedral angles
481:
476:
474:Characteristics
434:
427:
421:10 {3, 3}
420:
413:
411:Five tetrahedra
406:
361:
355:
319:
312:
305:
298:
260:
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210:
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196:
189:
182:
137:
131:
129:Platonic solids
115:; four regular
112:Platonic solids
105:There are five
103:
87:Platonic solids
67:SchlÀfli symbol
49:face-transitive
41:edge-transitive
17:
12:
11:
5:
3382:
3372:
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3330:
3329:External links
3327:
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3295:
3271:
3254:
3253:
3237:
3231:
3211:
3199:
3178:
3166:
3142:
3127:
3101:
3094:
3078:Virus Taxonomy
3076:"Myoviridae".
3068:
3034:
3024:(12): e81749.
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833:star polyhedra
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798:square pyramid
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754:H.S.M. Coxeter
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435:5 {3, 4}
432:Five octahedra
429:
428:5 {4, 3}
422:
418:Ten tetrahedra
415:
414:5 {3, 3}
408:
407:2 {3, 3}
404:Two tetrahedra
400:
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357:Main article:
354:
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320:{3, 5/2}
314:
313:{5/2, 3}
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135:Platonic solid
133:Main article:
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117:star polyhedra
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91:star polyhedra
29:symmetry group
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1308:
1305:
1301:
1298:
1294:
1291:
1287:
1286:
1272:
1269:
1263:
1262:Coxeter plane
1258:
1252:
1250:
1246:
1242:
1238:
1234:
1230:
1225:
1223:
1220:
1216:
1212:
1206:
1189:
1185:
1180:
1176:
1171:
1167:
1161:
1158:
1156:
1152:
1148:
1144:
1140:
1136:
1132:
1131:vertex figure
1128:
1124:
1118:
1108:
1100:
1098:
1092:
1090:
1086:
1082:
1077:
1073:
1069:
1065:
1056:
1052:
1048:
1041:
1036:
1032:
1031:
1027:
1020:
1015:
1011:
1010:
1006:
999:
994:
993:
992:
990:
986:
982:
978:
974:
970:
966:
962:
958:
957:Ernst Haeckel
954:
950:
949:
945:
940:
922:
918:
913:
911:
907:
903:
899:
895:
890:
882:
879:
877:
873:
869:
864:
862:
858:
854:
850:
846:
842:
838:
837:Louis Poinsot
834:
830:
826:
822:
818:
808:
805:
803:
799:
791:
788:
784:
783:
782:
779:
777:
773:
772:
767:
763:
759:
755:
751:
747:
743:
733:
731:
727:
723:
720:(in Northern
719:
715:
710:
708:
704:
699:
688:
678:
672:
668:
664:
661:
657:
653:
650:
646:
642:
639:
635:
631:
628:
624:
623:
622:
619:
617:
607:
605:
601:
591:
589:
579:
573:
570:
567:
566:
565:
563:
559:
546:
542:
539:
535:
532:
528:
527:
526:
515:
511:
508:
504:
500:
497:
493:
490:
486:
485:
484:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
438:
433:
430:
426:
423:
419:
416:
412:
409:
405:
402:
401:
397:
393:
390:
386:
383:
379:
376:
372:
369:
365:
364:
360:
345:
342:
339:
336:
333:
330:
327:
324:
318:
315:
311:
308:
304:
301:
297:
294:
289:
285:
282:
278:
275:
271:
268:
264:
259:
244:
241:
238:
235:
232:
229:
226:
223:
220:
219:
215:
209:
206:
202:
199:
195:
192:
188:
185:
181:
178:
173:
169:
166:
162:
159:
155:
152:
148:
145:
141:
136:
126:
124:
123:
118:
114:
113:
108:
98:
96:
92:
88:
84:
80:
76:
72:
69:of the form {
68:
63:
61:
57:
54:
50:
46:
42:
38:
34:
30:
26:
22:
3343:
3308:
3298:
3280:
3274:
3267:
3263:
3260:Bertrand, J.
3248:
3221:
3214:
3202:
3181:
3169:
3153:
3145:
3110:
3104:
3077:
3071:
3051:
3045:
3021:
3017:
3013:
3008:
2996:
2986:
2981:
2961:
2954:
2912:
2908:
2904:
2902:
2894:
2827:
2744:
2741:great circle
2732:
2716:
2701:
2695:
2690:
2682:
2678:
2670:
2656:
2651:
2647:
2643:
2637:
2561:
2557:
2555:
2461:Dodecahedron
2404:
2184:
2177:
2170:
2163:
2156:
2149:
2144:
2140:
2136:
2069:
1963:20 hexagrams
1947:12 pentagons
1922:12 pentagons
1827:
1823:
1819:
1708:
1678:
1677:in his book
1672:
1666:
1662:
1658:
1654:
1650:
1632:
1630:
1616:
1527:
1503:at a single
1496:
1493:angle defect
1474:
1422:
1408:
1401:
1393:
1371:30 vertices
1360:20 vertices
1264:projections
1226:
1208:
1150:
1146:
1142:
1138:
1134:
1120:
1106:
1093:
1064:Pythagoreans
1061:
1028:
1007:
976:
972:
968:
964:
961:radiolarians
946:
941:
914:
906:pyritohedron
891:
888:
880:
865:
825:Pythagoreans
814:
806:
795:
780:
775:
770:
762:Pythagoreans
741:
739:
726:dodecahedron
711:
695:
676:
649:dodecahedron
620:
613:
597:
585:
577:
555:
545:circumsphere
524:
514:solid angles
482:
464:
458:
452:
446:
440:
343:
337:
331:
325:
242:
236:
230:
224:
217:
211:{3, 5}
201:Dodecahedron
120:
110:
104:
82:
78:
74:
70:
64:
33:transitively
20:
18:
3066:(in german)
2669:. Allowing
2474:Icosahedron
2422:Tetrahedron
2183:(12,30,6),
2176:(20,30,6),
2076:regular map
2072:Petrie dual
2066:Petrie dual
2060:Petrie dual
1938:20 hexagons
1845:(20,60,20)
1723:Polyhedron
1685:J. M. Wills
1505:ideal point
1423:Studies of
1260:Orthogonal
1241:duocylinder
1217:faces, but
1076:Tycho Brahe
1026:radiolarian
953:micrometres
645:icosahedron
627:tetrahedron
568:Tetrahedral
558:symmetrical
208:Icosahedron
204:{5, 3}
197:{3, 4}
190:{4, 3}
183:{3, 3}
180:Tetrahedron
2946:References
2802:Pentagonal
2737:hemisphere
2710:degenerate
2696:A regular
2548:Hosohedron
2546:See also:
2448:Octahedron
2282:Animation
2169:(6,12,4),
2162:(8,12,4),
1878:{5}, {5/2}
1855:{5}, {5/2}
1842:(20,60,24)
1839:(24,60,20)
1836:(30,60,24)
1833:(24,60,30)
1786:Dual {5,4}
1518:hypercycle
1509:horosphere
1465:horosphere
1429:hyperbolic
1055:nanometers
921:fullerenes
868:stellation
746:Theaetetus
692:Prehistory
685:See also:
638:octahedron
604:tetrahedra
425:Five cubes
194:Octahedron
25:polyhedron
3345:MathWorld
3247:Coxeter,
3193:1005.4911
2962:Polyhedra
2816:Hexagonal
2675:hosohedra
2614:−
2155:(4,6,3),
1913:30 rhombi
1634:polytopes
1569:Hemi-cube
1534:hemi-cube
1412:vertices
1397:{4} faces
1387:576 edges
1376:576 edges
1247:in their
1237:duoprisms
872:facetting
821:pentagram
817:pentagram
730:soapstone
714:Etruscans
538:midsphere
77:}, where
53:congruent
3363:Category
3307:(1930).
3137:80803624
3018:PLoS One
2919:See also
2818:dihedron
2804:dihedron
2790:dihedron
2776:dihedron
2774:Trigonal
2762:dihedron
2743:. It is
2721:polygons
2698:dihedron
2646:â„ 3 and
2552:Dihedron
2535:{3,5/2}
2522:{5/2,3}
2509:{5,5/2}
2496:{5/2,5}
2324:figures
1659:faithful
1655:realised
1433:elliptic
1369:60 edges
1358:60 edges
1192:{6,6|3}
1183:{6,4|4}
1174:{4,6|4}
1145:defines
1137:regular
1047:myovirus
989:myovirus
900:nor the
894:crystals
750:Athenian
728:made of
698:Scotland
552:Symmetry
531:insphere
512:All the
501:All the
494:All the
3251:, p. 12
3150:Coxeter
2760:Digonal
2745:regular
2719:-sided
2560:,
2373:= {6,4}
2357:= {6,3}
2341:= {4,3}
2339:{4,3}/2
2322:Related
2115:Symbol
1974:Tiling
1892:(5.5/3)
1869:(5.5/2)
1705:toroids
1667:regular
1233:24-cell
1149:-gonal
1133:, with
1089:Neptune
1068:planets
1057:across.
981:viruses
771:Timaeus
742:written
681:History
35:on its
3321:
3287:
3229:
3160:
3135:
3125:
3092:
3058:
2969:
2887:{2,6}
2876:{2,5}
2865:{2,4}
2854:{2,3}
2843:{2,2}
2820:{6,2}
2806:{5,2}
2792:{4,2}
2788:Square
2778:{3,2}
2764:{2,2}
2663:digons
2477:{3,5}
2464:{5,3}
2451:{3,4}
2438:{4,3}
2425:{3,3}
2407:sphere
2385:{10,5}
2379:{10,3}
2242:Image
2192:Faces
2187:= â12
2185:χ
2178:χ
2171:χ
2164:χ
2157:χ
2150:χ
2130:{3,5}
2127:{5,3}
2124:{3,4}
2121:{4,3}
2118:{3,3}
2028:{6, 6}
2017:{5, 6}
2006:{6, 5}
1995:{5, 4}
1984:{4, 5}
1910:Faces
1611:{5,3}
1598:{3,5}
1585:{3,4}
1572:{4,3}
1544:, and
1499:as an
1485:facets
1229:5-cell
1085:Uranus
1051:capsid
917:carbon
787:convex
736:Greeks
489:sphere
119:, the
107:convex
60:vertex
27:whose
3188:arXiv
3133:S2CID
2713:prism
2375:(4,0)
2369:{6,4}
2359:(2,0)
2353:{6,3}
2343:(2,0)
2333:{4,3}
2094:Name
1810:{6,6}
1804:{5,6}
1792:{5,4}
1783:Type
1645:as a
1497:close
1475:In H
1405:edges
1385:faces
1367:faces
1356:faces
1245:torus
1151:holes
935:and C
758:Plato
722:Italy
718:Padua
614:In a
467:= 10
461:= â10
93:(the
37:flags
31:acts
23:is a
3319:ISBN
3285:ISBN
3227:ISBN
3158:ISBN
3123:ISBN
3090:ISBN
3056:ISBN
2967:ISBN
2890:...
2830:,2}
2823:...
2667:area
2638:The
2550:and
2435:Cube
2180:= â4
2173:= â2
2080:skew
2070:The
2051:â20
2048:â16
2045:â16
1663:flag
1487:and
1455:The
1431:and
1381:144
1087:and
1024:The
1003:The
975:and
876:dual
859:and
851:and
748:(an
669:and
665:The
658:and
654:The
647:and
643:The
636:and
634:cube
632:The
625:The
616:dual
449:= 10
346:= 2
334:= â6
328:= â6
245:= 2
187:Cube
47:and
3115:doi
3082:doi
3026:doi
2893:{2,
2700:, {
2409:):
2166:= 0
2159:= 1
2147:),
2042:â6
2039:â6
1383:{8}
1365:{6}
1363:20
1354:{4}
1352:30
985:HIV
937:960
933:480
931:, C
929:240
705:at
529:An
455:= 0
443:= 4
340:= 2
239:= 2
233:= 2
227:= 2
221:= 2
3365::
3342:.
3268:46
3266:,
3240:^
3152:,
3131:.
3121:.
3088:.
3037:^
3020:,
2897:}
2337:=
2085:.
1830:)
1699:ĂA
1691:ĂS
1540:,
1536:,
1479:,
1224:.
1045:A
991:.
971:,
967:,
925:60
863:.
778:.
764:,
606:.
543:A
73:,
62:.
43:,
19:A
3348:.
3196:.
3190::
3139:.
3117::
3098:.
3084::
3032:.
3028::
3022:8
3003:.
2975:.
2913:n
2909:n
2905:n
2895:n
2828:n
2826:{
2733:n
2717:n
2702:n
2691:n
2689:/
2687:Ï
2683:n
2679:n
2671:m
2652:m
2648:n
2644:m
2620:n
2617:m
2611:n
2608:2
2605:+
2602:m
2599:2
2594:n
2591:4
2585:=
2580:2
2576:N
2562:n
2558:m
2387:3
2381:5
2371:3
2355:3
2335:3
2145:f
2143:,
2141:e
2139:,
2137:v
2135:(
2035:Ï
1828:f
1826:,
1824:e
1822:,
1820:v
1818:(
1812:6
1806:4
1800:4
1794:6
1788:6
1709:n
1701:5
1697:2
1693:5
1689:2
1471:.
1427:(
1409:n
1402:n
1400:2
1394:n
1282:4
1280:F
1276:4
1274:A
1147:n
1143:n
1139:l
1135:m
509:.
491:.
465:Ï
459:Ï
453:Ï
447:Ï
441:Ï
344:Ï
338:Ï
332:Ï
326:Ï
243:Ï
237:Ï
231:Ï
225:Ï
218:Ï
83:m
79:n
75:m
71:n
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