6996:
8310:
5879:
6991:{\displaystyle {\begin{aligned}S_{}^{(2)}=S_{}^{(2)}=S_{}^{(2)}=S_{}^{(2)}=S_{}^{(2)}&=R^{2}+L^{2},\\S_{}^{(4)}=S_{}^{(4)}=S_{}^{(4)}=S_{}^{(4)}=S_{}^{(4)}&=\left(R^{2}+L^{2}\right)^{2}+{\frac {4}{3}}R^{2}L^{2},\\S_{}^{(6)}=S_{}^{(6)}=S_{}^{(6)}=S_{}^{(6)}&=\left(R^{2}+L^{2}\right)^{3}+4R^{2}L^{2}\left(R^{2}+L^{2}\right),\\S_{}^{(8)}=S_{}^{(8)}&=\left(R^{2}+L^{2}\right)^{4}+8R^{2}L^{2}\left(R^{2}+L^{2}\right)^{2}+{\frac {16}{5}}R^{4}L^{4},\\S_{}^{(10)}=S_{}^{(10)}&=\left(R^{2}+L^{2}\right)^{5}+{\frac {40}{3}}R^{2}L^{2}\left(R^{2}+L^{2}\right)^{3}+16R^{4}L^{4}\left(R^{2}+L^{2}\right).\end{aligned}}}
411:
12301:
11481:
8639:
8646:
8625:
8134:
8618:
2369:
329:
8632:
8956:
8222:
302:
8194:
320:
293:
2223:
8500:
8843:
8829:
7883:
7816:
311:
7340:
7765:
7329:
7318:
2201:
8528:
8521:
8514:
8507:
8836:
11468:
8949:
8401:
8580:
8573:
8208:
8587:
7943:
1349:
8566:
2177:
8942:
8559:
1378:
1291:
1262:
2166:
2155:
2144:
8390:
1320:
186:
8935:
206:
126:
4435:
2212:
166:
146:
9987:
399:, the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.
2190:
4119:
477:, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his
4243:
1768:. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other.
3959:
5738:
7628:
There are only three symmetry groups associated with the
Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual
5742:
Among the
Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to
264:
represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the
Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not
7400:
One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number
8235:
452:
in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five
Platonic solids enclosed within a sphere that
7664:
The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All
Platonic solids
379:) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used for arranging the constellations on the whole heaven".
2127:
The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five
Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question—one that requires an explicit construction.
2877:
2766:
4430:{\displaystyle {\frac {R}{r}}=\tan \left({\frac {\pi }{p}}\right)\tan \left({\frac {\pi }{q}}\right)={\frac {\sqrt {{\csc ^{2}}{\Bigl (}{\frac {\theta }{2}}{\Bigr )}-{\cos ^{2}}{\Bigl (}{\frac {\alpha }{2}}{\Bigr )}}}{\sin {\Bigl (}{\frac {\alpha }{2}}{\Bigr )}}}.}
4218:
1728:
5601:
5743:
that of a sphere of the same size (i.e. either the same surface area or the same volume). The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.
7138:
4550:
4114:{\displaystyle {\begin{aligned}R&={\frac {a}{2}}\tan \left({\frac {\pi }{q}}\right)\tan \left({\frac {\theta }{2}}\right)\\r&={\frac {a}{2}}\cot \left({\frac {\pi }{p}}\right)\tan \left({\frac {\theta }{2}}\right)\end{aligned}}}
7274:
8094:
subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral
2779:
2668:
8015:
of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.
3703:
5884:
7676:
The following table lists the various symmetry properties of the
Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parentheses (likewise for the number of symmetries).
3854:
3305:
9262:, which he calls "a long step towards Aristotle's theory", and he points out that Aristotle's ether is above the other four elements rather than on an equal footing with them, making the correspondence less apposite.
3565:
5869:
5370:
998:
The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign
2607:
2543:
7510:
2979:
5557:
4136:
3964:
1534:
5153:
4836:
3418:
284:, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
2470:
280:
with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to
5438:
2274:
At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be strictly less than 360°. The amount less than 360° is called an
8040:
10009:
8007:. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the
3056:
5317:
5219:
8479:, which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)
7003:
2294:
Each vertex of a regular triangle is 60°, so a shape may have three, four, or five triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
5276:
4468:
5042:
4772:
4742:
4712:
4616:
3235:
5581:
5394:
5177:
4860:
8429:, meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen
3513:
7158:
5511:
3447:
5483:
5247:
5117:
4800:
3880:
3793:
3767:
3729:
3648:
3591:
3383:
1125:
1115:
1091:
1081:
1042:
1014:
5089:
4939:
4914:
3482:
1466:
1513:
1135:
1101:
1062:
1052:
1034:
1024:
7601:. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the
5459:
3622:
2285:
or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. For these different shapes of faces the following holds:
1130:
1120:
1096:
1086:
1057:
1047:
1029:
1019:
4620:
The following table lists the various radii of the
Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length,
2399:
9842:, 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature. Talks about platonic solids.
4961:
3326:
3256:
5733:{\displaystyle \varphi =2\cos {\pi \over 5}={\frac {1+{\sqrt {5}}}{2}},\qquad \xi =2\sin {\pi \over 5}={\sqrt {\frac {5-{\sqrt {5}}}{2}}}={\sqrt {3-\varphi }}.}
5064:
5005:
4983:
4889:
387:(aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.
3357:
11517:
9374:
1010:
These coordinates reveal certain relationships between the
Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or
8425:, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both
7681:
is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference
Wythoff's symbol for each of the Platonic solids.
8672:
8152:
is often based on platonic solids. In the MERO system, Platonic solids are used for naming convention of various space frame configurations. For example,
10237:
5775:
9821:
2556:
2479:
7437:
3654:
2917:
9544:
8703:,2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.
3799:
3262:
11341:
10117:
10126:
8021:
3934:. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius
2392:
3519:
2872:{\displaystyle \tan \left({\frac {\theta }{2}}\right)={\frac {\cos \left({\frac {\pi }{q}}\right)}{\sin \left({\frac {\pi }{h}}\right)}}.}
2761:{\displaystyle \sin \left({\frac {\theta }{2}}\right)={\frac {\cos \left({\frac {\pi }{q}}\right)}{\sin \left({\frac {\pi }{p}}\right)}}.}
2413:
12516:
8445:, are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen
5323:
10941:
9567:
12521:
11510:
11419:
482:
10154:
12495:
10722:
10258:
9125:
as {3,3,...,4}. In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.
3007:
2385:
403:
has advocated the view that the construction of the five regular solids is the chief goal of the deductive system canonized in the
8675:
of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the
5517:
3094:
The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in
10230:
7397:}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.
11931:
11266:
10186:
8333:
that can be seen by the soul in the objects of the material world, but turned these shapes into more suitable for construction
5123:
10996:
9797:
8452:
The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of
4806:
3389:
12427:
11503:
10961:
478:
12621:
11329:
10732:
10105:
9400:
4579:
7609:
on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is
11396:
10223:
5407:
3087:
The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4
7363:. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.
10035:
9969:
9949:
9923:
9830:
9755:
9725:
9701:
9303:
7990:
4213:{\displaystyle \rho ={\frac {a}{2}}\cos \left({\frac {\pi }{p}}\right)\,{\csc }{\biggl (}{\frac {\pi }{h}}{\biggr )}}
1765:
8309:
7972:
1723:{\displaystyle V={\frac {4p}{4-(p-2)(q-2)}},\quad E={\frac {2pq}{4-(p-2)(q-2)}},\quad F={\frac {4q}{4-(p-2)(q-2)}}.}
7594:
2771:
5282:
5190:
12386:
7964:
2303:
Each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
1426:
17:
9334:
1479:
12511:
11365:
11298:
10931:
10811:
9776:
8683:
which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the
7968:
1216:
description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.
12554:
8371:
7670:
1146:
Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the
486:
2889:) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.
12534:
11434:
11191:
11079:
10254:
8475:
are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex
5253:
10095:
2312:
Each vertex is 108°; again, only one arrangement of three faces at a vertex is possible, the dodecahedron.
230:
have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher
12529:
12452:
11716:
11657:
11143:
11074:
9789:
7574:
of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the
5018:
4748:
4718:
4688:
3211:
502:
1158:
A convex polyhedron is a Platonic solid if and only if all three of the following requirements are met.
11924:
11746:
11706:
10770:
10586:
8354:
8322:
7602:
2896:
at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2
1140:
384:
281:
10561:
10180:
5563:
5376:
5159:
4842:
2243:
12381:
11741:
11736:
11305:
11276:
10636:
10491:
9209:
9165:
9160:
9083:
8992:
8978:
8963:
7678:
3488:
1073:
The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform
372:
364:
10120:
is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.
12616:
11439:
11173:
10716:
10017:
10013:
9997:
9715:
8418:
8298:
8127:
7953:
7579:
5489:
3424:
2365:
for the number of edges meeting at each vertex. Combining these equations one obtains the equation
444:
423:
376:
10123:
5465:
5225:
5095:
4778:
3860:
3773:
3745:
3709:
3628:
3571:
3363:
12581:
12576:
11847:
11842:
11721:
11627:
11401:
11377:
11251:
11186:
11127:
11064:
11054:
10790:
10709:
10571:
10481:
10361:
9294:
9180:
9170:
8679:. This is done by projecting each solid onto a concentric sphere. The faces project onto regular
8442:
8090:
virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical
7957:
5070:
4920:
4895:
3463:
3091:
steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.
368:
47:
10671:
7605:
of the symmetry group, as are the edges and faces. One says the action of the symmetry group is
7522:) is often convenient because the midsphere has the same relationship to both polyhedra. Taking
7133:{\displaystyle S_{}^{(4)}+{\frac {16}{9}}R^{4}=\left(S_{}^{(2)}+{\frac {2}{3}}R^{2}\right)^{2}.}
12571:
12566:
12347:
11711:
11652:
11642:
11587:
11424:
11372:
11271:
11099:
11049:
11034:
11029:
10800:
10601:
10536:
10526:
10476:
9252:
Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements in
8221:
7587:
3892:
Another virtue of regularity is that the Platonic solids all possess three concentric spheres:
1147:
1107:
10049:
7724:
5444:
4545:{\displaystyle A={\biggl (}{\frac {a}{2}}{\biggr )}^{2}Fp\cot \left({\frac {\pi }{p}}\right).}
3607:
410:
12561:
12544:
12341:
12236:
12144:
12109:
12081:
11997:
11917:
11731:
11647:
11602:
11550:
11472:
11324:
11168:
11109:
10986:
10909:
10857:
10659:
10566:
10416:
9959:
9538:
9368:
8968:
8193:
8172:
1163:
1004:
196:
192:
55:
8460:
for faces. These include all the polyhedra mentioned above together with an infinite set of
8293:
and K. Maki. In aluminum the icosahedral structure was discovered three years after this by
12447:
12185:
12130:
12067:
11969:
11962:
11691:
11617:
11565:
11449:
11389:
11353:
11196:
11019:
10966:
10936:
10926:
10835:
10698:
10591:
10506:
10461:
10441:
10286:
10271:
9865:
9653:
9498:
9464:
9086:. There are exactly six of these figures; five are analogous to the Platonic solids :
8438:
8375:
7878:
7760:
7571:
7368:
5183:
3897:
3597:
2330:
2261:
1471:
1342:
1248:
1074:
1067:
394:
216:
212:
136:
132:
95:
9430:
Schrek, D. J. E. (1950), "Prince Rupert's problem and its extension by Pieter Nieuwland",
4945:
3311:
3241:
1401:
All other combinatorial information about these solids, such as total number of vertices (
485:
rather than circles, changing the course of physics and astronomy. He also discovered the
8:
12594:
12300:
12278:
12178:
12171:
12095:
11976:
11857:
11726:
11701:
11686:
11622:
11570:
11429:
11348:
11336:
11317:
11281:
11201:
11119:
11104:
11094:
11044:
11039:
10981:
10850:
10742:
10606:
10596:
10496:
10466:
10406:
10381:
10306:
10296:
10281:
10166:
9739:
9681:
9432:
9185:
9155:
8869:
8330:
8048:
8017:
7811:
7559:
5048:
4989:
4967:
4873:
3735:
1517:
This can be proved in many ways. Together these three relationships completely determine
1371:
1255:
176:
172:
156:
152:
100:
80:
10005:
9869:
9657:
9502:
2619:
must both be at least 3, one can easily see that there are only five possibilities for {
407:. Much of the information in Book XIII is probably derived from the work of Theaetetus.
252:
The Platonic solids have been known since antiquity. It has been suggested that certain
12457:
12229:
12222:
12151:
12123:
12018:
11983:
11872:
11837:
11696:
11591:
11540:
11485:
11444:
11384:
11312:
11178:
11153:
10971:
10946:
10914:
10752:
10511:
10456:
10421:
10316:
9902:
9850:
9669:
9643:
9521:
9486:
9412:
9346:
9113:
In all dimensions higher than four, there are only three convex regular polytopes: the
8692:
8434:
7567:
4563:
3342:
2893:
2373:
1313:
1290:
351:
253:
236:
90:
43:
10192:
9391:
Jerrard, Richard P.; Wetzel, John E.; Yuan, Liping (April 2017). "Platonic Passages".
9079:
8110:, global numerical models of atmospheric flow are of increasing interest which employ
8063:, some of whose skeletons are shaped like various regular polyhedra. Examples include
7690:
7269:{\displaystyle 4\left(\sum _{i=1}^{n}d_{i}^{2}\right)^{2}=3n\sum _{i=1}^{n}d_{i}^{4}.}
1243:
1209:
1200:
is the number of faces (or, equivalently, edges) that meet at each vertex. This pair {
12317:
12271:
12137:
12046:
11852:
11662:
11637:
11581:
11480:
11158:
11069:
10919:
10845:
10818:
10581:
10401:
10391:
10326:
10246:
10205:
10196:
10140:
10078:
10059:
9965:
9945:
9919:
9906:
9826:
9793:
9772:
9751:
9744:
9721:
9697:
9690:
9673:
9526:
9416:
9299:
9135:
8973:
8855:
8680:
8430:
8422:
8405:
8251:
8004:
7734:
7654:
7634:
7606:
4230:= 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in
3170:
3065:
2297:
1228:
1177:
360:
241:
71:
7404:
More generally, one can dualize a Platonic solid with respect to a sphere of radius
2329:
proof can be made using only combinatorial information about the solids. The key is
1348:
12588:
12539:
12490:
12485:
12391:
12374:
12327:
12243:
12206:
12192:
12116:
12060:
12011:
11791:
11360:
11231:
11148:
10904:
10892:
10840:
10576:
10062:
9894:
9873:
9711:
9661:
9516:
9506:
9404:
9356:
9223:
9175:
9072:
8860:
8461:
7729:
7719:
7644:
7598:
3911:
3061:
1000:
458:
10201:
10146:
9476:
8766:. Likewise, a regular tessellation of the plane is characterized by the condition
8638:
1377:
1261:
12369:
12053:
11955:
11001:
10991:
10885:
10611:
10130:
9898:
9836:
9783:
9460:
9289:
8706:
Every regular tessellation of the sphere is characterized by a pair of integers {
8645:
8453:
8274:
8270:
8207:
7704:
7357:
7288:
1238:
1233:
1167:
435:
414:
400:
67:
59:
51:
39:
10210:
8624:
8269:
These shapes frequently show up in other games or puzzles. Puzzles similar to a
8168:
O+T refers to a configuration made of one half of octahedron and a tetrahedron.
1319:
12361:
12213:
12039:
11612:
11261:
11256:
11084:
10956:
10784:
10341:
10311:
10081:
9938:
9846:
9764:
9631:
9627:
9572:(1989). by Tamar Seideman, Reports on Progress in Physics, Volume 53, Number 6"
9451:
Scriba, Christoph J. (1968), "Das Problem des Prinzen Ruprecht von der Pfalz",
9285:
9205:
9122:
8617:
8346:
8290:
8286:
8138:
8033:
7711:
7697:
7614:
7563:
3137:
3002:, at the vertex of a Platonic solid is given in terms of the dihedral angle by
2886:
2648:
2376:
of the vertices of the five platonic solids – only the octahedron has an
328:
269:
257:
10111:
9665:
8631:
8126:
grid. This has the advantage of evenly distributed spatial resolution without
7153:
vertices of the Platonic solid to any point on its circumscribed sphere, then
3698:{\displaystyle \pi -\arctan \left({\frac {2}{11}}\right)\quad \approx 2.96174}
2368:
12610:
12477:
12462:
12442:
12322:
12032:
12025:
11817:
11673:
11607:
11221:
11089:
11059:
10880:
10688:
10631:
9877:
9812:
9408:
9150:
9140:
8850:
8472:
8446:
8414:
8394:
8350:
8294:
8263:
8227:
8180:
8115:
8111:
8056:
8025:
7348:
3069:
2987:
divided by the number of vertices (i.e. the total defect at all vertices is 4
2377:
1421:. Since any edge joins two vertices and has two adjacent faces we must have:
1213:
10215:
9887:
BSHM Bulletin: Journal of the British Society for the History of Mathematics
9511:
9360:
8955:
3849:{\displaystyle 2\pi -5\arcsin \left({2 \over 3}\right)\quad \approx 2.63455}
12549:
12416:
12264:
11163:
10951:
10626:
10386:
10376:
9933:
9685:
9530:
8499:
8457:
8314:
8250:. 6-sided dice are very common, but the other numbers are commonly used in
8199:
8184:
8039:
8008:
7864:
7622:
7618:
4440:
3300:{\displaystyle \arccos \left({\frac {23}{27}}\right)\quad \approx 0.551286}
3125:
2275:
2235:
2222:
510:
449:
418:
301:
9785:
Sources of Architectural Form: A Critical History of Western Design Theory
8130:(i.e. the poles) at the expense of somewhat greater numerical difficulty.
5751:
For an arbitrary point in the space of a Platonic solid with circumradius
319:
292:
12404:
10777:
10665:
10371:
10356:
10162:
10136:
9769:
The 2nd Scientific American Book of Mathematical Puzzles & Diversions
9145:
9082:
discovered the four-dimensional analogues of the Platonic solids, called
8476:
8413:
The next most regular convex polyhedra after the Platonic solids are the
8149:
8107:
8103:
7915:
7882:
7815:
7746:
7739:
5400:
4681:
3201:
3182:
2995:
1782:
9648:
8527:
8520:
8513:
8506:
7764:
2200:
310:
12467:
12333:
12250:
11882:
11770:
11560:
11527:
10763:
10682:
10621:
10616:
10556:
10541:
10486:
10471:
10426:
10366:
10351:
10331:
10301:
10266:
9482:
8842:
8835:
8828:
8379:
8060:
8044:
7848:
7339:
5011:
3453:
2651:
is the interior angle between any two face planes. The dihedral angle,
2240:
A 0° angle defect will fill the Euclidean plane with a regular tiling.
432:
277:
9215:
8579:
8572:
8133:
7328:
7317:
3560:{\displaystyle 4\arcsin \left({1 \over 3}\right)\quad \approx 1.35935}
1196:
is the number of edges (or, equivalently, vertices) of each face, and
12354:
12309:
12257:
12004:
11877:
11867:
11812:
11796:
11632:
10516:
10501:
10451:
10346:
10336:
10321:
10291:
10086:
10067:
9478:
9256:
but notes that this correspondence appears to have been forgotten in
9118:
9099:
8684:
8586:
8465:
8357:, was preoccupied with the architects' version of "Platonic solids".
8280:
8247:
8142:
8119:
3904:
3095:
2256:
The following geometric argument is very similar to the one given by
380:
9771:, University of Chicago Press, Chapter 1: The Five Platonic Solids,
8948:
8565:
8400:
8234:
8079:. The shapes of these creatures should be obvious from their names.
7942:
4226:
is the quantity used above in the definition of the dihedral angle (
12437:
12410:
12199:
12102:
11763:
11495:
10653:
10431:
10276:
10016:
external links, and converting useful links where appropriate into
9351:
9258:
9103:
9095:
9068:
8868:
In a similar manner, one can consider regular tessellations of the
8558:
8338:
8318:
8123:
7555:
5864:{\displaystyle S_{}^{(2m)}={\frac {1}{n}}\sum _{i=1}^{n}d_{i}^{2m}}
5365:{\displaystyle {\frac {20\varphi ^{3}}{\xi ^{2}}}\approx 61.304952}
2326:
2306:
2288:
2271:
Each vertex of the solid must be a vertex for at least three faces.
261:
227:
31:
7593:
The symmetry groups of the Platonic solids are a special class of
7530:
yields a dual solid with the same circumradius and inradius (i.e.
2176:
12285:
11887:
11862:
11226:
10551:
10546:
10446:
10436:
10411:
9114:
9107:
9091:
8696:
8289:, the existence of such symmetries was first proposed in 1981 by
8091:
8036:
also have molecular structures approximating regular icosahedra.
7291:
if a polyhedron of the same or larger size and the same shape as
2282:
2165:
2154:
2143:
474:
273:
9885:
Lloyd, David Robert (2012). "How old are the Platonic Solids?".
9208:
wrote a popular account of the five solids in his December 1958
9075:
being the equivalents of the three-dimensional Platonic solids.
8941:
8389:
5755:, whose distances to the centroid of the Platonic solid and its
2630:{3, 3}, {4, 3}, {3, 4}, {5, 3}, {3, 5}.
2602:{\displaystyle {\frac {1}{q}}+{\frac {1}{p}}>{\frac {1}{2}}.}
2538:{\displaystyle {1 \over q}+{1 \over p}={1 \over 2}+{1 \over E}.}
1752:
unchanged. For a geometric interpretation of this property, see
1038:, one of two sets of 4 vertices in dual positions, as h{4,3} or
12088:
10521:
10396:
9735:
9087:
8676:
8370:
There exist four regular polyhedra that are not convex, called
8334:
8213:
8176:
8096:
8087:
8012:
7505:{\displaystyle d^{2}=R^{\ast }r=r^{\ast }R=\rho ^{\ast }\rho .}
4555:
3919:
2257:
2211:
454:
439:
393:
completely mathematically described the Platonic solids in the
390:
272:
studied the Platonic solids extensively. Some sources (such as
185:
8934:
2974:{\displaystyle \delta =2\pi -q\pi \left(1-{2 \over p}\right).}
2770:
This is sometimes more conveniently expressed in terms of the
205:
125:
11990:
11940:
10897:
10875:
10531:
8083:
8003:
The tetrahedron, cube, and octahedron all occur naturally in
3907:
that is tangent to each edge at the midpoint of the edge, and
2644:
466:
462:
347:
231:
165:
145:
63:
9840:
Strena seu de nive sexangula (On the Six-Cornered Snowflake)
9593:
5552:{\displaystyle {\frac {20\varphi ^{2}}{3}}\approx 17.453560}
12074:
11555:
9295:
The Mechanical Universe: Introduction to Mechanics and Heat
8342:
8243:
7797:
4866:
3332:
2189:
1284:
470:
457:. The six spheres each corresponded to one of the planets (
85:
9961:
John Philoponus' Criticism of Aristotle's Theory of Aether
9581:
1003:), or, in the other cases, by exchanging two coordinates (
9487:"Why large icosahedral viruses need scaffolding proteins"
9335:"Cyclic Averages of Regular Polygons and Platonic Solids"
350:, their namesake. Plato wrote about them in the dialogue
11909:
9634:(2003). "Polyhedra in Physics, Chemistry and Geometry".
5148:{\displaystyle {\frac {\sqrt {128}}{3}}\approx 3.771236}
1470:
The other relationship between these values is given by
10152:
10057:
9918:. California: University of California Press Berkeley.
9845:
9283:
9067:
In more than three dimensions, polyhedra generalize to
7660:(which is also the symmetry group of the dodecahedron).
3914:
that is tangent to each face at the center of the face.
346:
The Platonic solids are prominent in the philosophy of
27:
Convex polyhedron with identical, regular polygon faces
12161:
9605:
7377:
The dodecahedron and the icosahedron form a dual pair.
4831:{\displaystyle {\frac {\sqrt {8}}{3}}\approx 0.942809}
3413:{\displaystyle {\frac {\pi }{2}}\quad \approx 1.57080}
1184:
Each Platonic solid can therefore be assigned a pair {
70:
congruent), and the same number of faces meet at each
9750:(2nd unabr. ed.). New York: Dover Publications.
8924:. There is an infinite family of such tessellations.
7440:
7161:
7006:
5882:
5778:
5604:
5566:
5520:
5492:
5468:
5447:
5410:
5379:
5326:
5285:
5256:
5228:
5193:
5162:
5126:
5098:
5073:
5051:
5021:
4992:
4970:
4948:
4923:
4898:
4876:
4845:
4809:
4781:
4751:
4722:
4692:
4582:
4471:
4246:
4139:
3962:
3864:
3802:
3777:
3748:
3713:
3657:
3632:
3610:
3575:
3522:
3492:
3466:
3428:
3392:
3367:
3345:
3314:
3265:
3244:
3215:
3010:
2920:
2782:
2671:
2559:
2482:
2416:
1537:
1482:
1429:
9746:
The Thirteen Books of Euclid's Elements, Books 10–13
2465:{\displaystyle {\frac {2E}{q}}-E+{\frac {2E}{p}}=2.}
2380:
or cycle, by extending its path with the dotted one
2372:
Orthographic projections and Schlegel diagrams with
2315:
Altogether this makes five possible Platonic solids.
448:, published in 1596, Kepler proposed a model of the
7650:(which is also the symmetry group of the cube), and
2994:The three-dimensional analog of a plane angle is a
1764:The elements of a polyhedron can be expressed in a
1007:with respect to any of the three diagonal planes).
501:For Platonic solids centered at the origin, simple
442:known at that time to the five Platonic solids. In
10076:
9937:
9743:
9689:
9318:Coxeter, Regular Polytopes, sec 1.8 Configurations
8360:
8281:Liquid crystals with symmetries of Platonic solids
7504:
7268:
7132:
6990:
5863:
5732:
5575:
5551:
5505:
5477:
5453:
5432:
5388:
5364:
5311:
5270:
5241:
5213:
5171:
5147:
5111:
5083:
5058:
5036:
4999:
4977:
4955:
4933:
4908:
4883:
4854:
4830:
4794:
4766:
4736:
4706:
4610:
4544:
4429:
4212:
4113:
3874:
3848:
3787:
3761:
3723:
3697:
3642:
3616:
3585:
3559:
3507:
3476:
3441:
3412:
3377:
3351:
3320:
3299:
3250:
3229:
3050:
2973:
2871:
2760:
2601:
2537:
2464:
1722:
1507:
1460:
1173:None of its faces intersect except at their edges.
505:of the vertices are given below. The Greek letter
10000:may not follow Knowledge's policies or guidelines
9851:"Lattice Textures in Cholesteric Liquid Crystals"
9390:
9373:: CS1 maint: DOI inactive as of September 2024 (
9265:
8822:The three regular tilings of the Euclidean plane
8059:described (Haeckel, 1904) a number of species of
7629:and vice versa. The three polyhedral groups are:
5433:{\displaystyle {\frac {\varphi ^{2}}{\sqrt {3}}}}
4498:
4480:
4416:
4399:
4384:
4367:
4345:
4328:
4205:
4188:
359:360 B.C. in which he associated each of the four
12608:
9298:. Cambridge University Press. pp. 434–436.
9234:
9078:In the mid-19th century the Swiss mathematician
8928:Example regular tilings of the hyperbolic plane
2361:stands for the number of edges of each face and
234:, who hypothesized in one of his dialogues, the
9626:
9491:Proceedings of the National Academy of Sciences
7299:. All five Platonic solids have this property.
1066:. Both tetrahedral positions make the compound
9720:(3rd ed.). New York: Dover Publications.
9339:Communications in Mathematics and Applications
8329:Architects liked the idea of Plato's timeless
438:attempted to relate the five extraterrestrial
11925:
11511:
10245:
10231:
10124:Interactive Folding/Unfolding Platonic Solids
9944:. Princeton, NJ: Princeton University Press.
9680:
8114:that are based on an icosahedron (refined by
7374:The cube and the octahedron form a dual pair.
3051:{\displaystyle \Omega =q\theta -(q-2)\pi .\,}
2983:By a theorem of Descartes, this is equal to 4
2393:
1176:The same number of faces meet at each of its
10211:How to make four platonic solids from a cube
9620:
9543:: CS1 maint: multiple names: authors list (
9332:
8262:is the number of faces (d8, d20, etc.); see
8032:icosahedra within their crystal structures.
7570:) which leave the polyhedron invariant. The
5312:{\displaystyle 12{\sqrt {25+10{\sqrt {5}}}}}
5214:{\displaystyle {\frac {\varphi ^{2}}{\xi }}}
50:. Being a regular polyhedron means that the
9106:as {5,3,3}, and a sixth one, the self-dual
8872:. These are characterized by the condition
8437:with polyhedral symmetry. Their duals, the
8285:For the intermediate material phase called
8246:, because dice of these shapes can be made
7971:. Unsourced material may be challenged and
7932:
7566:, which is the set of all transformations (
1153:
11932:
11918:
11518:
11504:
10238:
10224:
4455:} is easily computed as area of a regular
3887:
2400:
2386:
10036:Learn how and when to remove this message
9781:
9647:
9599:
9587:
9520:
9510:
9386:
9384:
9350:
8382:of the dodecahedron and the icosahedron.
8254:. Such dice are commonly referred to as d
7991:Learn how and when to remove this message
7514:Dualizing with respect to the midsphere (
5264:
5055:
4996:
4974:
4952:
4880:
4180:
3047:
2647:associated with each Platonic solid. The
2474:Simple algebraic manipulation then gives
1504:
1457:
12517:List of manuscripts of Plato's dialogues
9328:
9326:
9324:
8482:
8421:of the cube and the octahedron, and the
8308:
8233:
8132:
8038:
2904:, at any vertex of the Platonic solids {
2367:
2234:A vertex needs at least 3 faces, and an
496:
409:
9710:
9611:
9228:The Stanford Encyclopedia of Philosophy
8349:. In particular, one of the leaders of
8242:Platonic solids are often used to make
7371:(i.e. its dual is another tetrahedron).
2633:
337:Assignment to the elements in Kepler's
14:
12609:
11267:Latin translations of the 12th century
10189:teacher instructions for making models
9964:. Walter de Gruyter. pp. 11–12.
9734:
9450:
9429:
9381:
7408:concentric with the solid. The radii (
6998:For all five Platonic solids, we have
4570:-gon and whose height is the inradius
12496:List of speakers in Plato's dialogues
11913:
11499:
10997:Straightedge and compass construction
10219:
10077:
10058:
9884:
9321:
9240:
7562:. Every polyhedron has an associated
7381:If a polyhedron has Schläfli symbol {
5271:{\displaystyle {\sqrt {3}}\,\varphi }
4127:is the dihedral angle. The midradius
3900:that passes through all the vertices,
1759:
1224:
11525:
10962:Incircle and excircles of a triangle
9980:
9932:
9913:
9271:
9221:
8986:
8691:} with 2 vertices at the poles, and
8365:
8118:) instead of the more commonly used
8043:Circogonia icosahedra, a species of
7969:adding citations to reliable sources
7936:
7420:) of a solid and those of its dual (
7361:with faces and vertices interchanged
2320:
74:There are only five such polyhedra:
10137:Paper models of the Platonic solids
9819:. Available as Haeckel, E. (1998);
9401:Mathematical Association of America
7679:Wythoff's kaleidoscope construction
5037:{\displaystyle {\sqrt {2 \over 3}}}
4767:{\displaystyle {\sqrt {3 \over 2}}}
4737:{\displaystyle 1 \over {\sqrt {2}}}
4707:{\displaystyle 1 \over {\sqrt {6}}}
4611:{\displaystyle V={\frac {1}{3}}rA.}
3230:{\displaystyle 1 \over {\sqrt {2}}}
1753:
598:20 (8 + 4 × 3)
244:were made of these regular solids.
24:
9570:The liquid-crystalline blue phases
8304:
7549:
7278:
3011:
2551:is strictly positive we must have
2345: = 2, and the fact that
2131:
483:the orbits of planets are ellipses
25:
12633:
9976:
9071:, with higher-dimensional convex
8818:. There are three possibilities:
8468:, and 53 other non-convex forms.
8175:have been synthesised, including
7669:meaning they are preserved under
7389:}, then its dual has the symbol {
7307:
5746:
2246:, the number of vertices is 720°/
2122:
48:three-dimensional Euclidean space
12299:
11479:
11466:
10139:created using nets generated by
9985:
8954:
8947:
8940:
8933:
8841:
8834:
8827:
8644:
8637:
8630:
8623:
8616:
8585:
8578:
8571:
8564:
8557:
8526:
8519:
8512:
8505:
8498:
8399:
8388:
8220:
8206:
8192:
7941:
7881:
7814:
7763:
7558:is studied with the notion of a
7338:
7327:
7316:
5576:{\displaystyle \approx 2.181695}
5389:{\displaystyle \approx 7.663119}
5172:{\displaystyle \approx 0.471404}
4855:{\displaystyle \approx 0.117851}
3922:of these spheres are called the
2221:
2210:
2199:
2188:
2175:
2164:
2153:
2142:
1376:
1347:
1318:
1289:
1260:
1133:
1128:
1123:
1118:
1113:
1099:
1094:
1089:
1084:
1079:
1060:
1055:
1050:
1045:
1040:
1032:
1027:
1022:
1017:
1012:
431:In the 16th century, the German
327:
318:
309:
300:
291:
204:
184:
164:
144:
124:
10153:Grime, James; Steckles, Katie.
10053:at Encyclopaedia of Mathematics
9560:
9551:
9470:
9444:
9423:
8361:Related polyhedra and polytopes
7554:In mathematics, the concept of
5658:
4459:-gon times the number of faces
3839:
3688:
3550:
3508:{\displaystyle {2\pi } \over 3}
3403:
3290:
1660:
1597:
12522:Cultural influence of Plato's
11299:A History of Greek Mathematics
10812:The Quadrature of the Parabola
9312:
9277:
9246:
9198:
8273:come in all five shapes – see
7595:three-dimensional point groups
7088:
7082:
7077:
7071:
7029:
7023:
7018:
7012:
6797:
6791:
6786:
6780:
6767:
6761:
6756:
6750:
6587:
6581:
6576:
6570:
6557:
6551:
6546:
6540:
6417:
6411:
6406:
6400:
6387:
6381:
6376:
6370:
6357:
6351:
6346:
6340:
6327:
6321:
6316:
6310:
6213:
6207:
6202:
6196:
6183:
6177:
6172:
6166:
6153:
6147:
6142:
6136:
6123:
6117:
6112:
6106:
6093:
6087:
6082:
6076:
6029:
6023:
6018:
6012:
5999:
5993:
5988:
5982:
5969:
5963:
5958:
5952:
5939:
5933:
5928:
5922:
5909:
5903:
5898:
5892:
5804:
5795:
5790:
5784:
3038:
3026:
1711:
1699:
1696:
1684:
1651:
1639:
1636:
1624:
1588:
1576:
1573:
1561:
1220:Properties of Platonic solids
481:, the first of which was that
479:three laws of orbital dynamics
58:(identical in shape and size)
13:
1:
9333:Meskhishvili, Mamuka (2020).
7671:reflection through the origin
5506:{\displaystyle 20{\sqrt {3}}}
3442:{\displaystyle 2\pi \over 3}
2137:Polygon nets around a vertex
356:
12535:Platonism in the Renaissance
12387:Plato's political philosophy
11898:Degenerate polyhedra are in
11080:Intersecting secants theorem
9958:Wildberg, Christian (1988).
9916:Polyhedra: A visual approach
9899:10.1080/17498430.2012.670845
9192:
7358:dual (or "polar") polyhedron
5478:{\displaystyle \xi \varphi }
5242:{\displaystyle \varphi ^{2}}
5112:{\displaystyle 8{\sqrt {3}}}
4864:
4795:{\displaystyle 4{\sqrt {3}}}
3875:{\displaystyle \pi \over 5}
3788:{\displaystyle \pi \over 3}
3762:{\displaystyle \varphi ^{2}}
3724:{\displaystyle \pi \over 3}
3643:{\displaystyle \pi \over 5}
3586:{\displaystyle \pi \over 2}
3378:{\displaystyle \pi \over 2}
2104:
2093:
2082:
2062:
2051:
2040:
2020:
2009:
1998:
1978:
1967:
1956:
1936:
1925:
1914:
1881:
1864:
1840:
1369:
1340:
1311:
1282:
1253:
417:Platonic solid model of the
7:
12530:Neoplatonism and Gnosticism
11717:pentagonal icositetrahedron
11658:truncated icosidodecahedron
11075:Intersecting chords theorem
10942:Doctrine of proportionality
9790:Manchester University Press
9128:
8055:In the early 20th century,
7665:except the tetrahedron are
7302:
7295:can pass through a hole in
7149:are the distances from the
5084:{\displaystyle {\sqrt {2}}}
4934:{\displaystyle {\sqrt {3}}}
4909:{\displaystyle {\sqrt {2}}}
3477:{\displaystyle {\sqrt {2}}}
1461:{\displaystyle pF=2E=qV.\,}
10:
12638:
12622:Multi-dimensional geometry
11747:pentagonal hexecontahedron
11707:deltoidal icositetrahedron
10771:On the Sphere and Cylinder
10724:On the Sizes and Distances
9084:convex regular 4-polytopes
8990:
8488:Regular spherical tilings
5596:in the above are given by
2663:} is given by the formula
1508:{\displaystyle V-E+F=2.\,}
1413:), can be determined from
1141:compound of two icosahedra
247:
76:
12504:
12476:
12426:
12308:
12297:
11947:
11939:
11896:
11830:
11805:
11787:
11780:
11755:
11742:disdyakis triacontahedron
11737:deltoidal hexecontahedron
11671:
11579:
11534:
11473:Ancient Greece portal
11462:
11412:
11290:
11277:Philosophy of mathematics
11247:
11240:
11214:
11192:Ptolemy's table of chords
11136:
11118:
11017:
11010:
10866:
10828:
10645:
10253:
10247:Ancient Greek mathematics
10193:Frames of Platonic Solids
9666:10.1007/s00032-003-0014-1
9621:General and cited sources
9210:Mathematical Games column
9166:List of regular polytopes
9161:Kepler-Poinsot polyhedron
8993:List of regular polytopes
8610:
8551:
8492:
8238:A set of polyhedral dice.
8137:Icosahedron as a part of
7907:
7902:
7899:
7887:
7877:
7840:
7835:
7832:
7820:
7810:
7710:
7703:
7696:
7689:
7686:
4647:
4640:
4637:
4629:
2638:
2233:
1816:
1810:
1776:
1370:
1341:
1312:
1283:
1254:
1247:
1242:
1237:
1232:
1227:
629:
597:
594:
585:
582:
577:
574:
565:
562:
557:
554:
545:
542:
509:is used to represent the
453:represented the orbit of
11144:Aristarchus's inequality
10717:On Conoids and Spheroids
10149:Free paper models (nets)
10106:Interactive 3D Polyhedra
9878:10.1002/prop.19810290503
9782:Gelernter, Mark (1995).
9692:A History of Mathematics
9557:Kleinert and Maki (1981)
9409:10.4169/math.mag.90.2.87
9121:as {4,3,...,3}, and the
8372:Kepler–Poinsot polyhedra
8299:Nobel Prize in Chemistry
8077:Circorrhegma dodecahedra
8011:(named for the group of
7933:In nature and technology
5454:{\displaystyle \varphi }
4566:whose base is a regular
4562:times the volume of the
3617:{\displaystyle \varphi }
1777:Platonic configurations
1154:Combinatorial properties
445:Mysterium Cosmographicum
424:Mysterium Cosmographicum
11848:gyroelongated bipyramid
11722:rhombic triacontahedron
11628:truncated cuboctahedron
11252:Ancient Greek astronomy
11065:Inscribed angle theorem
11055:Greek geometric algebra
10710:Measurement of a Circle
9858:Fortschritte der Physik
9849:& Maki, K. (1981).
9696:(2nd ed.). Wiley.
9512:10.1073/pnas.1807706115
9361:10.26713/cma.v11i3.1420
9212:in Scientific American.
9181:Regular skew polyhedron
8443:rhombic triacontahedron
8433:, which are the convex
8378:and may be obtained as
8297:, which earned him the
7714:(reflection, rotation)
7356:Every polyhedron has a
4447:, of a Platonic solid {
3888:Radii, area, and volume
383:added a fifth element,
11843:truncated trapezohedra
11712:disdyakis dodecahedron
11678:(duals of Archimedean)
11653:rhombicosidodecahedron
11643:truncated dodecahedron
11486:Mathematics portal
11272:Non-Euclidean geometry
11227:Mouseion of Alexandria
11100:Tangent-secant theorem
11050:Geometric mean theorem
11035:Exterior angle theorem
11030:Angle bisector theorem
10734:On Sizes and Distances
10187:Teaching Math with Art
10183:student-created models
10181:Teaching Math with Art
9914:Pugh, Anthony (1976).
9481:, Alex Travesset, and
9363:(inactive 2024-09-18).
8326:
8239:
8145:
8073:Lithocubus geometricus
8052:
7586:, which includes only
7506:
7270:
7247:
7191:
7134:
6992:
5865:
5842:
5734:
5577:
5553:
5507:
5479:
5455:
5434:
5390:
5366:
5313:
5272:
5243:
5215:
5173:
5149:
5113:
5085:
5060:
5038:
5001:
4979:
4957:
4935:
4910:
4885:
4856:
4832:
4796:
4768:
4738:
4708:
4612:
4546:
4431:
4214:
4115:
3876:
3850:
3789:
3763:
3725:
3699:
3644:
3618:
3587:
3561:
3509:
3478:
3443:
3414:
3379:
3353:
3322:
3301:
3252:
3231:
3068:and the fact that the
3060:This follows from the
3052:
2975:
2873:
2762:
2643:There are a number of
2603:
2539:
2466:
2408:
1724:
1509:
1462:
1148:compound of five cubes
428:
12486:The Academy in Athens
12342:Platonic epistemology
11732:pentakis dodecahedron
11648:truncated icosahedron
11603:truncated tetrahedron
11174:Pappus's area theorem
11110:Theorem of the gnomon
10987:Quadratrix of Hippias
10910:Circles of Apollonius
10858:Problem of Apollonius
10836:Constructible numbers
10660:Archimedes Palimpsest
9817:Kunstformen der Natur
9453:Praxis der Mathematik
9399:(2). Washington, DC:
9222:Zeyl, Donald (2019).
8991:Further information:
8673:regular tessellations
8483:Regular tessellations
8464:, an infinite set of
8355:Étienne-Louis Boullée
8323:Étienne-Louis Boullée
8312:
8237:
8173:Platonic hydrocarbons
8136:
8069:Circogonia icosahedra
8065:Circoporus octahedrus
8042:
7613:if and only if it is
7584:proper symmetry group
7507:
7271:
7227:
7171:
7135:
6993:
5866:
5822:
5735:
5578:
5554:
5508:
5480:
5456:
5435:
5391:
5367:
5314:
5273:
5244:
5216:
5174:
5150:
5114:
5086:
5061:
5039:
5002:
4980:
4958:
4936:
4911:
4886:
4857:
4833:
4797:
4769:
4739:
4709:
4613:
4547:
4432:
4215:
4116:
3877:
3851:
3790:
3764:
3726:
3700:
3645:
3619:
3588:
3562:
3510:
3479:
3444:
3415:
3380:
3354:
3323:
3302:
3253:
3232:
3053:
2976:
2874:
2763:
2604:
2540:
2467:
2371:
1754:§ Dual polyhedra
1725:
1510:
1463:
1192:} of integers, where
1162:All of its faces are
503:Cartesian coordinates
497:Cartesian coordinates
413:
11692:rhombic dodecahedron
11618:truncated octahedron
11390:prehistoric counting
11187:Ptolemy's inequality
11128:Apollonius's theorem
10967:Method of exhaustion
10937:Diophantine equation
10927:Circumscribed circle
10744:On the Moving Sphere
10006:improve this article
9393:Mathematics Magazine
9171:Prince Rupert's cube
9117:as {3,3,...,3}, the
8695:faces, and the dual
8439:rhombic dodecahedron
8376:icosahedral symmetry
8028:, include discrete B
7965:improve this section
7667:centrally symmetric,
7568:Euclidean isometries
7438:
7287:is said to have the
7159:
7004:
5880:
5776:
5602:
5564:
5518:
5490:
5466:
5445:
5408:
5377:
5324:
5283:
5254:
5226:
5191:
5160:
5124:
5096:
5071:
5049:
5019:
4990:
4968:
4956:{\displaystyle 24\,}
4946:
4921:
4896:
4874:
4843:
4807:
4779:
4749:
4719:
4689:
4624:, to be equal to 2.
4580:
4469:
4244:
4137:
3960:
3898:circumscribed sphere
3861:
3800:
3774:
3746:
3710:
3655:
3629:
3608:
3572:
3520:
3489:
3464:
3425:
3390:
3364:
3343:
3321:{\displaystyle \pi }
3312:
3263:
3251:{\displaystyle \pi }
3242:
3212:
3008:
2918:
2780:
2669:
2634:Geometric properties
2611:Using the fact that
2557:
2480:
2414:
2281:Regular polygons of
1766:configuration matrix
1535:
1480:
1427:
1249:Vertex configuration
1075:truncated octahedron
1068:stellated octahedron
595:12 (4 × 3)
265:always symmetrical.
12595:Poitier Meets Plato
12512:Unwritten doctrines
11727:triakis icosahedron
11702:tetrakis hexahedron
11687:triakis tetrahedron
11623:rhombicuboctahedron
11476: •
11282:Neusis construction
11202:Spiral of Theodorus
11095:Pythagorean theorem
11040:Euclidean algorithm
10982:Lune of Hippocrates
10851:Squaring the circle
10607:Theon of Alexandria
10282:Aristaeus the Elder
10206:formula derivations
10114:in Visual Polyhedra
10018:footnote references
9870:1981ForPh..29..219K
9822:Art forms in nature
9658:2003math.ph...3071A
9602:, pp. 172–173.
9503:2018PNAS..11510971L
9497:(43): 10971–10976.
9440:: 73–80 and 261–267
9433:Scripta Mathematica
9186:Toroidal polyhedron
9156:Goldberg polyhedron
8929:
8823:
8611:Regular hosohedral
8489:
8049:regular icosahedron
8018:Allotropes of boron
7576:full symmetry group
7367:The tetrahedron is
7262:
7206:
7092:
7033:
6801:
6771:
6591:
6561:
6421:
6391:
6361:
6331:
6217:
6187:
6157:
6127:
6097:
6033:
6003:
5973:
5943:
5913:
5860:
5808:
5059:{\displaystyle 1\,}
5000:{\displaystyle 1\,}
4978:{\displaystyle 8\,}
4884:{\displaystyle 1\,}
3950:} with edge length
3072:of the polyhedron {
2998:. The solid angle,
2331:Euler's observation
2138:
2080:
2038:
1996:
1954:
1912:
1256:Regular tetrahedron
1221:
589:6 (2 × 3)
539:
493:regular polyhedra.
12562:Oxyrhynchus Papyri
11697:triakis octahedron
11582:Archimedean solids
11169:Menelaus's theorem
11159:Irrational numbers
10972:Parallel postulate
10947:Euclidean geometry
10915:Apollonian circles
10457:Isidore of Miletus
10197:algebraic surfaces
10129:2007-02-09 at the
10079:Weisstein, Eric W.
10060:Weisstein, Eric W.
9009:regular polytopes
8927:
8821:
8681:spherical polygons
8487:
8431:Archimedean solids
8327:
8266:for more details.
8252:role-playing games
8240:
8146:
8053:
8005:crystal structures
7560:mathematical group
7502:
7432:*) are related by
7266:
7248:
7192:
7130:
7066:
7007:
6988:
6986:
6775:
6745:
6565:
6535:
6395:
6365:
6335:
6305:
6191:
6161:
6131:
6101:
6071:
6007:
5977:
5947:
5917:
5887:
5861:
5843:
5779:
5770:respectively, and
5730:
5573:
5549:
5503:
5475:
5451:
5430:
5386:
5362:
5309:
5268:
5239:
5211:
5169:
5145:
5109:
5081:
5056:
5034:
4997:
4975:
4953:
4931:
4906:
4881:
4852:
4828:
4792:
4764:
4726:
4696:
4608:
4542:
4427:
4210:
4111:
4109:
3868:
3846:
3781:
3759:
3717:
3695:
3636:
3614:
3579:
3557:
3501:
3474:
3435:
3410:
3371:
3349:
3318:
3297:
3248:
3219:
3048:
2971:
2894:angular deficiency
2869:
2758:
2599:
2535:
2462:
2409:
2374:Hamiltonian cycles
2244:Descartes' theorem
2136:
2078:
2036:
1994:
1952:
1910:
1760:As a configuration
1720:
1505:
1458:
1314:Regular octahedron
1219:
1139:, and seen in the
537:
429:
361:classical elements
254:carved stone balls
242:classical elements
66:congruent and all
44:regular polyhedron
12604:
12603:
12318:Euthyphro dilemma
12295:
12294:
12272:Second Alcibiades
11907:
11906:
11826:
11825:
11663:snub dodecahedron
11638:icosidodecahedron
11493:
11492:
11458:
11457:
11210:
11209:
11197:Ptolemy's theorem
11070:Intercept theorem
10920:Apollonian gasket
10846:Doubling the cube
10819:The Sand Reckoner
10155:"Platonic Solids"
10118:Solid Body Viewer
10046:
10045:
10038:
9799:978-0-7190-4129-7
9717:Regular Polytopes
9712:Coxeter, H. S. M.
9590:, pp. 50–51.
9224:"Plato's Timaeus"
9136:Archimedean solid
9073:regular polytopes
9065:
9064:
8987:Higher dimensions
8984:
8983:
8866:
8865:
8669:
8668:
8552:Regular dihedral
8435:uniform polyhedra
8423:icosidodecahedron
8411:
8410:
8406:icosidodecahedron
8374:. These all have
8366:Uniform polyhedra
8313:A project of the
8001:
8000:
7993:
7930:
7929:
7655:icosahedral group
7635:tetrahedral group
7599:polyhedral groups
7578:, which includes
7104:
7045:
6860:
6716:
6276:
5820:
5725:
5709:
5708:
5702:
5682:
5653:
5647:
5628:
5586:
5585:
5541:
5501:
5428:
5427:
5354:
5307:
5305:
5262:
5209:
5137:
5133:
5107:
5079:
5032:
5031:
4929:
4904:
4820:
4816:
4790:
4762:
4761:
4734:
4732:
4704:
4702:
4597:
4533:
4493:
4422:
4412:
4389:
4380:
4341:
4302:
4278:
4255:
4201:
4174:
4154:
4101:
4077:
4057:
4029:
4005:
3985:
3938:and the inradius
3885:
3884:
3872:
3833:
3785:
3721:
3682:
3640:
3583:
3544:
3505:
3472:
3439:
3401:
3375:
3352:{\displaystyle 1}
3284:
3227:
3225:
3066:spherical polygon
2961:
2864:
2857:
2831:
2801:
2753:
2746:
2720:
2690:
2594:
2581:
2568:
2530:
2517:
2504:
2491:
2454:
2430:
2321:Topological proof
2254:
2253:
2120:
2119:
2116:
2115:
2074:
2073:
2032:
2031:
1990:
1989:
1948:
1947:
1906:
1905:
1715:
1655:
1592:
1399:
1398:
996:
995:
225:
224:
220:
200:
180:
160:
140:
16:(Redirected from
12629:
12555:and Christianity
12540:Middle Platonism
12491:Socratic problem
12453:The Divided Line
12392:Philosopher king
12375:Form of the Good
12328:Cardinal virtues
12303:
12159:
12158:
12012:First Alcibiades
11934:
11927:
11920:
11911:
11910:
11785:
11784:
11781:Dihedral uniform
11756:Dihedral regular
11679:
11595:
11544:
11520:
11513:
11506:
11497:
11496:
11484:
11483:
11471:
11470:
11469:
11245:
11244:
11232:Platonic Academy
11179:Problem II.8 of
11149:Crossbar theorem
11105:Thales's theorem
11045:Euclid's theorem
11015:
11014:
10932:Commensurability
10893:Axiomatic system
10841:Angle trisection
10806:
10796:
10758:
10748:
10738:
10728:
10704:
10694:
10677:
10240:
10233:
10226:
10217:
10216:
10177:
10175:
10174:
10165:. Archived from
10092:
10091:
10073:
10072:
10063:"Platonic solid"
10041:
10034:
10030:
10027:
10021:
9989:
9988:
9981:
9955:
9943:
9929:
9910:
9881:
9855:
9837:Kepler. Johannes
9809:
9807:
9806:
9761:
9749:
9740:Heath, Thomas L.
9731:
9707:
9695:
9677:
9651:
9615:
9609:
9603:
9597:
9591:
9585:
9579:
9578:
9576:
9564:
9558:
9555:
9549:
9548:
9542:
9534:
9524:
9514:
9485:(October 2018).
9474:
9468:
9467:
9448:
9442:
9441:
9427:
9421:
9420:
9388:
9379:
9378:
9372:
9364:
9354:
9330:
9319:
9316:
9310:
9309:
9290:Goodstein, D. L.
9284:Olenick, R. P.;
9281:
9275:
9269:
9263:
9250:
9244:
9238:
9232:
9231:
9219:
9213:
9204:Gardner (1987):
9202:
9176:Regular polytope
9102:as {4,3,3}, and
9008:
9007:Number of convex
9002:
8997:
8996:
8958:
8951:
8944:
8937:
8930:
8926:
8923:
8921:
8920:
8917:
8914:
8908: <
8907:
8905:
8904:
8899:
8896:
8889:
8887:
8886:
8881:
8878:
8870:hyperbolic plane
8845:
8838:
8831:
8824:
8820:
8817:
8815:
8814:
8811:
8808:
8801:
8799:
8798:
8793:
8790:
8783:
8781:
8780:
8775:
8772:
8765:
8763:
8762:
8759:
8756:
8750: >
8749:
8747:
8746:
8741:
8738:
8731:
8729:
8728:
8723:
8720:
8648:
8641:
8634:
8627:
8620:
8589:
8582:
8575:
8568:
8561:
8530:
8523:
8516:
8509:
8502:
8490:
8486:
8403:
8392:
8385:
8384:
8224:
8210:
8196:
8167:
8165:
8164:
8161:
8158:
8047:, shaped like a
7996:
7989:
7985:
7982:
7976:
7945:
7937:
7885:
7818:
7767:
7684:
7683:
7645:octahedral group
7511:
7509:
7508:
7503:
7495:
7494:
7479:
7478:
7463:
7462:
7450:
7449:
7342:
7331:
7320:
7275:
7273:
7272:
7267:
7261:
7256:
7246:
7241:
7217:
7216:
7211:
7207:
7205:
7200:
7190:
7185:
7139:
7137:
7136:
7131:
7126:
7125:
7120:
7116:
7115:
7114:
7105:
7097:
7091:
7080:
7056:
7055:
7046:
7038:
7032:
7021:
6997:
6995:
6994:
6989:
6987:
6980:
6976:
6975:
6974:
6962:
6961:
6947:
6946:
6937:
6936:
6921:
6920:
6915:
6911:
6910:
6909:
6897:
6896:
6881:
6880:
6871:
6870:
6861:
6853:
6848:
6847:
6842:
6838:
6837:
6836:
6824:
6823:
6800:
6789:
6770:
6759:
6737:
6736:
6727:
6726:
6717:
6709:
6704:
6703:
6698:
6694:
6693:
6692:
6680:
6679:
6664:
6663:
6654:
6653:
6638:
6637:
6632:
6628:
6627:
6626:
6614:
6613:
6590:
6579:
6560:
6549:
6527:
6523:
6522:
6521:
6509:
6508:
6494:
6493:
6484:
6483:
6468:
6467:
6462:
6458:
6457:
6456:
6444:
6443:
6420:
6409:
6390:
6379:
6360:
6349:
6330:
6319:
6297:
6296:
6287:
6286:
6277:
6269:
6264:
6263:
6258:
6254:
6253:
6252:
6240:
6239:
6216:
6205:
6186:
6175:
6156:
6145:
6126:
6115:
6096:
6085:
6063:
6062:
6050:
6049:
6032:
6021:
6002:
5991:
5972:
5961:
5942:
5931:
5912:
5901:
5870:
5868:
5867:
5862:
5859:
5851:
5841:
5836:
5821:
5813:
5807:
5793:
5739:
5737:
5736:
5731:
5726:
5715:
5710:
5704:
5703:
5698:
5689:
5688:
5683:
5675:
5654:
5649:
5648:
5643:
5634:
5629:
5621:
5582:
5580:
5579:
5574:
5558:
5556:
5555:
5550:
5542:
5537:
5536:
5535:
5522:
5512:
5510:
5509:
5504:
5502:
5497:
5484:
5482:
5481:
5476:
5460:
5458:
5457:
5452:
5439:
5437:
5436:
5431:
5429:
5423:
5422:
5421:
5412:
5395:
5393:
5392:
5387:
5371:
5369:
5368:
5363:
5355:
5353:
5352:
5343:
5342:
5341:
5328:
5318:
5316:
5315:
5310:
5308:
5306:
5301:
5290:
5277:
5275:
5274:
5269:
5263:
5258:
5248:
5246:
5245:
5240:
5238:
5237:
5220:
5218:
5217:
5212:
5210:
5205:
5204:
5195:
5178:
5176:
5175:
5170:
5154:
5152:
5151:
5146:
5138:
5129:
5128:
5118:
5116:
5115:
5110:
5108:
5103:
5090:
5088:
5087:
5082:
5080:
5075:
5065:
5063:
5062:
5057:
5043:
5041:
5040:
5035:
5033:
5024:
5023:
5006:
5004:
5003:
4998:
4984:
4982:
4981:
4976:
4962:
4960:
4959:
4954:
4940:
4938:
4937:
4932:
4930:
4925:
4915:
4913:
4912:
4907:
4905:
4900:
4890:
4888:
4887:
4882:
4861:
4859:
4858:
4853:
4837:
4835:
4834:
4829:
4821:
4812:
4811:
4801:
4799:
4798:
4793:
4791:
4786:
4773:
4771:
4770:
4765:
4763:
4754:
4753:
4743:
4741:
4740:
4735:
4733:
4728:
4721:
4713:
4711:
4710:
4705:
4703:
4698:
4691:
4627:
4626:
4617:
4615:
4614:
4609:
4598:
4590:
4551:
4549:
4548:
4543:
4538:
4534:
4526:
4508:
4507:
4502:
4501:
4494:
4486:
4484:
4483:
4436:
4434:
4433:
4428:
4423:
4421:
4420:
4419:
4413:
4405:
4403:
4402:
4388:
4387:
4381:
4373:
4371:
4370:
4364:
4363:
4362:
4349:
4348:
4342:
4334:
4332:
4331:
4325:
4324:
4323:
4313:
4312:
4307:
4303:
4295:
4283:
4279:
4271:
4256:
4248:
4219:
4217:
4216:
4211:
4209:
4208:
4202:
4194:
4192:
4191:
4185:
4179:
4175:
4167:
4155:
4147:
4120:
4118:
4117:
4112:
4110:
4106:
4102:
4094:
4082:
4078:
4070:
4058:
4050:
4034:
4030:
4022:
4010:
4006:
3998:
3986:
3978:
3912:inscribed sphere
3881:
3879:
3878:
3873:
3863:
3855:
3853:
3852:
3847:
3838:
3834:
3826:
3794:
3792:
3791:
3786:
3776:
3768:
3766:
3765:
3760:
3758:
3757:
3730:
3728:
3727:
3722:
3712:
3704:
3702:
3701:
3696:
3687:
3683:
3675:
3649:
3647:
3646:
3641:
3631:
3623:
3621:
3620:
3615:
3592:
3590:
3589:
3584:
3574:
3566:
3564:
3563:
3558:
3549:
3545:
3537:
3514:
3512:
3511:
3506:
3500:
3491:
3483:
3481:
3480:
3475:
3473:
3468:
3448:
3446:
3445:
3440:
3427:
3419:
3417:
3416:
3411:
3402:
3394:
3384:
3382:
3381:
3376:
3366:
3358:
3356:
3355:
3350:
3327:
3325:
3324:
3319:
3306:
3304:
3303:
3298:
3289:
3285:
3277:
3257:
3255:
3254:
3249:
3236:
3234:
3233:
3228:
3226:
3221:
3214:
3167:
3165:
3164:
3161:
3158:
3131:
3130:
3123:
3121:
3120:
3117:
3114:
3113:
3112:
3090:
3062:spherical excess
3057:
3055:
3054:
3049:
2990:
2986:
2980:
2978:
2977:
2972:
2967:
2963:
2962:
2954:
2899:
2878:
2876:
2875:
2870:
2865:
2863:
2862:
2858:
2850:
2837:
2836:
2832:
2824:
2811:
2806:
2802:
2794:
2767:
2765:
2764:
2759:
2754:
2752:
2751:
2747:
2739:
2726:
2725:
2721:
2713:
2700:
2695:
2691:
2683:
2655:, of the solid {
2608:
2606:
2605:
2600:
2595:
2587:
2582:
2574:
2569:
2561:
2544:
2542:
2541:
2536:
2531:
2523:
2518:
2510:
2505:
2497:
2492:
2484:
2471:
2469:
2468:
2463:
2455:
2450:
2442:
2431:
2426:
2418:
2407:
2402:
2395:
2388:
2225:
2214:
2203:
2192:
2179:
2168:
2157:
2146:
2139:
2135:
2081:
2077:
2039:
2035:
1997:
1993:
1955:
1951:
1913:
1909:
1826:
1825:
1771:
1770:
1729:
1727:
1726:
1721:
1716:
1714:
1676:
1668:
1656:
1654:
1616:
1605:
1593:
1591:
1553:
1545:
1514:
1512:
1511:
1506:
1467:
1465:
1464:
1459:
1380:
1351:
1322:
1293:
1264:
1222:
1218:
1168:regular polygons
1138:
1137:
1136:
1132:
1131:
1127:
1126:
1122:
1121:
1117:
1116:
1105:, also called a
1104:
1103:
1102:
1098:
1097:
1093:
1092:
1088:
1087:
1083:
1082:
1065:
1064:
1063:
1059:
1058:
1054:
1053:
1049:
1048:
1044:
1043:
1037:
1036:
1035:
1031:
1030:
1026:
1025:
1021:
1020:
1016:
1015:
1001:central symmetry
987:
986:
982:
980:
979:
974:
971:
962:
960:
956:
954:
953:
948:
945:
932:
930:
928:
927:
922:
919:
908:
902:
897:
895:
893:
892:
887:
884:
877:
867:
865:
857:
855:
854:
849:
846:
837:
831:
829:
828:
823:
820:
813:
807:
802:
796:
790:
788:
778:
772:
765:
763:
753:
751:
741:
735:
728:
723:
721:
717:
711:
709:
705:
699:
697:
693:
686:
684:
680:
674:
672:
668:
662:
658:
653:
651:
645:
641:
637:
540:
536:
533:
531:
530:
527:
524:
523:
522:
489:, which are two
358:
339:Harmonices Mundi
331:
322:
313:
304:
295:
210:
208:
190:
188:
170:
168:
150:
148:
130:
128:
77:
60:regular polygons
21:
12637:
12636:
12632:
12631:
12630:
12628:
12627:
12626:
12617:Platonic solids
12607:
12606:
12605:
12600:
12500:
12472:
12429:
12422:
12370:Theory of Forms
12304:
12291:
12163:
12157:
11943:
11938:
11908:
11903:
11892:
11831:Dihedral others
11822:
11801:
11776:
11751:
11680:
11677:
11676:
11667:
11596:
11585:
11584:
11575:
11538:
11536:Platonic solids
11530:
11524:
11494:
11489:
11478:
11467:
11465:
11454:
11420:Arabian/Islamic
11408:
11397:numeral systems
11286:
11236:
11206:
11154:Heron's formula
11132:
11114:
11006:
11002:Triangle center
10992:Regular polygon
10869:and definitions
10868:
10862:
10824:
10804:
10794:
10756:
10746:
10736:
10726:
10702:
10692:
10675:
10641:
10612:Theon of Smyrna
10257:
10249:
10244:
10202:Platonic Solids
10172:
10170:
10147:Platonic Solids
10131:Wayback Machine
10112:Platonic Solids
10051:Platonic solids
10042:
10031:
10025:
10022:
10003:
9994:This section's
9990:
9986:
9979:
9952:
9926:
9853:
9847:Kleinert, Hagen
9825:, Prestel USA.
9804:
9802:
9800:
9765:Gardner, Martin
9758:
9728:
9704:
9649:math-ph/0303071
9632:Sutcliffe, Paul
9628:Atiyah, Michael
9623:
9618:
9610:
9606:
9598:
9594:
9586:
9582:
9574:
9566:
9565:
9561:
9556:
9552:
9536:
9535:
9475:
9471:
9449:
9445:
9428:
9424:
9389:
9382:
9366:
9365:
9331:
9322:
9317:
9313:
9306:
9282:
9278:
9270:
9266:
9251:
9247:
9239:
9235:
9220:
9216:
9203:
9199:
9195:
9190:
9131:
9080:Ludwig Schläfli
9006:
9000:
8995:
8989:
8918:
8915:
8912:
8911:
8909:
8900:
8897:
8894:
8893:
8891:
8882:
8879:
8876:
8875:
8873:
8812:
8809:
8806:
8805:
8803:
8794:
8791:
8788:
8787:
8785:
8776:
8773:
8770:
8769:
8767:
8760:
8757:
8754:
8753:
8751:
8742:
8739:
8736:
8735:
8733:
8724:
8721:
8718:
8717:
8715:
8485:
8404:
8393:
8368:
8363:
8307:
8305:In architecture
8287:liquid crystals
8283:
8275:magic polyhedra
8230:
8225:
8216:
8211:
8202:
8197:
8162:
8159:
8156:
8155:
8153:
8034:Carborane acids
8031:
8022:boron compounds
7997:
7986:
7980:
7977:
7962:
7946:
7935:
7909:
7904:
7900:
7894:
7893:
7842:
7837:
7833:
7827:
7826:
7791:
7786:
7782:
7776:
7775:
7706:
7699:
7692:
7552:
7550:Symmetry groups
7490:
7486:
7474:
7470:
7458:
7454:
7445:
7441:
7439:
7436:
7435:
7354:
7353:
7352:
7351:
7345:
7344:
7343:
7334:
7333:
7332:
7323:
7322:
7321:
7310:
7305:
7289:Rupert property
7281:
7279:Rupert property
7257:
7252:
7242:
7231:
7212:
7201:
7196:
7186:
7175:
7170:
7166:
7165:
7160:
7157:
7156:
7147:
7121:
7110:
7106:
7096:
7081:
7070:
7065:
7061:
7060:
7051:
7047:
7037:
7022:
7011:
7005:
7002:
7001:
6985:
6984:
6970:
6966:
6957:
6953:
6952:
6948:
6942:
6938:
6932:
6928:
6916:
6905:
6901:
6892:
6888:
6887:
6883:
6882:
6876:
6872:
6866:
6862:
6852:
6843:
6832:
6828:
6819:
6815:
6814:
6810:
6809:
6802:
6790:
6779:
6760:
6749:
6742:
6741:
6732:
6728:
6722:
6718:
6708:
6699:
6688:
6684:
6675:
6671:
6670:
6666:
6665:
6659:
6655:
6649:
6645:
6633:
6622:
6618:
6609:
6605:
6604:
6600:
6599:
6592:
6580:
6569:
6550:
6539:
6532:
6531:
6517:
6513:
6504:
6500:
6499:
6495:
6489:
6485:
6479:
6475:
6463:
6452:
6448:
6439:
6435:
6434:
6430:
6429:
6422:
6410:
6399:
6380:
6369:
6350:
6339:
6320:
6309:
6302:
6301:
6292:
6288:
6282:
6278:
6268:
6259:
6248:
6244:
6235:
6231:
6230:
6226:
6225:
6218:
6206:
6195:
6176:
6165:
6146:
6135:
6116:
6105:
6086:
6075:
6068:
6067:
6058:
6054:
6045:
6041:
6034:
6022:
6011:
5992:
5981:
5962:
5951:
5932:
5921:
5902:
5891:
5883:
5881:
5878:
5877:
5852:
5847:
5837:
5826:
5812:
5794:
5783:
5777:
5774:
5773:
5768:
5749:
5714:
5697:
5690:
5687:
5674:
5642:
5635:
5633:
5620:
5603:
5600:
5599:
5565:
5562:
5561:
5531:
5527:
5523:
5521:
5519:
5516:
5515:
5496:
5491:
5488:
5487:
5467:
5464:
5463:
5446:
5443:
5442:
5417:
5413:
5411:
5409:
5406:
5405:
5378:
5375:
5374:
5348:
5344:
5337:
5333:
5329:
5327:
5325:
5322:
5321:
5300:
5289:
5284:
5281:
5280:
5257:
5255:
5252:
5251:
5233:
5229:
5227:
5224:
5223:
5200:
5196:
5194:
5192:
5189:
5188:
5161:
5158:
5157:
5127:
5125:
5122:
5121:
5102:
5097:
5094:
5093:
5074:
5072:
5069:
5068:
5050:
5047:
5046:
5022:
5020:
5017:
5016:
4991:
4988:
4987:
4969:
4966:
4965:
4947:
4944:
4943:
4924:
4922:
4919:
4918:
4899:
4897:
4894:
4893:
4875:
4872:
4871:
4844:
4841:
4840:
4810:
4808:
4805:
4804:
4785:
4780:
4777:
4776:
4752:
4750:
4747:
4746:
4727:
4720:
4717:
4716:
4697:
4690:
4687:
4686:
4642:
4635: = 2
4631:
4589:
4581:
4578:
4577:
4558:is computed as
4525:
4521:
4503:
4497:
4496:
4495:
4485:
4479:
4478:
4470:
4467:
4466:
4415:
4414:
4404:
4398:
4397:
4390:
4383:
4382:
4372:
4366:
4365:
4358:
4354:
4353:
4344:
4343:
4333:
4327:
4326:
4319:
4315:
4314:
4311:
4294:
4290:
4270:
4266:
4247:
4245:
4242:
4241:
4204:
4203:
4193:
4187:
4186:
4181:
4166:
4162:
4146:
4138:
4135:
4134:
4108:
4107:
4093:
4089:
4069:
4065:
4049:
4042:
4036:
4035:
4021:
4017:
3997:
3993:
3977:
3970:
3963:
3961:
3958:
3957:
3890:
3862:
3859:
3858:
3825:
3821:
3801:
3798:
3797:
3775:
3772:
3771:
3753:
3749:
3747:
3744:
3743:
3711:
3708:
3707:
3674:
3670:
3656:
3653:
3652:
3630:
3627:
3626:
3609:
3606:
3605:
3573:
3570:
3569:
3536:
3532:
3521:
3518:
3517:
3493:
3490:
3487:
3486:
3467:
3465:
3462:
3461:
3426:
3423:
3422:
3393:
3391:
3388:
3387:
3365:
3362:
3361:
3344:
3341:
3340:
3313:
3310:
3309:
3276:
3272:
3264:
3261:
3260:
3243:
3240:
3239:
3220:
3213:
3210:
3209:
3195:
3193:
3173:
3162:
3159:
3154:
3153:
3151:
3142:
3139:
3118:
3115:
3110:
3108:
3106:
3105:
3103:
3098:. The constant
3088:
3080:} is a regular
3009:
3006:
3005:
2988:
2984:
2953:
2946:
2942:
2919:
2916:
2915:
2897:
2849:
2845:
2838:
2823:
2819:
2812:
2810:
2793:
2789:
2781:
2778:
2777:
2738:
2734:
2727:
2712:
2708:
2701:
2699:
2682:
2678:
2670:
2667:
2666:
2641:
2636:
2631:
2586:
2573:
2560:
2558:
2555:
2554:
2522:
2509:
2496:
2483:
2481:
2478:
2477:
2443:
2441:
2419:
2417:
2415:
2412:
2411:
2406:
2381:
2323:
2318:
2241:
2239:
2228:
2226:
2217:
2215:
2206:
2204:
2195:
2193:
2182:
2180:
2171:
2169:
2160:
2158:
2149:
2147:
2134:
2132:Geometric proof
2125:
1786:
1762:
1677:
1669:
1667:
1617:
1606:
1604:
1554:
1546:
1544:
1536:
1533:
1532:
1481:
1478:
1477:
1472:Euler's formula
1428:
1425:
1424:
1244:Schläfli symbol
1210:Schläfli symbol
1156:
1134:
1129:
1124:
1119:
1114:
1112:
1111:, as s{3,4} or
1108:snub octahedron
1100:
1095:
1090:
1085:
1080:
1078:
1061:
1056:
1051:
1046:
1041:
1039:
1033:
1028:
1023:
1018:
1013:
1011:
984:
975:
972:
969:
968:
966:
964:
963:
958:
949:
946:
943:
942:
940:
934:
933:
923:
920:
917:
916:
914:
906:
904:
903:
900:
888:
885:
882:
881:
879:
875:
869:
868:
863:
850:
847:
844:
843:
841:
839:
838:
824:
821:
818:
817:
815:
811:
809:
808:
805:
794:
792:
791:
786:
780:
779:
770:
768:
761:
755:
754:
749:
743:
742:
733:
731:
726:
719:
715:
713:
712:
707:
703:
701:
700:
695:
691:
689:
682:
678:
676:
675:
670:
666:
664:
663:
660:
659:
656:
649:
647:
646:
643:
642:
639:
638:
635:
631:
528:
525:
520:
518:
516:
515:
513:
499:
436:Johannes Kepler
401:Andreas Speiser
344:
343:
342:
341:
334:
333:
332:
324:
323:
315:
314:
306:
305:
297:
296:
256:created by the
250:
28:
23:
22:
18:Platonic solids
15:
12:
11:
5:
12635:
12625:
12624:
12619:
12602:
12601:
12599:
12598:
12591:
12586:
12585:
12584:
12579:
12574:
12569:
12559:
12558:
12557:
12547:
12542:
12537:
12532:
12527:
12519:
12514:
12508:
12506:
12502:
12501:
12499:
12498:
12493:
12488:
12482:
12480:
12474:
12473:
12471:
12470:
12465:
12460:
12455:
12450:
12445:
12440:
12434:
12432:
12424:
12423:
12421:
12420:
12413:
12408:
12401:
12399:Platonic solid
12396:
12395:
12394:
12384:
12382:Theory of soul
12379:
12378:
12377:
12367:
12366:
12365:
12358:
12351:
12339:
12338:
12337:
12325:
12320:
12314:
12312:
12306:
12305:
12298:
12296:
12293:
12292:
12290:
12289:
12282:
12275:
12268:
12261:
12254:
12247:
12240:
12233:
12226:
12219:
12218:
12217:
12214:Seventh Letter
12203:
12196:
12189:
12182:
12175:
12167:
12165:
12156:
12155:
12148:
12141:
12134:
12127:
12120:
12113:
12106:
12099:
12092:
12085:
12078:
12071:
12064:
12057:
12050:
12043:
12036:
12029:
12022:
12015:
12008:
12001:
11994:
11987:
11980:
11973:
11966:
11959:
11951:
11949:
11945:
11944:
11937:
11936:
11929:
11922:
11914:
11905:
11904:
11897:
11894:
11893:
11891:
11890:
11885:
11880:
11875:
11870:
11865:
11860:
11855:
11850:
11845:
11840:
11834:
11832:
11828:
11827:
11824:
11823:
11821:
11820:
11815:
11809:
11807:
11803:
11802:
11800:
11799:
11794:
11788:
11782:
11778:
11777:
11775:
11774:
11767:
11759:
11757:
11753:
11752:
11750:
11749:
11744:
11739:
11734:
11729:
11724:
11719:
11714:
11709:
11704:
11699:
11694:
11689:
11683:
11681:
11674:Catalan solids
11672:
11669:
11668:
11666:
11665:
11660:
11655:
11650:
11645:
11640:
11635:
11630:
11625:
11620:
11615:
11613:truncated cube
11610:
11605:
11599:
11597:
11580:
11577:
11576:
11574:
11573:
11568:
11563:
11558:
11553:
11547:
11545:
11532:
11531:
11523:
11522:
11515:
11508:
11500:
11491:
11490:
11463:
11460:
11459:
11456:
11455:
11453:
11452:
11447:
11442:
11437:
11432:
11427:
11422:
11416:
11414:
11413:Other cultures
11410:
11409:
11407:
11406:
11405:
11404:
11394:
11393:
11392:
11382:
11381:
11380:
11370:
11369:
11368:
11358:
11357:
11356:
11346:
11345:
11344:
11334:
11333:
11332:
11322:
11321:
11320:
11310:
11309:
11308:
11294:
11292:
11288:
11287:
11285:
11284:
11279:
11274:
11269:
11264:
11262:Greek numerals
11259:
11257:Attic numerals
11254:
11248:
11242:
11238:
11237:
11235:
11234:
11229:
11224:
11218:
11216:
11212:
11211:
11208:
11207:
11205:
11204:
11199:
11194:
11189:
11184:
11176:
11171:
11166:
11161:
11156:
11151:
11146:
11140:
11138:
11134:
11133:
11131:
11130:
11124:
11122:
11116:
11115:
11113:
11112:
11107:
11102:
11097:
11092:
11087:
11085:Law of cosines
11082:
11077:
11072:
11067:
11062:
11057:
11052:
11047:
11042:
11037:
11032:
11026:
11024:
11012:
11008:
11007:
11005:
11004:
10999:
10994:
10989:
10984:
10979:
10977:Platonic solid
10974:
10969:
10964:
10959:
10957:Greek numerals
10954:
10949:
10944:
10939:
10934:
10929:
10924:
10923:
10922:
10917:
10907:
10902:
10901:
10900:
10890:
10889:
10888:
10883:
10872:
10870:
10864:
10863:
10861:
10860:
10855:
10854:
10853:
10848:
10843:
10832:
10830:
10826:
10825:
10823:
10822:
10815:
10808:
10798:
10788:
10785:Planisphaerium
10781:
10774:
10767:
10760:
10750:
10740:
10730:
10720:
10713:
10706:
10696:
10686:
10679:
10669:
10662:
10657:
10649:
10647:
10643:
10642:
10640:
10639:
10634:
10629:
10624:
10619:
10614:
10609:
10604:
10599:
10594:
10589:
10584:
10579:
10574:
10569:
10564:
10559:
10554:
10549:
10544:
10539:
10534:
10529:
10524:
10519:
10514:
10509:
10504:
10499:
10494:
10489:
10484:
10479:
10474:
10469:
10464:
10459:
10454:
10449:
10444:
10439:
10434:
10429:
10424:
10419:
10414:
10409:
10404:
10399:
10394:
10389:
10384:
10379:
10374:
10369:
10364:
10359:
10354:
10349:
10344:
10339:
10334:
10329:
10324:
10319:
10314:
10309:
10304:
10299:
10294:
10289:
10284:
10279:
10274:
10269:
10263:
10261:
10255:Mathematicians
10251:
10250:
10243:
10242:
10235:
10228:
10220:
10214:
10213:
10208:
10199:
10190:
10184:
10178:
10150:
10144:
10134:
10121:
10115:
10109:
10103:
10093:
10074:
10055:
10044:
10043:
9998:external links
9993:
9991:
9984:
9978:
9977:External links
9975:
9974:
9973:
9956:
9950:
9930:
9924:
9911:
9893:(3): 131–140.
9882:
9864:(5): 219–259.
9843:
9834:
9813:Haeckel, Ernst
9810:
9798:
9779:
9762:
9756:
9732:
9726:
9708:
9702:
9678:
9622:
9619:
9617:
9616:
9614:, p. 136.
9604:
9600:Gelernter 1995
9592:
9588:Gelernter 1995
9580:
9559:
9550:
9469:
9459:(9): 241–246,
9443:
9422:
9380:
9320:
9311:
9304:
9286:Apostol, T. M.
9276:
9264:
9245:
9233:
9214:
9206:Martin Gardner
9196:
9194:
9191:
9189:
9188:
9183:
9178:
9173:
9168:
9163:
9158:
9153:
9148:
9143:
9138:
9132:
9130:
9127:
9123:cross-polytope
9063:
9062:
9059:
9055:
9054:
9051:
9047:
9046:
9041:
9035:
9034:
9031:
9027:
9026:
9023:
9019:
9018:
9015:
9011:
9010:
9004:
8988:
8985:
8982:
8981:
8976:
8971:
8966:
8960:
8959:
8952:
8945:
8938:
8864:
8863:
8858:
8853:
8847:
8846:
8839:
8832:
8667:
8666:
8663:
8660:
8657:
8654:
8650:
8649:
8642:
8635:
8628:
8621:
8613:
8612:
8608:
8607:
8604:
8601:
8598:
8595:
8591:
8590:
8583:
8576:
8569:
8562:
8554:
8553:
8549:
8548:
8545:
8542:
8539:
8536:
8532:
8531:
8524:
8517:
8510:
8503:
8495:
8494:
8484:
8481:
8473:Johnson solids
8447:Catalan solids
8409:
8408:
8397:
8367:
8364:
8362:
8359:
8347:square pyramid
8306:
8303:
8282:
8279:
8232:
8231:
8226:
8219:
8217:
8212:
8205:
8203:
8198:
8191:
8112:geodesic grids
8086:, such as the
8029:
7999:
7998:
7949:
7947:
7940:
7934:
7931:
7928:
7927:
7924:
7921:
7918:
7912:
7911:
7906:
7901:
7898:
7891:
7886:
7876:
7873:
7870:
7867:
7861:
7860:
7857:
7854:
7851:
7845:
7844:
7839:
7834:
7831:
7824:
7819:
7809:
7806:
7803:
7800:
7794:
7793:
7788:
7783:
7780:
7773:
7768:
7758:
7755:
7752:
7749:
7743:
7742:
7737:
7732:
7727:
7722:
7716:
7715:
7712:Symmetry group
7709:
7702:
7695:
7688:
7662:
7661:
7651:
7641:
7615:vertex-uniform
7564:symmetry group
7551:
7548:
7542:* =
7534:* =
7501:
7498:
7493:
7489:
7485:
7482:
7477:
7473:
7469:
7466:
7461:
7457:
7453:
7448:
7444:
7379:
7378:
7375:
7372:
7349:Dual compounds
7347:
7346:
7337:
7336:
7335:
7326:
7325:
7324:
7315:
7314:
7313:
7312:
7311:
7309:
7308:Dual polyhedra
7306:
7304:
7301:
7280:
7277:
7265:
7260:
7255:
7251:
7245:
7240:
7237:
7234:
7230:
7226:
7223:
7220:
7215:
7210:
7204:
7199:
7195:
7189:
7184:
7181:
7178:
7174:
7169:
7164:
7145:
7129:
7124:
7119:
7113:
7109:
7103:
7100:
7095:
7090:
7087:
7084:
7079:
7076:
7073:
7069:
7064:
7059:
7054:
7050:
7044:
7041:
7036:
7031:
7028:
7025:
7020:
7017:
7014:
7010:
6983:
6979:
6973:
6969:
6965:
6960:
6956:
6951:
6945:
6941:
6935:
6931:
6927:
6924:
6919:
6914:
6908:
6904:
6900:
6895:
6891:
6886:
6879:
6875:
6869:
6865:
6859:
6856:
6851:
6846:
6841:
6835:
6831:
6827:
6822:
6818:
6813:
6808:
6805:
6803:
6799:
6796:
6793:
6788:
6785:
6782:
6778:
6774:
6769:
6766:
6763:
6758:
6755:
6752:
6748:
6744:
6743:
6740:
6735:
6731:
6725:
6721:
6715:
6712:
6707:
6702:
6697:
6691:
6687:
6683:
6678:
6674:
6669:
6662:
6658:
6652:
6648:
6644:
6641:
6636:
6631:
6625:
6621:
6617:
6612:
6608:
6603:
6598:
6595:
6593:
6589:
6586:
6583:
6578:
6575:
6572:
6568:
6564:
6559:
6556:
6553:
6548:
6545:
6542:
6538:
6534:
6533:
6530:
6526:
6520:
6516:
6512:
6507:
6503:
6498:
6492:
6488:
6482:
6478:
6474:
6471:
6466:
6461:
6455:
6451:
6447:
6442:
6438:
6433:
6428:
6425:
6423:
6419:
6416:
6413:
6408:
6405:
6402:
6398:
6394:
6389:
6386:
6383:
6378:
6375:
6372:
6368:
6364:
6359:
6356:
6353:
6348:
6345:
6342:
6338:
6334:
6329:
6326:
6323:
6318:
6315:
6312:
6308:
6304:
6303:
6300:
6295:
6291:
6285:
6281:
6275:
6272:
6267:
6262:
6257:
6251:
6247:
6243:
6238:
6234:
6229:
6224:
6221:
6219:
6215:
6212:
6209:
6204:
6201:
6198:
6194:
6190:
6185:
6182:
6179:
6174:
6171:
6168:
6164:
6160:
6155:
6152:
6149:
6144:
6141:
6138:
6134:
6130:
6125:
6122:
6119:
6114:
6111:
6108:
6104:
6100:
6095:
6092:
6089:
6084:
6081:
6078:
6074:
6070:
6069:
6066:
6061:
6057:
6053:
6048:
6044:
6040:
6037:
6035:
6031:
6028:
6025:
6020:
6017:
6014:
6010:
6006:
6001:
5998:
5995:
5990:
5987:
5984:
5980:
5976:
5971:
5968:
5965:
5960:
5957:
5954:
5950:
5946:
5941:
5938:
5935:
5930:
5927:
5924:
5920:
5916:
5911:
5908:
5905:
5900:
5897:
5894:
5890:
5886:
5885:
5858:
5855:
5850:
5846:
5840:
5835:
5832:
5829:
5825:
5819:
5816:
5811:
5806:
5803:
5800:
5797:
5792:
5789:
5786:
5782:
5766:
5748:
5747:Point in space
5745:
5729:
5724:
5721:
5718:
5713:
5707:
5701:
5696:
5693:
5686:
5681:
5678:
5673:
5670:
5667:
5664:
5661:
5657:
5652:
5646:
5641:
5638:
5632:
5627:
5624:
5619:
5616:
5613:
5610:
5607:
5588:The constants
5584:
5583:
5572:
5569:
5559:
5548:
5545:
5540:
5534:
5530:
5526:
5513:
5500:
5495:
5485:
5474:
5471:
5461:
5450:
5440:
5426:
5420:
5416:
5403:
5397:
5396:
5385:
5382:
5372:
5361:
5358:
5351:
5347:
5340:
5336:
5332:
5319:
5304:
5299:
5296:
5293:
5288:
5278:
5267:
5261:
5249:
5236:
5232:
5221:
5208:
5203:
5199:
5186:
5180:
5179:
5168:
5165:
5155:
5144:
5141:
5136:
5132:
5119:
5106:
5101:
5091:
5078:
5066:
5054:
5044:
5030:
5027:
5014:
5008:
5007:
4995:
4985:
4973:
4963:
4951:
4941:
4928:
4916:
4903:
4891:
4879:
4869:
4863:
4862:
4851:
4848:
4838:
4827:
4824:
4819:
4815:
4802:
4789:
4784:
4774:
4760:
4757:
4744:
4731:
4725:
4714:
4701:
4695:
4684:
4678:
4677:
4674:
4669:
4663:
4657:
4650:
4649:
4646:
4641:Surface area,
4639:
4636:
4607:
4604:
4601:
4596:
4593:
4588:
4585:
4541:
4537:
4532:
4529:
4524:
4520:
4517:
4514:
4511:
4506:
4500:
4492:
4489:
4482:
4477:
4474:
4426:
4418:
4411:
4408:
4401:
4396:
4393:
4386:
4379:
4376:
4369:
4361:
4357:
4352:
4347:
4340:
4337:
4330:
4322:
4318:
4310:
4306:
4301:
4298:
4293:
4289:
4286:
4282:
4277:
4274:
4269:
4265:
4262:
4259:
4254:
4251:
4207:
4200:
4197:
4190:
4184:
4178:
4173:
4170:
4165:
4161:
4158:
4153:
4150:
4145:
4142:
4105:
4100:
4097:
4092:
4088:
4085:
4081:
4076:
4073:
4068:
4064:
4061:
4056:
4053:
4048:
4045:
4043:
4041:
4038:
4037:
4033:
4028:
4025:
4020:
4016:
4013:
4009:
4004:
4001:
3996:
3992:
3989:
3984:
3981:
3976:
3973:
3971:
3969:
3966:
3965:
3942:of the solid {
3916:
3915:
3908:
3901:
3889:
3886:
3883:
3882:
3871:
3867:
3856:
3845:
3842:
3837:
3832:
3829:
3824:
3820:
3817:
3814:
3811:
3808:
3805:
3795:
3784:
3780:
3769:
3756:
3752:
3741:
3738:
3732:
3731:
3720:
3716:
3705:
3694:
3691:
3686:
3681:
3678:
3673:
3669:
3666:
3663:
3660:
3650:
3639:
3635:
3624:
3613:
3603:
3600:
3594:
3593:
3582:
3578:
3567:
3556:
3553:
3548:
3543:
3540:
3535:
3531:
3528:
3525:
3515:
3504:
3499:
3496:
3484:
3471:
3459:
3456:
3450:
3449:
3438:
3434:
3431:
3420:
3409:
3406:
3400:
3397:
3385:
3374:
3370:
3359:
3348:
3338:
3335:
3329:
3328:
3317:
3307:
3296:
3293:
3288:
3283:
3280:
3275:
3271:
3268:
3258:
3247:
3237:
3224:
3218:
3207:
3204:
3198:
3197:
3190:
3179:
3168:
3148:
3135:
3064:formula for a
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
2970:
2966:
2960:
2957:
2952:
2949:
2945:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2900:. The defect,
2887:Coxeter number
2868:
2861:
2856:
2853:
2848:
2844:
2841:
2835:
2830:
2827:
2822:
2818:
2815:
2809:
2805:
2800:
2797:
2792:
2788:
2785:
2757:
2750:
2745:
2742:
2737:
2733:
2730:
2724:
2719:
2716:
2711:
2707:
2704:
2698:
2694:
2689:
2686:
2681:
2677:
2674:
2649:dihedral angle
2640:
2637:
2635:
2632:
2629:
2598:
2593:
2590:
2585:
2580:
2577:
2572:
2567:
2564:
2534:
2529:
2526:
2521:
2516:
2513:
2508:
2503:
2500:
2495:
2490:
2487:
2461:
2458:
2453:
2449:
2446:
2440:
2437:
2434:
2429:
2425:
2422:
2405:
2404:
2397:
2390:
2382:
2349: = 2
2322:
2319:
2317:
2316:
2314:
2313:
2310:
2304:
2301:
2295:
2292:
2279:
2272:
2268:
2252:
2251:
2231:
2230:
2219:
2208:
2197:
2185:
2184:
2173:
2162:
2151:
2133:
2130:
2124:
2123:Classification
2121:
2118:
2117:
2114:
2113:
2110:
2107:
2103:
2102:
2099:
2096:
2092:
2091:
2088:
2085:
2075:
2072:
2071:
2068:
2065:
2061:
2060:
2057:
2054:
2050:
2049:
2046:
2043:
2033:
2030:
2029:
2026:
2023:
2019:
2018:
2015:
2012:
2008:
2007:
2004:
2001:
1991:
1988:
1987:
1984:
1981:
1977:
1976:
1973:
1970:
1966:
1965:
1962:
1959:
1949:
1946:
1945:
1942:
1939:
1935:
1934:
1931:
1928:
1924:
1923:
1920:
1917:
1907:
1904:
1903:
1894:
1889:
1884:
1880:
1879:
1876:
1870:
1867:
1863:
1862:
1857:
1852:
1843:
1839:
1838:
1835:
1832:
1829:
1822:
1821:
1815:
1809:
1803:
1779:
1778:
1775:
1761:
1758:
1748:while leaving
1719:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1675:
1672:
1666:
1663:
1659:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1615:
1612:
1609:
1603:
1600:
1596:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1552:
1549:
1543:
1540:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1409:), and faces (
1397:
1396:
1393:
1390:
1387:
1384:
1381:
1374:
1368:
1367:
1364:
1361:
1358:
1355:
1352:
1345:
1339:
1338:
1335:
1332:
1329:
1326:
1323:
1316:
1310:
1309:
1306:
1303:
1300:
1297:
1294:
1287:
1281:
1280:
1277:
1274:
1271:
1268:
1265:
1258:
1252:
1251:
1246:
1241:
1236:
1231:
1226:
1208:}, called the
1182:
1181:
1174:
1171:
1155:
1152:
994:
993:
898:
803:
766:
729:
724:
687:
654:
633:
627:
626:
623:
620:
617:
614:
612:
610:
607:
604:
600:
599:
596:
593:
590:
587:
584:
580:
579:
576:
573:
570:
567:
564:
560:
559:
556:
553:
550:
547:
544:
498:
495:
336:
335:
326:
325:
317:
316:
308:
307:
299:
298:
290:
289:
288:
287:
286:
270:ancient Greeks
258:late Neolithic
249:
246:
223:
222:
202:
182:
162:
142:
121:
120:
117:
114:
111:
108:
104:
103:
98:
93:
88:
83:
36:Platonic solid
26:
9:
6:
4:
3:
2:
12634:
12623:
12620:
12618:
12615:
12614:
12612:
12597:
12596:
12592:
12590:
12589:Plato's Dream
12587:
12583:
12580:
12578:
12575:
12573:
12570:
12568:
12565:
12564:
12563:
12560:
12556:
12553:
12552:
12551:
12548:
12546:
12543:
12541:
12538:
12536:
12533:
12531:
12528:
12526:
12525:
12520:
12518:
12515:
12513:
12510:
12509:
12507:
12503:
12497:
12494:
12492:
12489:
12487:
12484:
12483:
12481:
12479:
12475:
12469:
12466:
12464:
12463:Ship of State
12461:
12459:
12456:
12454:
12451:
12449:
12446:
12444:
12443:Ring of Gyges
12441:
12439:
12436:
12435:
12433:
12431:
12430:and metaphors
12425:
12419:
12418:
12414:
12412:
12409:
12407:
12406:
12402:
12400:
12397:
12393:
12390:
12389:
12388:
12385:
12383:
12380:
12376:
12373:
12372:
12371:
12368:
12364:
12363:
12359:
12357:
12356:
12352:
12350:
12349:
12345:
12344:
12343:
12340:
12336:
12335:
12331:
12330:
12329:
12326:
12324:
12323:Platonic love
12321:
12319:
12316:
12315:
12313:
12311:
12307:
12302:
12288:
12287:
12283:
12281:
12280:
12276:
12274:
12273:
12269:
12267:
12266:
12262:
12260:
12259:
12255:
12253:
12252:
12248:
12246:
12245:
12241:
12239:
12238:
12234:
12232:
12231:
12227:
12225:
12224:
12220:
12216:
12215:
12211:
12210:
12209:
12208:
12204:
12202:
12201:
12197:
12195:
12194:
12190:
12188:
12187:
12183:
12181:
12180:
12176:
12174:
12173:
12169:
12168:
12166:
12160:
12154:
12153:
12149:
12147:
12146:
12142:
12140:
12139:
12135:
12133:
12132:
12128:
12126:
12125:
12121:
12119:
12118:
12114:
12112:
12111:
12107:
12105:
12104:
12100:
12098:
12097:
12093:
12091:
12090:
12086:
12084:
12083:
12079:
12077:
12076:
12072:
12070:
12069:
12065:
12063:
12062:
12058:
12056:
12055:
12051:
12049:
12048:
12044:
12042:
12041:
12037:
12035:
12034:
12033:Hippias Minor
12030:
12028:
12027:
12026:Hippias Major
12023:
12021:
12020:
12016:
12014:
12013:
12009:
12007:
12006:
12002:
12000:
11999:
11995:
11993:
11992:
11988:
11986:
11985:
11981:
11979:
11978:
11974:
11972:
11971:
11967:
11965:
11964:
11960:
11958:
11957:
11953:
11952:
11950:
11946:
11942:
11935:
11930:
11928:
11923:
11921:
11916:
11915:
11912:
11901:
11895:
11889:
11886:
11884:
11881:
11879:
11876:
11874:
11871:
11869:
11866:
11864:
11861:
11859:
11856:
11854:
11851:
11849:
11846:
11844:
11841:
11839:
11836:
11835:
11833:
11829:
11819:
11816:
11814:
11811:
11810:
11808:
11804:
11798:
11795:
11793:
11790:
11789:
11786:
11783:
11779:
11773:
11772:
11768:
11766:
11765:
11761:
11760:
11758:
11754:
11748:
11745:
11743:
11740:
11738:
11735:
11733:
11730:
11728:
11725:
11723:
11720:
11718:
11715:
11713:
11710:
11708:
11705:
11703:
11700:
11698:
11695:
11693:
11690:
11688:
11685:
11684:
11682:
11675:
11670:
11664:
11661:
11659:
11656:
11654:
11651:
11649:
11646:
11644:
11641:
11639:
11636:
11634:
11631:
11629:
11626:
11624:
11621:
11619:
11616:
11614:
11611:
11609:
11608:cuboctahedron
11606:
11604:
11601:
11600:
11598:
11593:
11589:
11583:
11578:
11572:
11569:
11567:
11564:
11562:
11559:
11557:
11554:
11552:
11549:
11548:
11546:
11542:
11537:
11533:
11529:
11521:
11516:
11514:
11509:
11507:
11502:
11501:
11498:
11488:
11487:
11482:
11475:
11474:
11461:
11451:
11448:
11446:
11443:
11441:
11438:
11436:
11433:
11431:
11428:
11426:
11423:
11421:
11418:
11417:
11415:
11411:
11403:
11400:
11399:
11398:
11395:
11391:
11388:
11387:
11386:
11383:
11379:
11376:
11375:
11374:
11371:
11367:
11364:
11363:
11362:
11359:
11355:
11352:
11351:
11350:
11347:
11343:
11340:
11339:
11338:
11335:
11331:
11328:
11327:
11326:
11323:
11319:
11316:
11315:
11314:
11311:
11307:
11303:
11302:
11301:
11300:
11296:
11295:
11293:
11289:
11283:
11280:
11278:
11275:
11273:
11270:
11268:
11265:
11263:
11260:
11258:
11255:
11253:
11250:
11249:
11246:
11243:
11239:
11233:
11230:
11228:
11225:
11223:
11220:
11219:
11217:
11213:
11203:
11200:
11198:
11195:
11193:
11190:
11188:
11185:
11183:
11182:
11177:
11175:
11172:
11170:
11167:
11165:
11162:
11160:
11157:
11155:
11152:
11150:
11147:
11145:
11142:
11141:
11139:
11135:
11129:
11126:
11125:
11123:
11121:
11117:
11111:
11108:
11106:
11103:
11101:
11098:
11096:
11093:
11091:
11090:Pons asinorum
11088:
11086:
11083:
11081:
11078:
11076:
11073:
11071:
11068:
11066:
11063:
11061:
11060:Hinge theorem
11058:
11056:
11053:
11051:
11048:
11046:
11043:
11041:
11038:
11036:
11033:
11031:
11028:
11027:
11025:
11023:
11022:
11016:
11013:
11009:
11003:
11000:
10998:
10995:
10993:
10990:
10988:
10985:
10983:
10980:
10978:
10975:
10973:
10970:
10968:
10965:
10963:
10960:
10958:
10955:
10953:
10950:
10948:
10945:
10943:
10940:
10938:
10935:
10933:
10930:
10928:
10925:
10921:
10918:
10916:
10913:
10912:
10911:
10908:
10906:
10903:
10899:
10896:
10895:
10894:
10891:
10887:
10884:
10882:
10879:
10878:
10877:
10874:
10873:
10871:
10865:
10859:
10856:
10852:
10849:
10847:
10844:
10842:
10839:
10838:
10837:
10834:
10833:
10831:
10827:
10821:
10820:
10816:
10814:
10813:
10809:
10807:
10803:
10799:
10797:
10793:
10789:
10787:
10786:
10782:
10780:
10779:
10775:
10773:
10772:
10768:
10766:
10765:
10761:
10759:
10755:
10751:
10749:
10745:
10741:
10739:
10735:
10731:
10729:
10727:(Aristarchus)
10725:
10721:
10719:
10718:
10714:
10712:
10711:
10707:
10705:
10701:
10697:
10695:
10691:
10687:
10685:
10684:
10680:
10678:
10674:
10670:
10668:
10667:
10663:
10661:
10658:
10656:
10655:
10651:
10650:
10648:
10644:
10638:
10635:
10633:
10632:Zeno of Sidon
10630:
10628:
10625:
10623:
10620:
10618:
10615:
10613:
10610:
10608:
10605:
10603:
10600:
10598:
10595:
10593:
10590:
10588:
10585:
10583:
10580:
10578:
10575:
10573:
10570:
10568:
10565:
10563:
10560:
10558:
10555:
10553:
10550:
10548:
10545:
10543:
10540:
10538:
10535:
10533:
10530:
10528:
10525:
10523:
10520:
10518:
10515:
10513:
10510:
10508:
10505:
10503:
10500:
10498:
10495:
10493:
10490:
10488:
10485:
10483:
10480:
10478:
10475:
10473:
10470:
10468:
10465:
10463:
10460:
10458:
10455:
10453:
10450:
10448:
10445:
10443:
10440:
10438:
10435:
10433:
10430:
10428:
10425:
10423:
10420:
10418:
10415:
10413:
10410:
10408:
10405:
10403:
10400:
10398:
10395:
10393:
10390:
10388:
10385:
10383:
10380:
10378:
10375:
10373:
10370:
10368:
10365:
10363:
10360:
10358:
10355:
10353:
10350:
10348:
10345:
10343:
10340:
10338:
10335:
10333:
10330:
10328:
10325:
10323:
10320:
10318:
10315:
10313:
10310:
10308:
10305:
10303:
10300:
10298:
10295:
10293:
10290:
10288:
10285:
10283:
10280:
10278:
10275:
10273:
10270:
10268:
10265:
10264:
10262:
10260:
10256:
10252:
10248:
10241:
10236:
10234:
10229:
10227:
10222:
10221:
10218:
10212:
10209:
10207:
10203:
10200:
10198:
10194:
10191:
10188:
10185:
10182:
10179:
10169:on 2018-10-23
10168:
10164:
10160:
10156:
10151:
10148:
10145:
10142:
10138:
10135:
10132:
10128:
10125:
10122:
10119:
10116:
10113:
10110:
10107:
10104:
10101:
10097:
10094:
10089:
10088:
10083:
10080:
10075:
10070:
10069:
10064:
10061:
10056:
10054:
10052:
10048:
10047:
10040:
10037:
10029:
10026:December 2019
10019:
10015:
10014:inappropriate
10011:
10007:
10001:
9999:
9992:
9983:
9982:
9971:
9970:9783110104462
9967:
9963:
9962:
9957:
9953:
9951:0-691-02374-3
9947:
9942:
9941:
9935:
9934:Weyl, Hermann
9931:
9927:
9925:0-520-03056-7
9921:
9917:
9912:
9908:
9904:
9900:
9896:
9892:
9888:
9883:
9879:
9875:
9871:
9867:
9863:
9859:
9852:
9848:
9844:
9841:
9838:
9835:
9832:
9831:3-7913-1990-6
9828:
9824:
9823:
9818:
9815:, E. (1904).
9814:
9811:
9801:
9795:
9791:
9787:
9786:
9780:
9778:
9774:
9770:
9766:
9763:
9759:
9757:0-486-60090-4
9753:
9748:
9747:
9741:
9737:
9733:
9729:
9727:0-486-61480-8
9723:
9719:
9718:
9713:
9709:
9705:
9703:0-471-54397-7
9699:
9694:
9693:
9687:
9686:Merzbach, Uta
9683:
9679:
9675:
9671:
9667:
9663:
9659:
9655:
9650:
9645:
9641:
9637:
9636:Milan J. Math
9633:
9629:
9625:
9624:
9613:
9608:
9601:
9596:
9589:
9584:
9573:
9571:
9563:
9554:
9546:
9540:
9532:
9528:
9523:
9518:
9513:
9508:
9504:
9500:
9496:
9492:
9488:
9484:
9480:
9473:
9466:
9462:
9458:
9455:(in German),
9454:
9447:
9439:
9435:
9434:
9426:
9418:
9414:
9410:
9406:
9402:
9398:
9394:
9387:
9385:
9376:
9370:
9362:
9358:
9353:
9348:
9344:
9340:
9336:
9329:
9327:
9325:
9315:
9307:
9305:0-521-30429-6
9301:
9297:
9296:
9291:
9287:
9280:
9274:, p. 74.
9273:
9268:
9261:
9260:
9255:
9249:
9242:
9237:
9229:
9225:
9218:
9211:
9207:
9201:
9197:
9187:
9184:
9182:
9179:
9177:
9174:
9172:
9169:
9167:
9164:
9162:
9159:
9157:
9154:
9152:
9151:Johnson solid
9149:
9147:
9144:
9142:
9141:Catalan solid
9139:
9137:
9134:
9133:
9126:
9124:
9120:
9116:
9111:
9109:
9105:
9101:
9097:
9093:
9089:
9085:
9081:
9076:
9074:
9070:
9060:
9057:
9056:
9052:
9049:
9048:
9045:
9042:
9040:
9037:
9036:
9032:
9029:
9028:
9024:
9021:
9020:
9016:
9013:
9012:
9005:
8999:
8998:
8994:
8980:
8977:
8975:
8972:
8970:
8967:
8965:
8962:
8961:
8957:
8953:
8950:
8946:
8943:
8939:
8936:
8932:
8931:
8925:
8903:
8890: +
8885:
8871:
8862:
8859:
8857:
8854:
8852:
8849:
8848:
8844:
8840:
8837:
8833:
8830:
8826:
8825:
8819:
8802: =
8797:
8784: +
8779:
8745:
8732: +
8727:
8713:
8709:
8704:
8702:
8698:
8694:
8690:
8686:
8682:
8678:
8674:
8664:
8661:
8658:
8655:
8652:
8651:
8647:
8643:
8640:
8636:
8633:
8629:
8626:
8622:
8619:
8615:
8614:
8609:
8605:
8602:
8599:
8596:
8593:
8592:
8588:
8584:
8581:
8577:
8574:
8570:
8567:
8563:
8560:
8556:
8555:
8550:
8546:
8543:
8540:
8537:
8534:
8533:
8529:
8525:
8522:
8518:
8515:
8511:
8508:
8504:
8501:
8497:
8496:
8491:
8480:
8478:
8474:
8469:
8467:
8463:
8459:
8458:star polygons
8455:
8450:
8448:
8444:
8440:
8436:
8432:
8428:
8427:quasi-regular
8424:
8420:
8419:rectification
8417:, which is a
8416:
8415:cuboctahedron
8407:
8402:
8398:
8396:
8395:cuboctahedron
8391:
8387:
8386:
8383:
8381:
8377:
8373:
8358:
8356:
8352:
8351:neoclassicism
8348:
8344:
8340:
8336:
8332:
8324:
8320:
8316:
8311:
8302:
8300:
8296:
8295:Dan Shechtman
8292:
8288:
8278:
8276:
8272:
8267:
8265:
8264:dice notation
8261:
8257:
8253:
8249:
8245:
8236:
8229:
8228:Dodecahedrane
8223:
8218:
8215:
8209:
8204:
8201:
8195:
8190:
8189:
8188:
8186:
8182:
8181:dodecahedrane
8178:
8174:
8169:
8151:
8144:
8140:
8135:
8131:
8129:
8128:singularities
8125:
8121:
8117:
8116:triangulation
8113:
8109:
8105:
8100:
8098:
8093:
8089:
8085:
8080:
8078:
8074:
8070:
8066:
8062:
8058:
8057:Ernst Haeckel
8050:
8046:
8041:
8037:
8035:
8027:
8026:boron carbide
8023:
8019:
8014:
8010:
8006:
7995:
7992:
7984:
7974:
7970:
7966:
7960:
7959:
7955:
7950:This section
7948:
7944:
7939:
7938:
7926:dodecahedron
7925:
7922:
7919:
7917:
7914:
7913:
7897:
7890:
7884:
7880:
7874:
7871:
7868:
7866:
7863:
7862:
7858:
7855:
7852:
7850:
7847:
7846:
7830:
7823:
7817:
7813:
7807:
7804:
7801:
7799:
7796:
7795:
7789:
7784:
7781:
7779:
7772:
7769:
7766:
7762:
7759:
7756:
7753:
7750:
7748:
7745:
7744:
7741:
7738:
7736:
7733:
7731:
7728:
7726:
7723:
7721:
7718:
7717:
7713:
7708:
7701:
7694:
7685:
7682:
7680:
7674:
7672:
7668:
7659:
7656:
7652:
7649:
7646:
7642:
7639:
7636:
7632:
7631:
7630:
7626:
7624:
7620:
7616:
7612:
7608:
7604:
7600:
7596:
7591:
7589:
7585:
7581:
7577:
7573:
7569:
7565:
7561:
7557:
7547:
7545:
7541:
7537:
7533:
7529:
7526: =
7525:
7521:
7518: =
7517:
7512:
7499:
7496:
7491:
7487:
7483:
7480:
7475:
7471:
7467:
7464:
7459:
7455:
7451:
7446:
7442:
7433:
7431:
7427:
7423:
7419:
7415:
7411:
7407:
7402:
7398:
7396:
7392:
7388:
7384:
7376:
7373:
7370:
7366:
7365:
7364:
7362:
7359:
7350:
7341:
7330:
7319:
7300:
7298:
7294:
7290:
7286:
7283:A polyhedron
7276:
7263:
7258:
7253:
7249:
7243:
7238:
7235:
7232:
7228:
7224:
7221:
7218:
7213:
7208:
7202:
7197:
7193:
7187:
7182:
7179:
7176:
7172:
7167:
7162:
7154:
7152:
7148:
7140:
7127:
7122:
7117:
7111:
7107:
7101:
7098:
7093:
7085:
7074:
7067:
7062:
7057:
7052:
7048:
7042:
7039:
7034:
7026:
7015:
7008:
6999:
6981:
6977:
6971:
6967:
6963:
6958:
6954:
6949:
6943:
6939:
6933:
6929:
6925:
6922:
6917:
6912:
6906:
6902:
6898:
6893:
6889:
6884:
6877:
6873:
6867:
6863:
6857:
6854:
6849:
6844:
6839:
6833:
6829:
6825:
6820:
6816:
6811:
6806:
6804:
6794:
6783:
6776:
6772:
6764:
6753:
6746:
6738:
6733:
6729:
6723:
6719:
6713:
6710:
6705:
6700:
6695:
6689:
6685:
6681:
6676:
6672:
6667:
6660:
6656:
6650:
6646:
6642:
6639:
6634:
6629:
6623:
6619:
6615:
6610:
6606:
6601:
6596:
6594:
6584:
6573:
6566:
6562:
6554:
6543:
6536:
6528:
6524:
6518:
6514:
6510:
6505:
6501:
6496:
6490:
6486:
6480:
6476:
6472:
6469:
6464:
6459:
6453:
6449:
6445:
6440:
6436:
6431:
6426:
6424:
6414:
6403:
6396:
6392:
6384:
6373:
6366:
6362:
6354:
6343:
6336:
6332:
6324:
6313:
6306:
6298:
6293:
6289:
6283:
6279:
6273:
6270:
6265:
6260:
6255:
6249:
6245:
6241:
6236:
6232:
6227:
6222:
6220:
6210:
6199:
6192:
6188:
6180:
6169:
6162:
6158:
6150:
6139:
6132:
6128:
6120:
6109:
6102:
6098:
6090:
6079:
6072:
6064:
6059:
6055:
6051:
6046:
6042:
6038:
6036:
6026:
6015:
6008:
6004:
5996:
5985:
5978:
5974:
5966:
5955:
5948:
5944:
5936:
5925:
5918:
5914:
5906:
5895:
5888:
5875:
5872:
5856:
5853:
5848:
5844:
5838:
5833:
5830:
5827:
5823:
5817:
5814:
5809:
5801:
5798:
5787:
5780:
5771:
5769:
5762:
5759:vertices are
5758:
5754:
5744:
5740:
5727:
5722:
5719:
5716:
5711:
5705:
5699:
5694:
5691:
5684:
5679:
5676:
5671:
5668:
5665:
5662:
5659:
5655:
5650:
5644:
5639:
5636:
5630:
5625:
5622:
5617:
5614:
5611:
5608:
5605:
5597:
5595:
5591:
5570:
5567:
5560:
5546:
5543:
5538:
5532:
5528:
5524:
5514:
5498:
5493:
5486:
5472:
5469:
5462:
5448:
5441:
5424:
5418:
5414:
5404:
5402:
5399:
5398:
5383:
5380:
5373:
5359:
5356:
5349:
5345:
5338:
5334:
5330:
5320:
5302:
5297:
5294:
5291:
5286:
5279:
5265:
5259:
5250:
5234:
5230:
5222:
5206:
5201:
5197:
5187:
5185:
5182:
5181:
5166:
5163:
5156:
5142:
5139:
5134:
5130:
5120:
5104:
5099:
5092:
5076:
5067:
5052:
5045:
5028:
5025:
5015:
5013:
5010:
5009:
4993:
4986:
4971:
4964:
4949:
4942:
4926:
4917:
4901:
4892:
4877:
4870:
4868:
4865:
4849:
4846:
4839:
4825:
4822:
4817:
4813:
4803:
4787:
4782:
4775:
4758:
4755:
4745:
4729:
4723:
4715:
4699:
4693:
4685:
4683:
4680:
4679:
4675:
4673:
4670:
4668:
4664:
4662:
4658:
4656:
4652:
4651:
4645:
4634:
4628:
4625:
4623:
4618:
4605:
4602:
4599:
4594:
4591:
4586:
4583:
4575:
4573:
4569:
4565:
4561:
4557:
4552:
4539:
4535:
4530:
4527:
4522:
4518:
4515:
4512:
4509:
4504:
4490:
4487:
4475:
4472:
4464:
4462:
4458:
4454:
4450:
4446:
4442:
4437:
4424:
4409:
4406:
4394:
4391:
4377:
4374:
4359:
4355:
4350:
4338:
4335:
4320:
4316:
4308:
4304:
4299:
4296:
4291:
4287:
4284:
4280:
4275:
4272:
4267:
4263:
4260:
4257:
4252:
4249:
4239:
4237:
4233:
4229:
4225:
4220:
4198:
4195:
4182:
4176:
4171:
4168:
4163:
4159:
4156:
4151:
4148:
4143:
4140:
4132:
4130:
4126:
4121:
4103:
4098:
4095:
4090:
4086:
4083:
4079:
4074:
4071:
4066:
4062:
4059:
4054:
4051:
4046:
4044:
4039:
4031:
4026:
4023:
4018:
4014:
4011:
4007:
4002:
3999:
3994:
3990:
3987:
3982:
3979:
3974:
3972:
3967:
3955:
3954:are given by
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3913:
3909:
3906:
3902:
3899:
3895:
3894:
3893:
3869:
3865:
3857:
3843:
3840:
3835:
3830:
3827:
3822:
3818:
3815:
3812:
3809:
3806:
3803:
3796:
3782:
3778:
3770:
3754:
3750:
3742:
3739:
3737:
3734:
3733:
3718:
3714:
3706:
3692:
3689:
3684:
3679:
3676:
3671:
3667:
3664:
3661:
3658:
3651:
3637:
3633:
3625:
3611:
3604:
3601:
3599:
3596:
3595:
3580:
3576:
3568:
3554:
3551:
3546:
3541:
3538:
3533:
3529:
3526:
3523:
3516:
3502:
3497:
3494:
3485:
3469:
3460:
3457:
3455:
3452:
3451:
3436:
3432:
3429:
3421:
3407:
3404:
3398:
3395:
3386:
3372:
3368:
3360:
3346:
3339:
3336:
3334:
3331:
3330:
3315:
3308:
3294:
3291:
3286:
3281:
3278:
3273:
3269:
3266:
3259:
3245:
3238:
3222:
3216:
3208:
3205:
3203:
3200:
3199:
3191:
3188:
3184:
3180:
3177:
3172:
3169:
3157:
3149:
3146:
3141:
3136:
3133:
3132:
3129:
3127:
3101:
3097:
3092:
3085:
3083:
3079:
3075:
3071:
3070:vertex figure
3067:
3063:
3058:
3044:
3041:
3035:
3032:
3029:
3023:
3020:
3017:
3014:
3003:
3001:
2997:
2992:
2981:
2968:
2964:
2958:
2955:
2950:
2947:
2943:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2913:
2911:
2907:
2903:
2895:
2890:
2888:
2884:
2881:The quantity
2879:
2866:
2859:
2854:
2851:
2846:
2842:
2839:
2833:
2828:
2825:
2820:
2816:
2813:
2807:
2803:
2798:
2795:
2790:
2786:
2783:
2775:
2773:
2768:
2755:
2748:
2743:
2740:
2735:
2731:
2728:
2722:
2717:
2714:
2709:
2705:
2702:
2696:
2692:
2687:
2684:
2679:
2675:
2672:
2664:
2662:
2658:
2654:
2650:
2646:
2628:
2626:
2622:
2618:
2614:
2609:
2596:
2591:
2588:
2583:
2578:
2575:
2570:
2565:
2562:
2552:
2550:
2545:
2532:
2527:
2524:
2519:
2514:
2511:
2506:
2501:
2498:
2493:
2488:
2485:
2475:
2472:
2459:
2456:
2451:
2447:
2444:
2438:
2435:
2432:
2427:
2423:
2420:
2403:
2398:
2396:
2391:
2389:
2384:
2383:
2379:
2378:Eulerian path
2375:
2370:
2366:
2364:
2360:
2356:
2353: =
2352:
2348:
2344:
2341: +
2340:
2337: −
2336:
2332:
2328:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2286:
2284:
2280:
2277:
2273:
2270:
2269:
2267:
2265:
2264:
2259:
2249:
2245:
2237:
2232:
2224:
2220:
2213:
2209:
2202:
2198:
2191:
2187:
2186:
2178:
2174:
2167:
2163:
2156:
2152:
2145:
2141:
2140:
2129:
2111:
2108:
2105:
2100:
2097:
2094:
2089:
2086:
2083:
2076:
2069:
2066:
2063:
2058:
2055:
2052:
2047:
2044:
2041:
2034:
2027:
2024:
2021:
2016:
2013:
2010:
2005:
2002:
1999:
1992:
1985:
1982:
1979:
1974:
1971:
1968:
1963:
1960:
1957:
1950:
1943:
1940:
1937:
1932:
1929:
1926:
1921:
1918:
1915:
1908:
1902:
1898:
1895:
1893:
1890:
1888:
1885:
1882:
1877:
1874:
1871:
1868:
1865:
1861:
1858:
1856:
1853:
1851:
1847:
1844:
1841:
1836:
1833:
1830:
1828:
1827:
1824:
1823:
1819:
1813:
1807:
1804:
1801:
1797:
1793:
1789:
1784:
1781:
1780:
1773:
1772:
1769:
1767:
1757:
1755:
1751:
1747:
1743:
1740:interchanges
1739:
1735:
1730:
1717:
1708:
1705:
1702:
1693:
1690:
1687:
1681:
1678:
1673:
1670:
1664:
1661:
1657:
1648:
1645:
1642:
1633:
1630:
1627:
1621:
1618:
1613:
1610:
1607:
1601:
1598:
1594:
1585:
1582:
1579:
1570:
1567:
1564:
1558:
1555:
1550:
1547:
1541:
1538:
1530:
1528:
1524:
1520:
1515:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1475:
1473:
1468:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1422:
1420:
1416:
1412:
1408:
1404:
1394:
1391:
1388:
1385:
1382:
1379:
1375:
1373:
1365:
1362:
1359:
1356:
1353:
1350:
1346:
1344:
1336:
1333:
1330:
1327:
1324:
1321:
1317:
1315:
1307:
1304:
1301:
1298:
1295:
1292:
1288:
1286:
1278:
1275:
1272:
1269:
1266:
1263:
1259:
1257:
1250:
1245:
1240:
1235:
1230:
1223:
1217:
1215:
1214:combinatorial
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1179:
1175:
1172:
1169:
1165:
1161:
1160:
1159:
1151:
1149:
1144:
1142:
1110:
1109:
1076:
1071:
1069:
1008:
1006:
1002:
991:
978:
952:
938:
926:
912:
899:
891:
873:
861:
853:
835:
827:
804:
800:
784:
776:
767:
759:
747:
739:
730:
725:
688:
655:
634:
628:
624:
621:
618:
615:
613:
611:
608:
605:
602:
601:
591:
588:
581:
571:
568:
561:
558:Dodecahedron
551:
548:
541:
535:
512:
508:
504:
494:
492:
488:
487:Kepler solids
484:
480:
476:
472:
468:
464:
460:
456:
451:
447:
446:
441:
437:
434:
426:
425:
420:
416:
412:
408:
406:
402:
398:
397:
392:
388:
386:
382:
378:
374:
370:
366:
362:
355:
354:
349:
340:
330:
321:
312:
303:
294:
285:
283:
279:
275:
271:
266:
263:
259:
255:
245:
243:
239:
238:
233:
229:
221:
218:
214:
207:
203:
201:
198:
194:
187:
183:
181:
178:
174:
167:
163:
161:
158:
154:
147:
143:
141:
138:
134:
127:
123:
122:
119:Twenty faces
118:
116:Twelve faces
115:
112:
109:
106:
105:
102:
99:
97:
94:
92:
89:
87:
84:
82:
79:
78:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
12593:
12550:Neoplatonism
12545:Commentaries
12523:
12417:Hyperuranion
12415:
12403:
12398:
12360:
12353:
12346:
12332:
12284:
12277:
12270:
12265:Rival Lovers
12263:
12256:
12249:
12242:
12235:
12228:
12221:
12212:
12205:
12198:
12191:
12184:
12177:
12170:
12164:authenticity
12150:
12143:
12136:
12129:
12122:
12115:
12108:
12101:
12094:
12087:
12080:
12073:
12066:
12059:
12052:
12045:
12038:
12031:
12024:
12017:
12010:
12003:
11996:
11989:
11982:
11975:
11968:
11961:
11954:
11899:
11818:trapezohedra
11769:
11762:
11566:dodecahedron
11535:
11477:
11464:
11306:Thomas Heath
11297:
11180:
11164:Law of sines
11020:
10976:
10952:Golden ratio
10817:
10810:
10801:
10795:(Theodosius)
10791:
10783:
10776:
10769:
10762:
10753:
10743:
10737:(Hipparchus)
10733:
10723:
10715:
10708:
10699:
10689:
10681:
10676:(Apollonius)
10672:
10664:
10652:
10627:Zeno of Elea
10387:Eratosthenes
10377:Dionysodorus
10171:. Retrieved
10167:the original
10158:
10099:
10098:of Euclid's
10085:
10066:
10050:
10032:
10023:
10008:by removing
9995:
9960:
9939:
9915:
9890:
9886:
9861:
9857:
9839:
9820:
9816:
9803:. Retrieved
9784:
9768:
9745:
9716:
9691:
9639:
9635:
9612:Coxeter 1973
9607:
9595:
9583:
9569:
9562:
9553:
9539:cite journal
9494:
9490:
9472:
9456:
9452:
9446:
9437:
9431:
9425:
9396:
9392:
9369:cite journal
9342:
9338:
9314:
9293:
9279:
9267:
9257:
9253:
9248:
9236:
9227:
9217:
9200:
9112:
9098:as {3,3,5},
9094:as {3,3,4},
9090:as {3,3,3},
9077:
9066:
9043:
9038:
8901:
8883:
8867:
8795:
8777:
8743:
8725:
8711:
8707:
8705:
8700:
8688:
8670:
8470:
8451:
8426:
8412:
8369:
8328:
8315:Isaac Newton
8284:
8271:Rubik's Cube
8268:
8259:
8255:
8241:
8200:Tetrahedrane
8185:tetrahedrane
8170:
8150:space frames
8148:Geometry of
8147:
8141:monument in
8101:
8081:
8076:
8072:
8068:
8064:
8054:
8009:pyritohedron
8002:
7987:
7981:October 2018
7978:
7963:Please help
7951:
7923:5 | 2 3
7895:
7888:
7875:icosahedron
7872:3 | 2 5
7865:dodecahedron
7856:4 | 2 3
7828:
7821:
7805:3 | 2 4
7777:
7770:
7757:tetrahedron
7754:3 | 2 3
7675:
7666:
7663:
7657:
7647:
7637:
7627:
7623:face-uniform
7619:edge-uniform
7610:
7592:
7583:
7575:
7553:
7543:
7539:
7535:
7531:
7527:
7523:
7519:
7515:
7513:
7434:
7429:
7425:
7421:
7417:
7413:
7409:
7405:
7403:
7399:
7394:
7390:
7386:
7382:
7380:
7360:
7355:
7296:
7292:
7284:
7282:
7155:
7150:
7143:
7141:
7000:
5876:
5873:
5772:
5764:
5760:
5756:
5752:
5750:
5741:
5598:
5593:
5589:
5587:
5184:dodecahedron
4671:
4666:
4660:
4654:
4643:
4632:
4630:Polyhedron,
4621:
4619:
4576:
4571:
4567:
4559:
4553:
4465:
4460:
4456:
4452:
4448:
4444:
4441:surface area
4438:
4240:
4235:
4231:
4227:
4223:
4221:
4133:
4131:is given by
4128:
4124:
4122:
3956:
3951:
3947:
3943:
3939:
3935:
3931:
3927:
3924:circumradius
3923:
3917:
3891:
3598:dodecahedron
3186:
3175:
3155:
3144:
3126:golden ratio
3099:
3093:
3086:
3081:
3077:
3073:
3059:
3004:
2999:
2993:
2982:
2914:
2909:
2905:
2901:
2891:
2885:(called the
2882:
2880:
2776:
2769:
2665:
2660:
2656:
2652:
2642:
2624:
2620:
2616:
2612:
2610:
2553:
2548:
2546:
2476:
2473:
2410:
2362:
2358:
2354:
2350:
2346:
2342:
2338:
2334:
2324:
2276:angle defect
2262:
2255:
2247:
2236:angle defect
2161:Defect 120°
2150:Defect 180°
2126:
1900:
1896:
1891:
1886:
1872:
1859:
1854:
1849:
1845:
1817:
1811:
1805:
1799:
1795:
1791:
1787:
1763:
1749:
1745:
1741:
1737:
1733:
1731:
1531:
1526:
1522:
1518:
1516:
1476:
1469:
1423:
1418:
1414:
1410:
1406:
1402:
1400:
1343:dodecahedron
1205:
1201:
1197:
1193:
1189:
1185:
1183:
1157:
1145:
1106:
1077:, t{3,4} or
1072:
1009:
997:
989:
976:
950:
936:
924:
910:
901:(±1, ±1, ±1)
889:
871:
859:
851:
833:
825:
806:(±1, ±1, ±1)
798:
782:
774:
757:
745:
737:
727:(±1, ±1, ±1)
657:(−1, −1, −1)
632:coordinates
511:golden ratio
506:
500:
490:
450:Solar System
443:
430:
422:
419:Solar System
404:
395:
389:
352:
345:
338:
267:
251:
235:
226:
209:
189:
169:
149:
129:
113:Eight faces
96:Dodecahedron
35:
29:
12405:Anima mundi
12362:Theia mania
12179:Definitions
12162:Of doubtful
11588:semiregular
11571:icosahedron
11551:tetrahedron
11373:mathematics
11181:Arithmetica
10778:Ostomachion
10747:(Autolycus)
10666:Arithmetica
10442:Hippocrates
10372:Dinostratus
10357:Dicaearchus
10287:Aristarchus
10163:Brady Haran
10159:Numberphile
10082:"Isohedron"
9682:Boyer, Carl
9345:: 335–355.
9146:Deltahedron
9110:, {3,4,3}.
9003:dimensions
8380:stellations
8291:H. Kleinert
8108:climatology
8104:meteorology
7916:icosahedron
7879:Icosahedral
7808:octahedron
7761:Tetrahedral
7747:tetrahedron
7687:Polyhedron
7580:reflections
5401:icosahedron
4682:tetrahedron
4676:Unit edges
4574:. That is,
4463:. This is:
3736:icosahedron
3202:tetrahedron
3183:solid angle
3134:Polyhedron
2996:solid angle
2327:topological
2218:Defect 36°
2196:Defect 90°
2172:Defect 60°
1783:Group order
1372:icosahedron
1225:Polyhedron
644:(−1, 1, −1)
640:(1, −1, −1)
555:Icosahedron
546:Tetrahedron
538:Parameters
240:, that the
107:Four faces
101:Icosahedron
81:Tetrahedron
12611:Categories
12468:Myth of Er
12428:Allegories
12334:Sophrosyne
12310:Philosophy
12251:On Justice
12237:Hipparchus
12145:Theaetetus
12110:Protagoras
12082:Parmenides
11998:Euthydemus
11883:prismatoid
11813:bipyramids
11797:antiprisms
11771:hosohedron
11561:octahedron
11425:Babylonian
11325:arithmetic
11291:History of
11120:Apollonius
10805:(Menelaus)
10764:On Spirals
10683:Catoptrics
10622:Xenocrates
10617:Thymaridas
10602:Theodosius
10587:Theaetetus
10567:Simplicius
10557:Pythagoras
10542:Posidonius
10527:Philonides
10487:Nicomachus
10482:Metrodorus
10472:Menaechmus
10427:Hipparchus
10417:Heliodorus
10367:Diophantus
10352:Democritus
10332:Chrysippus
10302:Archimedes
10297:Apollonius
10267:Anaxagoras
10259:(timeline)
10204:with some
10195:images of
10173:2013-04-13
9805:2024-02-12
9777:0226282538
9483:Roya Zandi
9352:2010.12340
9241:Lloyd 2012
8671:The three
8477:deltahedra
8466:antiprisms
8061:Radiolaria
8045:radiolaria
8024:, such as
7849:octahedron
7812:Octahedral
7720:Polyhedral
7707:polyhedron
7607:transitive
7582:, and the
7401:of edges.
5012:octahedron
3930:, and the
3454:octahedron
3096:steradians
2307:Pentagonal
2289:Triangular
2229:Defect 0°
2207:Defect 0°
2183:Defect 0°
1405:), edges (
1395:3.3.3.3.3
1212:, gives a
1005:reflection
661:(−1, 1, 1)
549:Octahedron
534:≈ 1.6180.
433:astronomer
282:Theaetetus
278:Pythagoras
260:people of
110:Six faces
91:Octahedron
12355:Peritrope
12258:On Virtue
12186:Demodocus
12138:Symposium
12131:Statesman
12068:Menexenus
12005:Euthyphro
11970:Clitophon
11963:Charmides
11878:birotunda
11868:bifrustum
11633:snub cube
11528:polyhedra
10886:Inscribed
10646:Treatises
10637:Zenodorus
10597:Theodorus
10572:Sosigenes
10517:Philolaus
10502:Oenopides
10497:Nicoteles
10492:Nicomedes
10452:Hypsicles
10347:Ctesibius
10337:Cleomedes
10322:Callippus
10307:Autolycus
10292:Aristotle
10272:Anthemius
10096:Book XIII
10087:MathWorld
10068:MathWorld
10010:excessive
9907:119544202
9674:119725110
9642:: 33–58.
9479:Polly Roy
9477:Siyu Li,
9417:218542147
9403:: 87–98.
9272:Weyl 1952
9193:Citations
9119:hypercube
9100:tesseract
9069:polytopes
9001:Number of
8685:hosohedra
8665:{2,6}...
8606:{6,2}...
8493:Platonic
8301:in 2011.
8143:Amsterdam
8120:longitude
8020:and many
7952:does not
7597:known as
7588:rotations
7497:ρ
7492:∗
7488:ρ
7476:∗
7460:∗
7369:self-dual
7229:∑
7173:∑
5824:∑
5723:φ
5720:−
5695:−
5677:π
5672:
5660:ξ
5623:π
5618:
5606:φ
5568:≈
5547:17.453560
5544:≈
5529:φ
5473:φ
5470:ξ
5449:φ
5415:φ
5381:≈
5360:61.304952
5357:≈
5346:ξ
5335:φ
5266:φ
5231:φ
5207:ξ
5198:φ
5164:≈
5140:≈
4847:≈
4823:≈
4665:Circum-,
4528:π
4519:
4407:α
4395:
4375:α
4351:−
4336:θ
4297:π
4288:
4273:π
4264:
4196:π
4169:π
4160:
4141:ρ
4096:θ
4087:
4072:π
4063:
4024:θ
4015:
4000:π
3991:
3928:midradius
3905:midsphere
3866:π
3841:≈
3819:
3810:−
3807:π
3779:π
3751:φ
3715:π
3690:≈
3668:
3662:−
3659:π
3634:π
3612:φ
3577:π
3552:≈
3530:
3498:π
3433:π
3405:≈
3396:π
3369:π
3316:π
3292:≈
3270:
3246:π
3150:tan
3138:Dihedral
3042:π
3033:−
3024:−
3021:θ
3012:Ω
2951:−
2940:π
2934:−
2931:π
2922:δ
2852:π
2843:
2826:π
2817:
2796:θ
2787:
2741:π
2732:
2715:π
2706:
2685:θ
2676:
2433:−
2325:A purely
1732:Swapping
1706:−
1691:−
1682:−
1646:−
1631:−
1622:−
1583:−
1568:−
1559:−
1487:−
1164:congruent
648:(−1, −1,
636:(1, 1, 1)
583:Vertices
491:nonconvex
381:Aristotle
276:) credit
228:Geometers
213:Animation
193:Animation
173:Animation
153:Animation
133:Animation
56:congruent
12524:Republic
12448:The Cave
12438:Atlantis
12411:Demiurge
12348:Amanesis
12279:Sisyphus
12207:Epistles
12200:Epinomis
12193:Epigrams
12172:Axiochus
12117:Republic
12103:Philebus
12096:Phaedrus
11977:Cratylus
11858:bicupola
11838:pyramids
11764:dihedron
11450:Japanese
11435:Egyptian
11378:timeline
11366:timeline
11354:timeline
11349:geometry
11342:timeline
11337:calculus
11330:timeline
11318:timeline
11021:Elements
10867:Concepts
10829:Problems
10802:Spherics
10792:Spherics
10757:(Euclid)
10703:(Euclid)
10700:Elements
10693:(Euclid)
10654:Almagest
10562:Serenus
10537:Porphyry
10477:Menelaus
10432:Hippasus
10407:Eutocius
10382:Domninus
10277:Archytas
10143:software
10127:Archived
10100:Elements
9940:Symmetry
9936:(1952).
9767:(1987).
9738:(1956).
9714:(1973).
9688:(1989).
9531:30301797
9292:(1986).
9259:Epinomis
9129:See also
9104:120-cell
9096:600-cell
9033:∞
8339:cylinder
8319:cenotaph
8183:and not
8171:Several
8124:latitude
8013:minerals
7691:Schläfli
7556:symmetry
7428:*,
7424:*,
7303:Symmetry
5874:we have
5571:2.181695
5384:7.663119
5167:0.471404
5143:3.771236
4850:0.117851
4826:0.942809
3932:inradius
3295:0.551286
2357:, where
2263:Elements
1337:3.3.3.3
1229:Vertices
1178:vertices
736:0, ±1, ±
603:Position
415:Kepler's
405:Elements
396:Elements
262:Scotland
217:3D model
197:3D model
177:3D model
157:3D model
137:3D model
32:geometry
12458:The Sun
12286:Theages
12230:Halcyon
12223:Eryxias
12152:Timaeus
12124:Sophist
12019:Gorgias
11984:Critias
11956:Apology
11900:italics
11888:scutoid
11873:rotunda
11863:frustum
11592:uniform
11541:regular
11526:Convex
11430:Chinese
11385:numbers
11313:algebra
11241:Related
11215:Centers
11011:Results
10881:Central
10552:Ptolemy
10547:Proclus
10512:Perseus
10467:Marinus
10447:Hypatia
10437:Hippias
10412:Geminus
10402:Eudoxus
10392:Eudemus
10362:Diocles
10133:in Java
10108:in Java
10004:Please
9996:use of
9866:Bibcode
9742:(ed.).
9654:Bibcode
9522:6205497
9499:Bibcode
9465:0497615
9254:Timaeus
9115:simplex
9108:24-cell
9092:16-cell
8922:
8910:
8906:
8892:
8888:
8874:
8816:
8804:
8800:
8786:
8782:
8768:
8764:
8752:
8748:
8734:
8730:
8716:
8714:} with
8710:,
8697:dihedra
8454:regular
8325:, 1784)
8166:
8154:
8139:Spinoza
8092:protein
8084:viruses
7973:removed
7958:sources
7698:Wythoff
7611:regular
7416:,
7412:,
7393:,
7385:,
4648:Volume
4638:Radius
4564:pyramid
4451:,
3946:,
3844:2.63455
3740:138.19°
3693:2.96174
3602:116.57°
3555:1.35935
3458:109.47°
3408:1.57080
3181:Vertex
3166:
3152:
3124:is the
3122:
3109:√
3104:
2772:tangent
2623:,
2260:in the
1794:/(4 − (
1204:,
1188:,
1166:convex
985:
981:
967:
959:
955:
941:
929:
915:
907:
894:
880:
876:
864:
856:
842:
830:
816:
812:
795:
787:
771:
762:
750:
734:
720:
716:
708:
706:0, ±1,
704:
696:
692:
683:
679:
671:
669:1, −1,
667:
650:
630:Vertex
543:Figure
532:
519:√
514:
475:Jupiter
459:Mercury
440:planets
353:Timaeus
274:Proclus
248:History
237:Timaeus
72:vertex.
12505:Legacy
12089:Phaedo
12047:Laches
11853:cupola
11806:duals:
11792:prisms
11445:Indian
11222:Cyrene
10754:Optics
10673:Conics
10592:Theano
10582:Thales
10577:Sporus
10522:Philon
10507:Pappus
10397:Euclid
10327:Carpus
10317:Bryson
10141:Stella
9968:
9948:
9922:
9905:
9829:
9796:
9775:
9754:
9736:Euclid
9724:
9700:
9672:
9529:
9519:
9463:
9415:
9302:
9088:5-cell
9058:> 4
8979:{3, 7}
8974:{7, 3}
8969:{4, 5}
8964:{5, 4}
8861:{6, 3}
8856:{3, 6}
8851:{4, 4}
8677:sphere
8662:{2,5}
8659:{2,4}
8656:{2,3}
8653:{2,2}
8603:{5,2}
8600:{4,2}
8597:{3,2}
8594:{2,2}
8547:{3,5}
8544:{5,3}
8541:{3,4}
8538:{4,3}
8535:{3,3}
8462:prisms
8345:, and
8335:sphere
8258:where
8214:Cubane
8177:cubane
8097:genome
8088:herpes
7920:{3, 5}
7869:{5, 3}
7853:{3, 4}
7802:{4, 3}
7751:{3, 3}
7725:Schön.
7700:symbol
7693:symbol
7621:, and
7603:action
4659:Mid-,
4556:volume
4222:where
4123:where
3926:, the
3816:arcsin
3665:arctan
3527:arcsin
3267:arccos
3206:70.53°
3196:angle
3194:solid
3171:Defect
3084:-gon.
2645:angles
2639:Angles
2547:Since
2298:Square
2258:Euclid
2248:defect
2079:{5,3}
2037:{3,5}
1995:{4,3}
1953:{3,4}
1911:{3,3}
1820:= 120
1802:− 2))
1774:{p,q}
1525:, and
1392:{3, 5}
1366:5.5.5
1363:{5, 3}
1334:{3, 4}
1308:4.4.4
1305:{4, 3}
1279:3.3.3
1276:{3, 3}
785:, ±1,
764:0, ±1)
744:(±1, ±
722:0, ±1)
685:1, −1)
563:Faces
455:Saturn
427:(1596)
391:Euclid
385:aither
375:, and
64:angles
40:convex
12244:Minos
12061:Lysis
11991:Crito
11948:Works
11941:Plato
11440:Incan
11361:logic
11137:Other
10905:Chord
10898:Axiom
10876:Angle
10532:Plato
10422:Heron
10342:Conon
9903:S2CID
9854:(PDF)
9670:S2CID
9644:arXiv
9575:(PDF)
9413:S2CID
9347:arXiv
8687:, {2,
8331:forms
8082:Many
7859:cube
7740:Order
7572:order
4653:In-,
3920:radii
3192:Face
3140:angle
2912:} is
2333:that
2309:faces
2300:faces
2291:faces
2227:{6,3}
2216:{5,3}
2205:{4,4}
2194:{4,3}
2181:{3,6}
2170:{3,5}
2159:{3,4}
2148:{3,3}
1814:= 48
1808:= 24
1798:− 2)(
1239:Faces
1234:Edges
793:(±1,
777:, ±1)
690:(±1,
467:Earth
463:Venus
421:from
373:water
365:earth
348:Plato
232:Plato
68:edges
62:(all
52:faces
38:is a
12478:Life
12075:Meno
12054:Laws
11556:cube
11402:list
10690:Data
10462:Leon
10312:Bion
9966:ISBN
9946:ISBN
9920:ISBN
9827:ISBN
9794:ISBN
9773:ISBN
9752:ISBN
9722:ISBN
9698:ISBN
9545:link
9527:PMID
9375:link
9300:ISBN
8693:lune
8471:The
8441:and
8343:cone
8248:fair
8244:dice
8179:and
8106:and
8075:and
7956:any
7954:cite
7905:532
7903:*532
7838:432
7836:*432
7798:cube
7787:332
7785:*332
7735:Orb.
7730:Cox.
7705:Dual
7653:the
7643:the
7633:the
7538:and
5763:and
5592:and
4867:cube
4554:The
4439:The
4234:and
3918:The
3910:the
3903:the
3896:the
3333:cube
3107:1 +
2892:The
2615:and
2584:>
1744:and
1736:and
1417:and
1285:cube
988:0, ±
909:0, ±
878:0, ±
814:0, ±
797:0, ±
773:0, ±
552:Cube
517:1 +
471:Mars
377:fire
268:The
86:Cube
54:are
34:, a
12582:229
12577:228
12040:Ion
11590:or
11304:by
11018:In
10012:or
9895:doi
9874:doi
9662:doi
9517:PMC
9507:doi
9495:115
9405:doi
9357:doi
8699:, {
8456:or
8317:'s
8102:In
7967:by
7910:60
7908:120
7843:24
7792:12
7546:).
7142:If
5669:sin
5615:cos
5131:128
4516:cot
4392:sin
4356:cos
4317:csc
4285:tan
4261:tan
4183:csc
4157:cos
4084:tan
4060:cot
4012:tan
3988:tan
3337:90°
2991:).
2840:sin
2814:cos
2784:tan
2774:by
2729:sin
2703:cos
2673:sin
2627:}:
2283:six
2242:By
2112:12
2070:20
1790:= 8
939:, ±
913:, ±
858:, ±
832:, ±
718:0,
694:0,
681:1,
578:12
369:air
46:in
30:In
12613::
12572:24
12567:23
10161:.
10157:.
10084:.
10065:.
9901:.
9891:27
9889:.
9872:.
9862:29
9860:.
9856:.
9792:.
9788:.
9684:;
9668:.
9660:.
9652:.
9640:71
9638:.
9630:;
9541:}}
9537:{{
9525:.
9515:.
9505:.
9493:.
9489:.
9461:MR
9457:10
9438:16
9436:,
9411:.
9397:90
9395:.
9383:^
9371:}}
9367:{{
9355:.
9343:11
9341:.
9337:.
9323:^
9288:;
9226:.
9061:3
9053:6
9025:1
9017:1
8449:.
8353:,
8341:,
8337:,
8277:.
8187:.
8099:.
8071:,
8067:,
8030:12
7841:48
7790:24
7673:.
7625:.
7617:,
7590:.
7528:Rr
7040:16
6926:16
6855:40
6795:10
6784:20
6765:10
6754:12
6711:16
6574:20
6544:12
6404:20
6374:12
6200:20
6170:12
6016:20
5986:12
5871:,
5525:20
5494:20
5331:20
5298:10
5292:25
5287:12
4950:24
4443:,
4238::
3680:11
3282:27
3279:23
3189:)
3178:)
3147:)
3128:.
3102:=
2460:2.
2355:qV
2347:pF
2266::
2250:.
2238:.
2101:2
2098:30
2090:3
2084:20
2059:2
2056:30
2048:5
2042:12
2028:6
2017:2
2014:12
2006:3
1986:8
1975:2
1972:12
1964:4
1944:4
1933:2
1922:3
1899:/2
1883:f
1878:2
1875:/4
1866:e
1848:/2
1842:v
1837:f
1792:pq
1785::
1756:.
1529::
1521:,
1502:2.
1474::
1389:20
1386:30
1383:12
1360:12
1357:30
1354:20
1328:12
1299:12
1150:.
1143:.
1070:.
992:)
965:(±
961:0)
957:,
935:(±
874:,
870:(±
866:0)
862:,
840:(±
789:0)
781:(±
760:,
756:(±
752:0)
748:,
710:0)
698:0)
673:1)
652:1)
625:2
575:20
473:,
469:,
465:,
461:,
371:,
367:,
357:c.
215:,
195:,
175:,
155:,
135:,
42:,
11933:e
11926:t
11919:v
11902:.
11594:)
11586:(
11543:)
11539:(
11519:e
11512:t
11505:v
10239:e
10232:t
10225:v
10176:.
10102:.
10090:.
10071:.
10039:)
10033:(
10028:)
10024:(
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10002:.
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9419:.
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9308:.
9243:.
9230:.
9050:4
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9030:2
9022:1
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8919:2
8916:/
8913:1
8902:q
8898:/
8895:1
8884:p
8880:/
8877:1
8813:2
8810:/
8807:1
8796:q
8792:/
8789:1
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8774:/
8771:1
8761:2
8758:/
8755:1
8744:q
8740:/
8737:1
8726:p
8722:/
8719:1
8712:q
8708:p
8701:n
8689:n
8321:(
8260:n
8256:n
8163:2
8160:/
8157:1
8122:/
8051:.
7994:)
7988:(
7983:)
7979:(
7975:.
7961:.
7896:I
7892:h
7889:I
7829:O
7825:h
7822:O
7778:T
7774:d
7771:T
7658:I
7648:O
7640:,
7638:T
7544:r
7540:r
7536:R
7532:R
7524:d
7520:ρ
7516:d
7500:.
7484:=
7481:R
7472:r
7468:=
7465:r
7456:R
7452:=
7447:2
7443:d
7430:r
7426:ρ
7422:R
7418:r
7414:ρ
7410:R
7406:d
7395:p
7391:q
7387:q
7383:p
7297:P
7293:P
7285:P
7264:.
7259:4
7254:i
7250:d
7244:n
7239:1
7236:=
7233:i
7225:n
7222:3
7219:=
7214:2
7209:)
7203:2
7198:i
7194:d
7188:n
7183:1
7180:=
7177:i
7168:(
7163:4
7151:n
7146:i
7144:d
7128:.
7123:2
7118:)
7112:2
7108:R
7102:3
7099:2
7094:+
7089:)
7086:2
7083:(
7078:]
7075:n
7072:[
7068:S
7063:(
7058:=
7053:4
7049:R
7043:9
7035:+
7030:)
7027:4
7024:(
7019:]
7016:n
7013:[
7009:S
6982:.
6978:)
6972:2
6968:L
6964:+
6959:2
6955:R
6950:(
6944:4
6940:L
6934:4
6930:R
6923:+
6918:3
6913:)
6907:2
6903:L
6899:+
6894:2
6890:R
6885:(
6878:2
6874:L
6868:2
6864:R
6858:3
6850:+
6845:5
6840:)
6834:2
6830:L
6826:+
6821:2
6817:R
6812:(
6807:=
6798:)
6792:(
6787:]
6781:[
6777:S
6773:=
6768:)
6762:(
6757:]
6751:[
6747:S
6739:,
6734:4
6730:L
6724:4
6720:R
6714:5
6706:+
6701:2
6696:)
6690:2
6686:L
6682:+
6677:2
6673:R
6668:(
6661:2
6657:L
6651:2
6647:R
6643:8
6640:+
6635:4
6630:)
6624:2
6620:L
6616:+
6611:2
6607:R
6602:(
6597:=
6588:)
6585:8
6582:(
6577:]
6571:[
6567:S
6563:=
6558:)
6555:8
6552:(
6547:]
6541:[
6537:S
6529:,
6525:)
6519:2
6515:L
6511:+
6506:2
6502:R
6497:(
6491:2
6487:L
6481:2
6477:R
6473:4
6470:+
6465:3
6460:)
6454:2
6450:L
6446:+
6441:2
6437:R
6432:(
6427:=
6418:)
6415:6
6412:(
6407:]
6401:[
6397:S
6393:=
6388:)
6385:6
6382:(
6377:]
6371:[
6367:S
6363:=
6358:)
6355:6
6352:(
6347:]
6344:8
6341:[
6337:S
6333:=
6328:)
6325:6
6322:(
6317:]
6314:6
6311:[
6307:S
6299:,
6294:2
6290:L
6284:2
6280:R
6274:3
6271:4
6266:+
6261:2
6256:)
6250:2
6246:L
6242:+
6237:2
6233:R
6228:(
6223:=
6214:)
6211:4
6208:(
6203:]
6197:[
6193:S
6189:=
6184:)
6181:4
6178:(
6173:]
6167:[
6163:S
6159:=
6154:)
6151:4
6148:(
6143:]
6140:8
6137:[
6133:S
6129:=
6124:)
6121:4
6118:(
6113:]
6110:6
6107:[
6103:S
6099:=
6094:)
6091:4
6088:(
6083:]
6080:4
6077:[
6073:S
6065:,
6060:2
6056:L
6052:+
6047:2
6043:R
6039:=
6030:)
6027:2
6024:(
6019:]
6013:[
6009:S
6005:=
6000:)
5997:2
5994:(
5989:]
5983:[
5979:S
5975:=
5970:)
5967:2
5964:(
5959:]
5956:8
5953:[
5949:S
5945:=
5940:)
5937:2
5934:(
5929:]
5926:6
5923:[
5919:S
5915:=
5910:)
5907:2
5904:(
5899:]
5896:4
5893:[
5889:S
5857:m
5854:2
5849:i
5845:d
5839:n
5834:1
5831:=
5828:i
5818:n
5815:1
5810:=
5805:)
5802:m
5799:2
5796:(
5791:]
5788:n
5785:[
5781:S
5767:i
5765:d
5761:L
5757:n
5753:R
5728:.
5717:3
5712:=
5706:2
5700:5
5692:5
5685:=
5680:5
5666:2
5663:=
5656:,
5651:2
5645:5
5640:+
5637:1
5631:=
5626:5
5612:2
5609:=
5594:ξ
5590:φ
5539:3
5533:2
5499:3
5425:3
5419:2
5350:2
5339:3
5303:5
5295:+
5260:3
5235:2
5202:2
5135:3
5105:3
5100:8
5077:2
5053:1
5029:3
5026:2
4994:1
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4927:3
4902:2
4878:1
4818:3
4814:8
4788:3
4783:4
4759:2
4756:3
4730:2
4724:1
4700:6
4694:1
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4667:R
4661:ρ
4655:r
4644:A
4633:a
4622:a
4606:.
4603:A
4600:r
4595:3
4592:1
4587:=
4584:V
4572:r
4568:p
4560:F
4540:.
4536:)
4531:p
4523:(
4513:p
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4505:2
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4457:p
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4449:p
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4425:.
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4410:2
4400:(
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4378:2
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4360:2
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4329:(
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4300:q
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4276:p
4268:(
4258:=
4253:r
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4199:h
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4177:)
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4152:2
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4144:=
4129:ρ
4125:θ
4104:)
4099:2
4091:(
4080:)
4075:p
4067:(
4055:2
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4047:=
4040:r
4032:)
4027:2
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3274:(
3223:2
3217:1
3187:Ω
3185:(
3176:δ
3174:(
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3156:θ
3145:θ
3143:(
3119:2
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3111:5
3100:φ
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3082:q
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3074:p
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3039:)
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3000:Ω
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2985:π
2969:.
2965:)
2959:p
2956:2
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2910:q
2908:,
2906:p
2902:δ
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2883:h
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2808:=
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2278:.
2109:5
2106:5
2095:2
2087:3
2067:3
2064:3
2053:2
2045:5
2025:4
2022:4
2011:2
2003:3
2000:8
1983:3
1980:3
1969:2
1961:4
1958:6
1941:3
1938:3
1930:6
1927:2
1919:3
1916:4
1901:p
1897:g
1892:p
1887:p
1873:g
1869:2
1860:q
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1834:e
1831:v
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1812:g
1806:g
1800:q
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1750:E
1746:V
1742:F
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1709:2
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1556:4
1551:p
1548:4
1542:=
1539:V
1527:F
1523:E
1519:V
1499:=
1496:F
1493:+
1490:E
1484:V
1455:.
1452:V
1449:q
1446:=
1443:E
1440:2
1437:=
1434:F
1431:p
1419:q
1415:p
1411:F
1407:E
1403:V
1331:8
1325:6
1302:6
1296:8
1273:4
1270:6
1267:4
1206:q
1202:p
1198:q
1194:p
1190:q
1186:p
1180:.
1170:.
999:(
990:φ
983:,
977:φ
973:/
970:1
951:φ
947:/
944:1
937:φ
931:)
925:φ
921:/
918:1
911:φ
905:(
896:)
890:φ
886:/
883:1
872:φ
860:φ
852:φ
848:/
845:1
836:)
834:φ
826:φ
822:/
819:1
810:(
801:)
799:φ
783:φ
775:φ
769:(
758:φ
746:φ
740:)
738:φ
732:(
714:(
702:(
677:(
665:(
622:1
619:2
616:1
609:2
606:1
592:8
586:4
572:6
569:8
566:4
529:2
526:/
521:5
507:φ
363:(
219:)
211:(
199:)
191:(
179:)
171:(
159:)
151:(
139:)
131:(
20:)
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