1856:
1425:
2234:
5352:
2124:
2133:
1017:
2258:
1961:
2021:
1864:
5345:
2085:
3508:
3501:
2297:
3494:
3473:
2099:
2092:
2028:
2426:
2419:
2412:
2405:
2398:
2384:
358:
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3480:
2014:
461:
445:
311:
5545:, a video game where one plays a character who can see beyond the two dimensions other characters can see, and must use this ability to solve platforming puzzles. Features "Dot", a tesseract who helps the player navigate the world and tells how to use abilities, fitting the theme of seeing beyond human perception of known dimensional space.
2447:
2211:
2440:
2433:
2391:
971:
2192:
2283:
1843:
1010:
Two parallel cubes ABCDEFGH and IJKLMNOP separated by a distance of AB can be connected to become a tesseract, with the corners marked as ABCDEFGHIJKLMNOP. However, this parallel positioning of two cubes such that their 8 corresponding pairs of vertices are each separated by a distance of AB can only
1707:
The nondiagonal numbers say how many of the column's element occur in or at the row's element. For example, the 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet
1146:
circumscribed about a regular polytope is the distance from the polytope's center to one of the vertices, and for the tesseract this radius is equal to its edge length; the diameter of the sphere, the length of the diagonal between opposite vertices of the tesseract, is twice the edge length. Only a
704:. Each pair of non-parallel hyperplanes intersects to form 24 square faces. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, a tesseract consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.
1913:. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto six rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.
1415:
854:
51:
5010:
2196:
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2200:
2198:
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4602:
5268:
5218:
2197:
4414:
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4864:
4465:
4116:
1699:
represents the tesseract. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole tesseract. The diagonal reduces to the
4351:
4304:
4790:
4257:
3945:
5553:
has been adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube; see
4062:
1645:
1598:
5122:
1315:
1223:
754:
3978:
1871:
forms the convex hull of the tesseract's vertex-first parallel-projection. The number of vertices in the layers of this projection is 1 4 6 4 1βthe fourth row in
1685:
5428:
4217:
4185:
4153:
1307:
1277:
1250:
1551:
917:
1510:
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2193:
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4488:
6226:
5168:
5145:
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4024:
4001:
3900:
3877:
3854:
3831:
3808:
3785:
1838:{\displaystyle {\begin{bmatrix}{\begin{matrix}16&4&6&4\\2&32&3&3\\4&4&24&2\\8&12&6&8\end{matrix}}\end{bmatrix}}}
1185:
663:
it can be represented by composite SchlΓ€fli symbol { } Γ { } Γ { } Γ { } or { }, with symmetry order 16.
1711:
The bottom row defines they facets, here cubes, have f-vector (8,12,6). The next row left of diagonal is ridge elements (facet of cube), here a square, (4,4).
1891:
envelope. The nearest and farthest cells are projected onto the cube, and the remaining six cells are projected onto the six square faces of the cube.
4872:
4547:
7398:
5490:
Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and science fiction. Notable examples include:
5226:
5176:
1016:
6087:
6025:
4364:
1902:
envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the four remaining cells project to the side faces.
4691:
732:
for 4-dimensional space has two opposite vertices at coordinates and , and other vertices with coordinates at all possible combinations of
4496:
5727:
4821:
4422:
6833:
1852:
It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space.
1004:
Two parallel squares ABCD and EFGH separated by a distance of AB can be connected to become a cube, with the corners marked as ABCDEFGH.
998:
Two parallel line segments AB and CD separated by a distance of AB can be connected to become a square, with the corners marked as ABCD.
647:
made of two parallel cubes, it can be named as a composite SchlΓ€fli symbol {4,3} Γ { }, with symmetry order 96. As a 4-4
4070:
6219:
5955:
5622:
2173:
683:
5463:
5453:
5387:
4312:
4265:
922:
Another commonly convenient tesseract is the
Cartesian product of the closed interval in each axis, with vertices at coordinates
4749:
2873:
2834:
2795:
2756:
2717:
2678:
292:
282:
272:
262:
254:
244:
224:
206:
186:
178:
148:
110:
6117:
5518:
5397:
2903:
2893:
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2825:
2815:
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2786:
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2727:
2708:
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2160:
1100:
1090:
1080:
1070:
234:
216:
196:
168:
158:
140:
130:
120:
464:
The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space.
6427:
6137:
6060:
5999:
5781:
545:
5458:
5392:
2898:
2888:
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2859:
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2839:
2820:
2810:
2800:
2781:
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2722:
2703:
2693:
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1095:
1085:
1075:
287:
277:
267:
249:
239:
229:
211:
201:
191:
173:
163:
153:
135:
125:
115:
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3913:
1859:
Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)
7420:
6212:
5495:
2248:
of the tesseract's vertex-centered central projection. Four of 8 cubic cells are shown. The 16th vertex is projected to
2508:
The tesseract (8-cell) is the third in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).
7415:
4032:
2210:
946:. There are 261 distinct nets of the tesseract. The unfoldings of the tesseract can be counted by mapping the nets to
729:
689:
Each edge of a regular tesseract is of the same length. This is of interest when using tesseracts as the basis for a
2168:
about a plane in 4-dimensional space. The plane bisects the figure from front-left to back-right and top to bottom.
1410:{\displaystyle {\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}.}
849:{\displaystyle {\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}.}
6482:
5554:
5506:'s "The No-Sided Professor", published in 1946, are among the first in science fiction to introduce readers to the
5300:
1608:
1561:
31:
5092:
6037:
5661:
1190:
570:
515:
61:
5522:, a 1954 oil painting by Salvador DalΓ featuring a four-dimensional hypercube unfolded into a three-dimensional
1855:
611:
498:. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square
6466:
6457:
6434:
6105:
6003:
5373:
2262:
1462:
970:
3953:
615:'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as
6826:
5666:
5315:
2375:
1973:
1440:
1424:
1034:
1022:
The 8 cells of the tesseract may be regarded (three different ways) as two interlocked rings of four cubes.
697:: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.
2282:
2233:
5351:
2220:
481:
5502:'s 1940 science fiction story featuring a building in the form of a four-dimensional hypercube. This and
1936:
of the tesseract. This projection is also the one with maximal volume. One set of projection vectors are
1652:
7370:
7363:
7356:
5610:
2534:
2353:
2330:
1925:
640:
579:
5533:, a monument and building near Paris, France, completed in 1989. According to the monument's engineer,
1025:
The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two
751:
Sometimes a unit tesseract is centered at the origin, so that its coordinates are the more symmetrical
5404:
7027:
6974:
6475:
6420:
6028:
5907:
4193:
4161:
4129:
2635:
2526:
1921:
1884:
1872:
1868:
1130:
the tesseract through the actions of the group, by reflecting itself in its own bounding facets (its
584:
2123:
1909:
parallel projection of the tesseract into three-dimensional space has an envelope in the shape of a
7382:
7281:
7031:
6009:
5566:
5296:
2621:
2514:
1924:
envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of
1444:
1428:
1282:
1163:. In particular, the tesseract is the only hypercube (other than a zero-dimensional point) that is
951:
2132:
1255:
1228:
7251:
7201:
7151:
7108:
7078:
7038:
7001:
6819:
6235:
6126:
6075:
5925:
5879:
5871:
5762:
5736:
5499:
5281:
1696:
1111:
694:
433:
2645:
2334:
636:
68:
5005:{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}
2257:
1517:
7390:
6083:
6021:
5951:
5618:
2249:
2035:
859:
741:
656:
652:
491:
341:
324:
5943:
5674:
5079:{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}
4673:{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}
1482:
7394:
6959:
6948:
6937:
6926:
6917:
6908:
6895:
6873:
6861:
6847:
6843:
5915:
5898:
5863:
5790:
5746:
5541:
5366:, with 16 vertices and 8 4-edges, with the 8 4-edges shown here as 4 red and 4 blue squares
2542:
1312:
An axis-aligned tesseract inscribed in a unit-radius 3-sphere has vertices with coordinates
959:
955:
943:
690:
628:
453:
422:
5804:
5758:
6984:
6969:
6155:
6112:
5800:
5754:
5326:
2670:
2326:
1960:
1910:
1432:
507:
499:
426:
418:
414:
410:
331:
102:
6174:
1718:, here tetrahedra, (4,6,4). The next row is vertex figure ridge, here a triangle, (3,3).
5911:
5854:
Fowler, David (2010), "Mathematics in
Science Fiction: Mathematics as Science Fiction",
4473:
659:, it can be named by a composite SchlΓ€fli symbol {4}Γ{4}, with symmetry order 64. As an
7334:
6491:
6450:
6443:
6097:
6065:
Conference Board of the
Mathematical Sciences Regional Conference Series in Mathematics
5503:
5153:
5130:
4798:
4009:
3986:
3885:
3862:
3839:
3816:
3793:
3770:
3715:
2372:
2342:
1602:
1170:
1055:
such triangulations and that the fewest 4-dimensional simplices in any of them is 16.
369:
5482:, order 16. This is the symmetry if the red and blue 4-edges are considered distinct.
2020:
1863:
7409:
7351:
7239:
7232:
7225:
7189:
7182:
7175:
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6856:
5883:
5795:
5655:
5311:
2521:
1980:
1715:
1555:
1156:
1115:
1030:
1026:
745:
675:
667:
589:
395:
381:
351:
5766:
5344:
7291:
5929:
5586:
5534:
5530:
5511:
5507:
3675:
2454:
2323:
2146:
2084:
1929:
1436:
992:
Two points A and B can be connected to become a line, giving a new line segment AB.
561:
without a dimension reference is frequently treated as a synonym for this specific
503:
6199:
2296:
5725:
Pournin, Lionel (2013), "The flip-Graph of the 4-dimensional cube is connected",
7300:
7261:
7211:
7161:
7118:
7088:
7020:
7006:
5523:
5308:
3507:
3500:
2569:
2550:
2301:
2287:
2245:
2241:
1476:
1143:
1119:
1063:
1059:
721:
671:
511:
362:
4922:{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}
4597:{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}
460:
7286:
7270:
7220:
7170:
7127:
7097:
7011:
6193:
includes animated tutorials on several different aspects of the tesseract, by
6185:
6181:
5750:
2583:
2319:
701:
644:
449:
6016:
F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss (1995)
7342:
7256:
7206:
7156:
7113:
7083:
6190:
5263:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}
5213:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}
3742:
2365:
2305:
1148:
982:
975:
660:
488:
484:
17:
5970:
3493:
1045:) that share their vertices with the tesseract. It is known that there are
50:
5867:
5537:, the Grande Arche was designed to resemble the projection of a hypercube.
3472:
2425:
2418:
2411:
2404:
2397:
2383:
2098:
2091:
2027:
7316:
7071:
7067:
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3707:
3700:
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3538:
2629:
2615:
2268:
1701:
939:
648:
562:
469:
5875:
2172:
1920:
parallel projection of the tesseract into three-dimensional space has a
1898:
parallel projection of the tesseract into three-dimensional space has a
682:
with SchlΓ€fli symbol {3,3,4}, with which it can be combined to form the
357:
7325:
7295:
7062:
7057:
7048:
6989:
6278:
6273:
6130:
5292:
4409:{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}
3735:
3728:
3693:
3683:
3633:
3621:
2605:
2577:
2499:
1160:
1152:
1104:) is the most basic direct construction of the tesseract possible. The
1038:
679:
444:
400:
374:
310:
6204:
4736:{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}
3486:
3479:
2013:
7265:
7215:
7165:
7122:
7092:
7043:
6979:
6263:
5688:
4534:{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}
3756:
3749:
3721:
3627:
3615:
2591:
2563:
2494:
2489:
2484:
2479:
2474:
2469:
2459:
1899:
1106:
1042:
318:
300:
6160:
4859:{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}
4460:{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}
6194:
2446:
5944:"Knowledge Visualization and Visual Literacy in Science Education"
5920:
5779:
Cottle, Richard W. (1982), "Minimal triangulation of the 4-cube",
5741:
2439:
2432:
2390:
2190:
2178:
2165:
2159:
1959:
1862:
1854:
969:
459:
443:
1932:, giving a total of eight possible rhombohedra, each a projected
1151:
have this property, including the four-dimensional tesseract and
7015:
6102:
On the
Regular and Semi-Regular Figures in Space of n Dimensions
5948:
5323:
2597:
2464:
1933:
1888:
666:
Since each vertex of a tesseract is adjacent to four edges, the
632:
495:
306:
6403:
6246:
6208:
4111:{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}
30:
This article is about the geometric shape. For other uses, see
6115:(1893). "The projection of fourfold figures on a three-flat".
604:
592:
4346:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}
4299:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}
2232:
4785:{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}
6071:, Providence, Rhode Island: American Mathematical Society
2204:
3D Projection of three tesseracts with and without faces
5449:, order 32. It also has a lower symmetry construction,
5231:
5181:
5097:
5047:
5028:
4947:
4884:
4833:
4761:
4703:
4622:
4559:
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4435:
4376:
4318:
4271:
4231:
4198:
4166:
4134:
4075:
4037:
3919:
2149:
as dash lines, and the tesseract without hidden lines.
1736:
1732:
1386:
1368:
1350:
1331:
1058:
The dissection of the tesseract into instances of its
891:
875:
825:
807:
789:
770:
5844:, second edition, Cambridge University Press, (1991).
5407:
5229:
5179:
5156:
5133:
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4801:
4752:
4694:
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4035:
4012:
3989:
3956:
3916:
3888:
3865:
3842:
3819:
3796:
3773:
2356:
of equal-sized spheres, in any number of dimensions.
1726:
1655:
1611:
1564:
1520:
1485:
1318:
1285:
1258:
1231:
1193:
1173:
862:
856:
This is the
Cartesian product of the closed interval
757:
5896:
Kemp, Martin (1 January 1998), "Dali's dimensions",
5331:
4252:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}
3940:{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}
2228:
2186:
2181:
about two orthogonal planes in 4-dimensional space.
2154:
1887:
of the tesseract into three-dimensional space has a
1964:
Animation showing each individual cube within the B
926:. This tesseract has side length 2 and hypervolume
432:
406:
394:
380:
368:
350:
340:
330:
317:
299:
101:
67:
57:
39:
6200:Tesseract animation with hidden volume elimination
6018:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
5654:
5422:
5262:
5212:
5162:
5139:
5116:
5078:
5004:
4921:
4858:
4807:
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4672:
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4459:
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4018:
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3779:
1837:
1679:
1639:
1592:
1545:
1504:
1409:
1301:
1271:
1244:
1217:
1179:
1126:. The tesseract's characteristic simplex directly
911:
848:
700:A tesseract is bounded by eight three-dimensional
6175:"4D uniform polytopes (polychora) x4o3o3o - tes"
5442:has 16 vertices, and 8 4-edges. Its symmetry is
5430:has a real representation as a tesseract or 4-4
4057:{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}
2349:
2333:consisting of 4 tesseracts around each face has
1011:be achieved in a space of 4 or more dimensions.
720:, and is typically taken as the basic unit for
6148:The Theory of Uniform Polytopes and Honeycombs
6078:, Heidi Burgiel, Chaim Goodman-Strauss (2008)
5640:, pp. 122β123, Β§7.2. illustration Fig 7.2
5301:sequence of regular 4-polytopes and honeycombs
1187:-dimensional hypercube of unit edge length is
1167:. The longest vertex-to-vertex diagonal of an
97:{2,2,2} or { }Γ{ }Γ{ }Γ{ }
6827:
6220:
5950:, Information Science Reference, p. 91,
5316:sequence of regular 4-polytope and honeycombs
2290:3D projection of a tesseract (parallel view)
1640:{\displaystyle d_{\mathrm {3} }={\sqrt {3}}s}
1593:{\displaystyle d_{\mathrm {2} }={\sqrt {2}}s}
1399:
1321:
904:
865:
838:
760:
8:
5615:The Semiregular Polytopes of the Hyperspaces
5117:{\displaystyle {\tfrac {2}{3}}\approx 0.667}
2177:A 3D projection of a tesseract performing a
2164:A 3D projection of a tesseract performing a
2223:and has 4 cubical cells meeting around it.
1928:a rhombic dodecahedron into four congruent
1218:{\displaystyle {\sqrt {n{\vphantom {t}}}},}
506:of the tesseract consists of eight cubical
6834:
6820:
6812:
6416:
6411:
6400:
6259:
6254:
6243:
6227:
6213:
6205:
6008:(3rd ed.). New York: Dover. pp.
5297:uniform 4-polytopes with the same symmetry
2510:
2354:unique regular body-centered cubic lattice
635:folded together around every edge, it has
5919:
5794:
5740:
5414:
5410:
5409:
5406:
5242:
5234:
5230:
5228:
5192:
5184:
5180:
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5132:
5096:
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4751:
4709:
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4693:
4650:
4634:
4621:
4612:
4574:
4558:
4549:
4507:
4498:
4475:
4433:
4424:
4382:
4375:
4366:
4324:
4316:
4314:
4277:
4269:
4267:
4229:
4227:
4197:
4195:
4165:
4163:
4133:
4131:
4091:
4085:
4074:
4072:
4036:
4034:
4011:
3988:
3957:
3955:
3917:
3915:
3887:
3864:
3841:
3818:
3795:
3772:
2230:
2188:
2156:
1968:Coxeter plane projection of the tesseract
1735:
1727:
1725:
1661:
1660:
1654:
1627:
1617:
1616:
1610:
1580:
1570:
1569:
1563:
1537:
1519:
1496:
1484:
1398:
1397:
1385:
1367:
1349:
1330:
1326:
1320:
1319:
1317:
1286:
1284:
1259:
1257:
1232:
1230:
1200:
1199:
1194:
1192:
1172:
903:
902:
890:
874:
870:
864:
863:
861:
837:
836:
824:
806:
788:
769:
765:
759:
758:
756:
3973:{\displaystyle {\sqrt {2}}\approx 1.414}
2370:
2277:
2252:and the four edges to it are not shown.
1971:
1423:
7399:List of regular polytopes and compounds
6090:(Chapter 26. pp. 409: Hemicubes: 1
5828:
5816:
5712:
5637:
5578:
2279:
2141:Orthographic projection Coxeter plane B
6052:Regular and Semi-Regular Polytopes III
5617:. Groningen: University of Groningen.
5587:"The Tesseract - a 4-dimensional cube"
5291:The regular tesseract, along with the
36:
6045:Regular and Semi-Regular Polytopes II
5728:Discrete & Computational Geometry
1714:The upper row is the f-vector of the
27:Four-dimensional analogue of the cube
7:
6034:Regular and Semi Regular Polytopes I
5675:participating institution membership
5314:, {3,3}. The tesseract is also in a
5299:. The tesseract {4,3,3} exists in a
2364:The tesseract is 4th in a series of
2219:. The red corner is the nearest in
1680:{\displaystyle d_{\mathrm {4} }=2s}
1107:characteristic 5-cell of the 4-cube
985:can be imagined the following way:
544:, taken as a unit for hypervolume.
6191:Some Notes on the Fourth Dimension
6184:A way to visualize hypercubes, by
5819:, p. 12, Β§1.8 Configurations.
643:of order 384. Constructed as a 4D
514:. The tesseract is one of the six
25:
6020:, Wiley-Interscience Publication
1468:For a tesseract with side length
684:compound of tesseract and 16-cell
6550:great grand stellated dodecaplex
5514:, and the hypercube (tesseract).
5461:
5456:
5451:
5423:{\displaystyle \mathbb {C} ^{2}}
5395:
5390:
5385:
5350:
5343:
3506:
3499:
3492:
3485:
3478:
3471:
2901:
2896:
2891:
2886:
2881:
2876:
2871:
2862:
2857:
2852:
2847:
2842:
2837:
2832:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2784:
2779:
2774:
2769:
2764:
2759:
2754:
2745:
2740:
2735:
2730:
2725:
2720:
2715:
2706:
2701:
2696:
2691:
2686:
2681:
2676:
2445:
2438:
2431:
2424:
2417:
2410:
2403:
2396:
2389:
2382:
2360:Related polytopes and honeycombs
2295:
2281:
2256:
2209:
2171:
2158:
2131:
2122:
2097:
2090:
2083:
2026:
2019:
2012:
1118:, the group which generates the
1098:
1093:
1088:
1083:
1078:
1073:
1068:
1015:
974:An animation of the shifting in
521:The tesseract is also called an
356:
309:
290:
285:
280:
275:
270:
265:
260:
252:
247:
242:
237:
232:
227:
222:
214:
209:
204:
199:
194:
189:
184:
176:
171:
166:
161:
156:
151:
146:
138:
133:
128:
123:
118:
113:
108:
91:{4,2,2} or {4}Γ{ }Γ{ }
49:
6118:American Journal of Mathematics
5519:Crucifixion (Corpus Hypercubus)
4212:{\displaystyle {\tfrac {1}{2}}}
4180:{\displaystyle {\tfrac {1}{2}}}
4148:{\displaystyle {\tfrac {1}{4}}}
693:to link multiple processors in
6063:(1970), "Twisted Honeycombs",
5971:"Dot (Character) - Giant Bomb"
2267:(Edges are projected onto the
1279:and only for the tesseract is
670:of the tesseract is a regular
1:
5282:sequence of uniform duoprisms
2341:. Hence, the tesseract has a
1302:{\displaystyle {\sqrt {4}}=2}
540:. It is the four-dimensional
5796:10.1016/0012-365X(82)90185-6
5496:And He Built a Crooked House
5280:, the tesseract exists in a
2452:
2380:
2104:
2078:
2046:
2033:
2007:
1978:
1272:{\displaystyle {\sqrt {3}},}
1245:{\displaystyle {\sqrt {2}},}
1201:
1114:of the tesseract's defining
605:
593:
588:. The term derives from the
6050:(Paper 24) H.S.M. Coxeter,
6043:(Paper 23) H.S.M. Coxeter,
6040:46 (1940) 380β407, MR 2,10]
6032:(Paper 22) H.S.M. Coxeter,
2352:makes its tessellation the
2350:radial equilateral symmetry
1138:Radial equilateral symmetry
730:Cartesian coordinate system
7437:
7388:
6815:
6706:grand stellated dodecaplex
6662:great stellated dodecaplex
5693:Unfolding.apperceptual.com
5555:Tesseract (disambiguation)
2515:Regular convex 4-polytopes
1159:, and the two-dimensional
610:
598:
516:convex regular 4-polytopes
32:Tesseract (disambiguation)
29:
6414:
6410:
6399:
6257:
6253:
6242:
6038:Mathematische Zeitschrift
5842:Regular Complex Polytopes
5751:10.1007/s00454-013-9488-y
5662:Oxford English Dictionary
5358:
2549:
2533:
2513:
2217:hidden volume elimination
2157:
1546:{\displaystyle SV=8s^{3}}
571:Oxford English Dictionary
79:{4,3,2} or {4,3}Γ{ }
62:Convex regular 4-polytope
48:
6106:Messenger of Mathematics
6080:The Symmetries of Things
5434:in 4-dimensional space.
5374:regular complex polytope
5295:, exists in a set of 15
2376:orthographic projections
2318:The tesseract, like all
2263:Stereographic projection
1974:Orthographic projections
1225:which for the square is
1155:, the three-dimensional
912:{\displaystyle {\bigl }}
724:in 4-dimensional space.
678:of the tesseract is the
641:hyperoctahedral symmetry
494:and a three-dimensional
5667:Oxford University Press
1514:Surface "volume" (3D):
1505:{\displaystyle H=s^{4}}
6734:great grand dodecaplex
6182:ken perlin's home page
5856:World Literature Today
5424:
5264:
5214:
5164:
5141:
5118:
5080:
5006:
4923:
4860:
4809:
4786:
4737:
4674:
4598:
4535:
4484:
4461:
4410:
4347:
4300:
4253:
4213:
4181:
4149:
4112:
4058:
4020:
3997:
3974:
3941:
3896:
3873:
3850:
3827:
3804:
3781:
2237:
2205:
1969:
1876:
1860:
1839:
1681:
1641:
1594:
1547:
1506:
1465:
1411:
1303:
1273:
1246:
1219:
1181:
1060:characteristic simplex
978:
913:
850:
465:
457:
5868:10.1353/wlt.2010.0188
5689:"Unfolding an 8-cell"
5425:
5265:
5215:
5165:
5142:
5119:
5081:
5007:
4924:
4861:
4810:
4787:
4738:
4675:
4599:
4536:
4485:
4462:
4411:
4348:
4301:
4254:
4214:
4182:
4150:
4113:
4059:
4021:
3998:
3975:
3942:
3897:
3874:
3851:
3828:
3805:
3782:
2331:tesseractic honeycomb
2236:
2203:
1963:
1866:
1858:
1840:
1682:
1642:
1595:
1548:
1507:
1427:
1412:
1304:
1274:
1247:
1220:
1182:
1066:with Coxeter diagram
973:
914:
851:
580:Charles Howard Hinton
487:, analogous to a two-
463:
447:
6578:stellated dodecaplex
6143:, Manuscript (1991)
5942:Ursyn, Anna (2016),
5782:Discrete Mathematics
5591:www.cut-the-knot.org
5405:
5227:
5177:
5154:
5131:
5093:
5024:
4936:
4873:
4822:
4799:
4750:
4692:
4611:
4548:
4497:
4474:
4423:
4365:
4313:
4266:
4226:
4194:
4162:
4130:
4071:
4033:
4010:
3987:
3954:
3914:
3886:
3863:
3840:
3817:
3794:
3771:
1922:rhombic dodecahedral
1869:rhombic dodecahedron
1724:
1697:configuration matrix
1653:
1609:
1562:
1518:
1483:
1316:
1283:
1256:
1229:
1206:
1191:
1171:
1165:radially equilateral
981:The construction of
860:
755:
728:unit tesseract in a
585:A New Era of Thought
557:polytope. The term
7421:Regular 4-polytopes
7383:pentagonal polytope
7282:Uniform 10-polytope
6842:Fundamental convex
6236:Regular 4-polytopes
6173:Klitzing, Richard.
5912:1998Natur.391...27K
5840:Coxeter, H. S. M.,
5665:(Online ed.).
5567:Mathematics and art
2378:
1976:
1447:and finding either
1429:Proof without words
1207:
1202:
1037:into 4-dimensional
7416:Algebraic topology
7252:Uniform 9-polytope
7202:Uniform 8-polytope
7152:Uniform 7-polytope
7109:Uniform 6-polytope
7079:Uniform 5-polytope
7039:Uniform polychoron
7002:Uniform polyhedron
6850:in dimensions 2β10
6430:stellated 120-cell
6361:hecatonicosachoron
5486:In popular culture
5475:{}, with symmetry
5420:
5260:
5252:
5210:
5202:
5160:
5137:
5114:
5106:
5076:
5058:
5039:
5002:
4990:
4919:
4907:
4856:
4844:
4805:
4782:
4770:
4733:
4721:
4670:
4658:
4594:
4582:
4531:
4518:
4483:{\displaystyle 24}
4480:
4457:
4444:
4406:
4394:
4343:
4334:
4296:
4287:
4249:
4240:
4209:
4207:
4177:
4175:
4145:
4143:
4108:
4100:
4054:
4046:
4016:
3993:
3970:
3937:
3928:
3892:
3869:
3846:
3823:
3800:
3777:
3708:irregular hexagons
2371:
2238:
2206:
2106:Dihedral symmetry
1972:
1970:
1877:
1861:
1835:
1829:
1825:
1691:As a configuration
1677:
1649:4-space diagonal:
1637:
1590:
1543:
1502:
1466:
1407:
1395:
1377:
1359:
1340:
1299:
1269:
1242:
1215:
1177:
1112:fundamental region
1033:). It can also be
979:
938:An unfolding of a
909:
900:
884:
846:
834:
816:
798:
779:
695:parallel computing
466:
458:
85:{4,2,4} or {4}Γ{4}
7404:
7403:
7391:Polytope families
6848:uniform polytopes
6810:
6809:
6806:
6805:
6802:
6801:
6797:
6796:
6395:
6394:
6391:
6390:
6386:
6385:
6195:Davide P. Cervone
6141:Uniform Polytopes
6088:978-1-56881-220-5
6026:978-0-471-01003-6
6005:Regular Polytopes
5673:(Subscription or
5643:
5370:
5369:
5274:
5273:
5270:
5251:
5245:
5237:
5220:
5201:
5195:
5187:
5170:
5163:{\displaystyle 2}
5147:
5140:{\displaystyle 1}
5124:
5105:
5086:
5057:
5053:
5038:
5034:
5012:
4989:
4986:
4964:
4929:
4906:
4890:
4866:
4843:
4839:
4815:
4808:{\displaystyle 8}
4792:
4769:
4743:
4720:
4714:
4680:
4657:
4641:
4639:
4604:
4581:
4565:
4541:
4519:
4517:
4490:
4467:
4445:
4443:
4416:
4393:
4387:
4353:
4335:
4333:
4306:
4288:
4286:
4259:
4241:
4239:
4219:
4206:
4187:
4174:
4155:
4142:
4118:
4099:
4096:
4064:
4045:
4026:
4019:{\displaystyle 1}
4003:
3996:{\displaystyle 1}
3980:
3962:
3947:
3929:
3927:
3902:
3895:{\displaystyle 1}
3879:
3872:{\displaystyle 1}
3856:
3849:{\displaystyle 1}
3833:
3826:{\displaystyle 1}
3810:
3803:{\displaystyle 1}
3787:
3780:{\displaystyle 1}
3689:4 rectangles x 4
2911:Mirror dihedrals
2505:
2504:
2311:
2310:
2276:
2275:
2227:
2226:
2215:Perspective with
2201:
2185:
2184:
2116:
2115:
2036:Dihedral symmetry
1873:Pascal's triangle
1632:
1585:
1445:Wagner's theorems
1394:
1376:
1358:
1339:
1291:
1264:
1237:
1210:
1180:{\displaystyle n}
1043:irregular 5-cells
899:
883:
833:
815:
797:
778:
742:Cartesian product
653:Cartesian product
442:
441:
16:(Redirected from
7428:
7395:Regular polytope
6956:
6945:
6934:
6893:
6836:
6829:
6822:
6813:
6786:
6784:
6783:
6780:
6777:
6758:
6756:
6755:
6752:
6749:
6730:
6728:
6727:
6724:
6721:
6702:
6700:
6699:
6696:
6693:
6686:
6684:
6683:
6680:
6677:
6658:
6656:
6655:
6652:
6649:
6634:grand dodecaplex
6630:
6628:
6627:
6624:
6621:
6606:great dodecaplex
6602:
6600:
6599:
6596:
6593:
6574:
6572:
6571:
6568:
6565:
6546:
6544:
6543:
6540:
6537:
6518:
6516:
6515:
6512:
6509:
6417:
6412:
6401:
6346:icositetrachoron
6260:
6255:
6244:
6229:
6222:
6215:
6206:
6178:
6134:
6113:Hall, T. Proctor
6072:
6013:
5986:
5985:
5983:
5981:
5967:
5961:
5960:
5939:
5933:
5932:
5923:
5893:
5887:
5886:
5851:
5845:
5838:
5832:
5826:
5820:
5814:
5808:
5807:
5798:
5776:
5770:
5769:
5744:
5722:
5716:
5710:
5704:
5703:
5701:
5699:
5685:
5679:
5678:
5670:
5658:
5651:
5645:
5641:
5635:
5629:
5628:
5607:
5601:
5600:
5598:
5597:
5583:
5466:
5465:
5464:
5460:
5459:
5455:
5454:
5429:
5427:
5426:
5421:
5419:
5418:
5413:
5400:
5399:
5398:
5394:
5393:
5389:
5388:
5354:
5347:
5332:
5269:
5267:
5266:
5261:
5253:
5247:
5246:
5243:
5238:
5235:
5232:
5223:
5219:
5217:
5216:
5211:
5203:
5197:
5196:
5193:
5188:
5185:
5182:
5173:
5169:
5167:
5166:
5161:
5150:
5146:
5144:
5143:
5138:
5127:
5123:
5121:
5120:
5115:
5107:
5098:
5089:
5085:
5083:
5082:
5077:
5069:
5068:
5063:
5059:
5049:
5048:
5040:
5030:
5029:
5020:
5011:
5009:
5008:
5003:
4995:
4991:
4988:
4987:
4982:
4980:
4979:
4966:
4965:
4960:
4948:
4932:
4928:
4926:
4925:
4920:
4912:
4908:
4905:
4904:
4903:
4886:
4885:
4869:
4865:
4863:
4862:
4857:
4849:
4845:
4835:
4834:
4818:
4814:
4812:
4811:
4806:
4795:
4791:
4789:
4788:
4783:
4775:
4771:
4762:
4746:
4742:
4740:
4739:
4734:
4726:
4722:
4716:
4715:
4710:
4704:
4688:
4679:
4677:
4676:
4671:
4663:
4659:
4656:
4655:
4654:
4640:
4635:
4624:
4623:
4607:
4603:
4601:
4600:
4595:
4587:
4583:
4580:
4579:
4578:
4561:
4560:
4544:
4540:
4538:
4537:
4532:
4524:
4520:
4510:
4508:
4493:
4489:
4487:
4486:
4481:
4470:
4466:
4464:
4463:
4458:
4450:
4446:
4436:
4434:
4419:
4415:
4413:
4412:
4407:
4399:
4395:
4389:
4388:
4383:
4377:
4361:
4352:
4350:
4349:
4344:
4336:
4329:
4328:
4319:
4317:
4309:
4305:
4303:
4302:
4297:
4289:
4282:
4281:
4272:
4270:
4262:
4258:
4256:
4255:
4250:
4242:
4232:
4230:
4222:
4218:
4216:
4215:
4210:
4208:
4199:
4190:
4186:
4184:
4183:
4178:
4176:
4167:
4158:
4154:
4152:
4151:
4146:
4144:
4135:
4126:
4117:
4115:
4114:
4109:
4101:
4098:
4097:
4092:
4090:
4089:
4076:
4067:
4063:
4061:
4060:
4055:
4047:
4038:
4029:
4025:
4023:
4022:
4017:
4006:
4002:
4000:
3999:
3994:
3983:
3979:
3977:
3976:
3971:
3963:
3958:
3950:
3946:
3944:
3943:
3938:
3930:
3920:
3918:
3910:
3901:
3899:
3898:
3893:
3882:
3878:
3876:
3875:
3870:
3859:
3855:
3853:
3852:
3847:
3836:
3832:
3830:
3829:
3824:
3813:
3809:
3807:
3806:
3801:
3790:
3786:
3784:
3783:
3778:
3767:
3658:675 in 120-cell
3655:120 in 120-cell
3604:120 dodecahedra
3558:1200 triangular
3533:600 tetrahedral
3530:120 icosahedral
3510:
3503:
3496:
3489:
3482:
3475:
3463:
3461:
3460:
3457:
3454:
3448:
3446:
3445:
3442:
3439:
3433:
3431:
3430:
3427:
3424:
3418:
3416:
3415:
3412:
3409:
3403:
3401:
3400:
3397:
3394:
3388:
3386:
3385:
3382:
3379:
3371:
3369:
3368:
3365:
3362:
3356:
3354:
3353:
3350:
3347:
3341:
3339:
3338:
3335:
3332:
3326:
3324:
3323:
3320:
3317:
3311:
3309:
3308:
3305:
3302:
3296:
3294:
3293:
3290:
3287:
3279:
3277:
3276:
3273:
3270:
3264:
3262:
3261:
3258:
3255:
3249:
3247:
3246:
3243:
3240:
3234:
3232:
3231:
3228:
3225:
3219:
3217:
3216:
3213:
3210:
3204:
3202:
3201:
3198:
3195:
3187:
3185:
3184:
3181:
3178:
3172:
3170:
3169:
3166:
3163:
3157:
3155:
3154:
3151:
3148:
3142:
3140:
3139:
3136:
3133:
3127:
3125:
3124:
3121:
3118:
3112:
3110:
3109:
3106:
3103:
3095:
3093:
3092:
3089:
3086:
3080:
3078:
3077:
3074:
3071:
3065:
3063:
3062:
3059:
3056:
3050:
3048:
3047:
3044:
3041:
3035:
3033:
3032:
3029:
3026:
3020:
3018:
3017:
3014:
3011:
3003:
3001:
3000:
2997:
2994:
2988:
2986:
2985:
2982:
2979:
2973:
2971:
2970:
2967:
2964:
2958:
2956:
2955:
2952:
2949:
2943:
2941:
2940:
2937:
2934:
2928:
2926:
2925:
2922:
2919:
2906:
2905:
2904:
2900:
2899:
2895:
2894:
2890:
2889:
2885:
2884:
2880:
2879:
2875:
2874:
2867:
2866:
2865:
2861:
2860:
2856:
2855:
2851:
2850:
2846:
2845:
2841:
2840:
2836:
2835:
2828:
2827:
2826:
2822:
2821:
2817:
2816:
2812:
2811:
2807:
2806:
2802:
2801:
2797:
2796:
2789:
2788:
2787:
2783:
2782:
2778:
2777:
2773:
2772:
2768:
2767:
2763:
2762:
2758:
2757:
2750:
2749:
2748:
2744:
2743:
2739:
2738:
2734:
2733:
2729:
2728:
2724:
2723:
2719:
2718:
2711:
2710:
2709:
2705:
2704:
2700:
2699:
2695:
2694:
2690:
2689:
2685:
2684:
2680:
2679:
2511:
2449:
2442:
2435:
2428:
2421:
2414:
2407:
2400:
2393:
2386:
2379:
2348:The tesseract's
2329:. The self-dual
2299:
2285:
2278:
2260:
2229:
2213:
2202:
2187:
2175:
2162:
2155:
2135:
2126:
2101:
2094:
2087:
2030:
2023:
2016:
1977:
1956:
1949:
1942:
1844:
1842:
1841:
1836:
1834:
1833:
1826:
1708:at each vertex.
1686:
1684:
1683:
1678:
1667:
1666:
1665:
1646:
1644:
1643:
1638:
1633:
1628:
1623:
1622:
1621:
1599:
1597:
1596:
1591:
1586:
1581:
1576:
1575:
1574:
1552:
1550:
1549:
1544:
1542:
1541:
1511:
1509:
1508:
1503:
1501:
1500:
1471:
1416:
1414:
1413:
1408:
1403:
1402:
1396:
1387:
1378:
1369:
1360:
1351:
1342:
1341:
1332:
1325:
1324:
1308:
1306:
1305:
1300:
1292:
1287:
1278:
1276:
1275:
1270:
1265:
1260:
1252:for the cube is
1251:
1249:
1248:
1243:
1238:
1233:
1224:
1222:
1221:
1216:
1211:
1209:
1208:
1195:
1186:
1184:
1183:
1178:
1142:The radius of a
1103:
1102:
1101:
1097:
1096:
1092:
1091:
1087:
1086:
1082:
1081:
1077:
1076:
1072:
1071:
1054:
1053:
1050:
1019:
956:perfect matching
954:together with a
929:
925:
924:(Β±1, Β±1, Β±1, Β±1)
918:
916:
915:
910:
908:
907:
901:
892:
886:
885:
876:
869:
868:
855:
853:
852:
847:
842:
841:
835:
826:
817:
808:
799:
790:
781:
780:
771:
764:
763:
739:
735:
719:
716:has side length
691:network topology
629:regular polytope
614:
608:
602:
596:
574:traces the word
556:
542:measure polytope
482:four-dimensional
360:
313:
295:
294:
293:
289:
288:
284:
283:
279:
278:
274:
273:
269:
268:
264:
263:
257:
256:
255:
251:
250:
246:
245:
241:
240:
236:
235:
231:
230:
226:
225:
219:
218:
217:
213:
212:
208:
207:
203:
202:
198:
197:
193:
192:
188:
187:
181:
180:
179:
175:
174:
170:
169:
165:
164:
160:
159:
155:
154:
150:
149:
143:
142:
141:
137:
136:
132:
131:
127:
126:
122:
121:
117:
116:
112:
111:
53:
37:
21:
7436:
7435:
7431:
7430:
7429:
7427:
7426:
7425:
7406:
7405:
7374:
7367:
7360:
7243:
7236:
7229:
7193:
7186:
7179:
7143:
7136:
6970:Regular polygon
6963:
6954:
6947:
6943:
6936:
6932:
6923:
6914:
6907:
6903:
6891:
6885:
6881:
6869:
6851:
6840:
6811:
6798:
6793:
6790:grand tetraplex
6781:
6778:
6775:
6774:
6772:
6765:
6762:great icosaplex
6753:
6750:
6747:
6746:
6744:
6737:
6725:
6722:
6719:
6718:
6716:
6709:
6697:
6694:
6691:
6690:
6688:
6681:
6678:
6675:
6674:
6672:
6665:
6653:
6650:
6647:
6646:
6644:
6637:
6625:
6622:
6619:
6618:
6616:
6609:
6597:
6594:
6591:
6590:
6588:
6581:
6569:
6566:
6563:
6562:
6560:
6553:
6541:
6538:
6535:
6534:
6532:
6525:
6513:
6510:
6507:
6506:
6504:
6493:
6486:
6484:
6477:
6470:
6468:
6461:
6459:
6452:
6445:
6438:
6436:
6429:
6422:
6406:
6387:
6382:
6367:
6352:
6337:
6322:
6307:
6249:
6238:
6233:
6172:
6169:
6156:Victor Schlegel
6111:
6093:
6061:Coxeter, H.S.M.
6059:
6000:Coxeter, H.S.M.
5998:
5995:
5990:
5989:
5979:
5977:
5969:
5968:
5964:
5958:
5941:
5940:
5936:
5895:
5894:
5890:
5853:
5852:
5848:
5839:
5835:
5827:
5823:
5815:
5811:
5778:
5777:
5773:
5724:
5723:
5719:
5711:
5707:
5697:
5695:
5687:
5686:
5682:
5672:
5653:
5652:
5648:
5636:
5632:
5625:
5609:
5608:
5604:
5595:
5593:
5585:
5584:
5580:
5575:
5563:
5500:Robert Heinlein
5488:
5481:
5478:
5474:
5470:
5462:
5457:
5452:
5450:
5448:
5445:
5441:
5437:
5408:
5403:
5402:
5396:
5391:
5386:
5384:
5382:
5378:
5365:
5361:
5233:
5225:
5224:
5183:
5175:
5174:
5152:
5151:
5129:
5128:
5091:
5090:
5042:
5041:
5022:
5021:
4971:
4967:
4949:
4942:
4934:
4933:
4895:
4891:
4879:
4871:
4870:
4828:
4820:
4819:
4797:
4796:
4756:
4748:
4747:
4705:
4698:
4690:
4689:
4646:
4642:
4617:
4609:
4608:
4570:
4566:
4554:
4546:
4545:
4503:
4495:
4494:
4472:
4471:
4429:
4421:
4420:
4378:
4371:
4363:
4362:
4320:
4311:
4310:
4273:
4264:
4263:
4224:
4223:
4192:
4191:
4160:
4159:
4128:
4127:
4081:
4080:
4069:
4068:
4031:
4030:
4008:
4007:
3985:
3984:
3952:
3951:
3912:
3911:
3884:
3883:
3861:
3860:
3838:
3837:
3815:
3814:
3792:
3791:
3769:
3768:
3716:Petrie polygons
3646:10-dodecahedron
3601:600 tetrahedra
3578:1200 triangles
3555:720 pentagonal
3524:16 tetrahedral
3458:
3455:
3452:
3451:
3449:
3443:
3440:
3437:
3436:
3434:
3428:
3425:
3422:
3421:
3419:
3413:
3410:
3407:
3406:
3404:
3398:
3395:
3392:
3391:
3389:
3383:
3380:
3377:
3376:
3374:
3366:
3363:
3360:
3359:
3357:
3351:
3348:
3345:
3344:
3342:
3336:
3333:
3330:
3329:
3327:
3321:
3318:
3315:
3314:
3312:
3306:
3303:
3300:
3299:
3297:
3291:
3288:
3285:
3284:
3282:
3274:
3271:
3268:
3267:
3265:
3259:
3256:
3253:
3252:
3250:
3244:
3241:
3238:
3237:
3235:
3229:
3226:
3223:
3222:
3220:
3214:
3211:
3208:
3207:
3205:
3199:
3196:
3193:
3192:
3190:
3182:
3179:
3176:
3175:
3173:
3167:
3164:
3161:
3160:
3158:
3152:
3149:
3146:
3145:
3143:
3137:
3134:
3131:
3130:
3128:
3122:
3119:
3116:
3115:
3113:
3107:
3104:
3101:
3100:
3098:
3090:
3087:
3084:
3083:
3081:
3075:
3072:
3069:
3068:
3066:
3060:
3057:
3054:
3053:
3051:
3045:
3042:
3039:
3038:
3036:
3030:
3027:
3024:
3023:
3021:
3015:
3012:
3009:
3008:
3006:
2998:
2995:
2992:
2991:
2989:
2983:
2980:
2977:
2976:
2974:
2968:
2965:
2962:
2961:
2959:
2953:
2950:
2947:
2946:
2944:
2938:
2935:
2932:
2931:
2929:
2923:
2920:
2917:
2916:
2914:
2902:
2897:
2892:
2887:
2882:
2877:
2872:
2870:
2863:
2858:
2853:
2848:
2843:
2838:
2833:
2831:
2824:
2819:
2814:
2809:
2804:
2799:
2794:
2792:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2753:
2746:
2741:
2736:
2731:
2726:
2721:
2716:
2714:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2675:
2671:Coxeter mirrors
2646:SchlΓ€fli symbol
2638:
2632:
2624:
2618:
2610:
2608:
2600:
2594:
2586:
2580:
2572:
2566:
2554:
2546:
2538:
2530:
2507:
2362:
2327:Euclidean space
2316:
2300:
2286:
2265:
2261:
2214:
2191:
2179:double rotation
2176:
2166:simple rotation
2163:
2153:
2152:
2151:
2150:
2144:
2138:
2137:
2136:
2128:
2127:
2075:
2071:
2065:
2061:
2057:
2004:
1998:
1994:
1988:
1967:
1951:
1944:
1937:
1911:hexagonal prism
1850:
1828:
1827:
1824:
1823:
1818:
1813:
1808:
1802:
1801:
1796:
1791:
1786:
1780:
1779:
1774:
1769:
1764:
1758:
1757:
1752:
1747:
1742:
1728:
1722:
1721:
1693:
1656:
1651:
1650:
1612:
1607:
1606:
1565:
1560:
1559:
1533:
1516:
1515:
1492:
1481:
1480:
1469:
1460:
1453:
1433:hypercube graph
1422:
1314:
1313:
1281:
1280:
1254:
1253:
1227:
1226:
1189:
1188:
1169:
1168:
1140:
1123:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1067:
1051:
1048:
1046:
968:
936:
927:
923:
858:
857:
753:
752:
737:
733:
717:
710:
637:SchlΓ€fli symbol
625:
555:
549:
530:
427:Hanner polytope
389:
361:
291:
286:
281:
276:
271:
266:
261:
259:
258:
253:
248:
243:
238:
233:
228:
223:
221:
220:
215:
210:
205:
200:
195:
190:
185:
183:
182:
177:
172:
167:
162:
157:
152:
147:
145:
144:
139:
134:
129:
124:
119:
114:
109:
107:
103:Coxeter diagram
96:
92:
90:
86:
84:
80:
78:
74:
69:SchlΓ€fli symbol
43:
41:
35:
28:
23:
22:
15:
12:
11:
5:
7434:
7432:
7424:
7423:
7418:
7408:
7407:
7402:
7401:
7386:
7385:
7376:
7372:
7365:
7358:
7354:
7345:
7328:
7319:
7308:
7307:
7305:
7303:
7298:
7289:
7284:
7278:
7277:
7275:
7273:
7268:
7259:
7254:
7248:
7247:
7245:
7241:
7234:
7227:
7223:
7218:
7209:
7204:
7198:
7197:
7195:
7191:
7184:
7177:
7173:
7168:
7159:
7154:
7148:
7147:
7145:
7141:
7134:
7130:
7125:
7116:
7111:
7105:
7104:
7102:
7100:
7095:
7086:
7081:
7075:
7074:
7065:
7060:
7055:
7046:
7041:
7035:
7034:
7025:
7023:
7018:
7009:
7004:
6998:
6997:
6992:
6987:
6982:
6977:
6972:
6966:
6965:
6961:
6957:
6952:
6941:
6930:
6921:
6912:
6905:
6899:
6889:
6883:
6877:
6871:
6865:
6859:
6853:
6852:
6841:
6839:
6838:
6831:
6824:
6816:
6808:
6807:
6804:
6803:
6800:
6799:
6795:
6794:
6792:
6791:
6788:
6768:
6766:
6764:
6763:
6760:
6740:
6738:
6736:
6735:
6732:
6712:
6710:
6708:
6707:
6704:
6668:
6666:
6664:
6663:
6660:
6640:
6638:
6636:
6635:
6632:
6612:
6610:
6608:
6607:
6604:
6584:
6582:
6580:
6579:
6576:
6556:
6554:
6552:
6551:
6548:
6528:
6526:
6524:
6523:
6520:
6500:
6497:
6496:
6489:
6480:
6473:
6464:
6455:
6448:
6441:
6432:
6425:
6415:
6408:
6407:
6404:
6397:
6396:
6393:
6392:
6389:
6388:
6384:
6383:
6381:
6380:
6377:
6376:hexacosichoron
6374:
6370:
6368:
6366:
6365:
6362:
6359:
6355:
6353:
6351:
6350:
6347:
6344:
6340:
6338:
6336:
6335:
6332:
6331:hexadecachoron
6329:
6325:
6323:
6321:
6320:
6317:
6314:
6310:
6308:
6306:
6305:
6302:
6299:
6295:
6292:
6291:
6286:
6281:
6276:
6271:
6266:
6258:
6251:
6250:
6247:
6240:
6239:
6234:
6232:
6231:
6224:
6217:
6209:
6203:
6202:
6197:
6188:
6179:
6168:
6167:External links
6165:
6164:
6163:
6153:
6152:
6151:
6150:, Ph.D. (1966)
6146:N.W. Johnson:
6138:Norman Johnson
6135:
6109:
6095:
6091:
6076:John H. Conway
6073:
6057:
6056:
6055:
6048:
6041:
6014:
5994:
5991:
5988:
5987:
5962:
5956:
5934:
5888:
5846:
5833:
5831:, p. 293.
5821:
5809:
5771:
5735:(3): 511β530,
5717:
5705:
5680:
5646:
5630:
5623:
5602:
5577:
5576:
5574:
5571:
5570:
5569:
5562:
5559:
5547:
5546:
5538:
5527:
5515:
5504:Martin Gardner
5487:
5484:
5479:
5476:
5472:
5468:
5446:
5443:
5439:
5435:
5417:
5412:
5380:
5376:
5368:
5367:
5363:
5359:
5356:
5355:
5348:
5340:
5339:
5336:
5312:vertex figures
5272:
5271:
5259:
5256:
5250:
5241:
5221:
5209:
5206:
5200:
5191:
5171:
5159:
5148:
5136:
5125:
5113:
5110:
5104:
5101:
5087:
5075:
5072:
5067:
5062:
5056:
5052:
5045:
5037:
5033:
5018:
5014:
5013:
5001:
4998:
4994:
4985:
4978:
4974:
4970:
4963:
4958:
4955:
4952:
4945:
4941:
4930:
4918:
4915:
4911:
4902:
4898:
4894:
4889:
4882:
4878:
4867:
4855:
4852:
4848:
4842:
4838:
4831:
4827:
4816:
4804:
4793:
4781:
4778:
4774:
4768:
4765:
4759:
4755:
4744:
4732:
4729:
4725:
4719:
4713:
4708:
4701:
4697:
4686:
4682:
4681:
4669:
4666:
4662:
4653:
4649:
4645:
4638:
4633:
4630:
4627:
4620:
4616:
4605:
4593:
4590:
4586:
4577:
4573:
4569:
4564:
4557:
4553:
4542:
4530:
4527:
4523:
4516:
4513:
4506:
4502:
4491:
4479:
4468:
4456:
4453:
4449:
4442:
4439:
4432:
4428:
4417:
4405:
4402:
4398:
4392:
4386:
4381:
4374:
4370:
4359:
4355:
4354:
4342:
4339:
4332:
4327:
4323:
4307:
4295:
4292:
4285:
4280:
4276:
4260:
4248:
4245:
4238:
4235:
4220:
4205:
4202:
4188:
4173:
4170:
4156:
4141:
4138:
4124:
4120:
4119:
4107:
4104:
4095:
4088:
4084:
4079:
4065:
4053:
4050:
4044:
4041:
4027:
4015:
4004:
3992:
3981:
3969:
3966:
3961:
3948:
3936:
3933:
3926:
3923:
3908:
3904:
3903:
3891:
3880:
3868:
3857:
3845:
3834:
3822:
3811:
3799:
3788:
3776:
3765:
3761:
3760:
3753:
3746:
3739:
3732:
3725:
3718:
3712:
3711:
3704:
3697:
3690:
3687:
3680:
3678:
3676:Great polygons
3672:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3649:
3648:
3642:
3640:30-tetrahedron
3636:
3630:
3624:
3618:
3612:
3606:
3605:
3602:
3599:
3596:
3593:
3592:16 tetrahedra
3590:
3587:
3583:
3582:
3581:720 pentagons
3579:
3576:
3573:
3570:
3567:
3564:
3560:
3559:
3556:
3553:
3552:96 triangular
3550:
3549:32 triangular
3547:
3544:
3543:10 triangular
3541:
3535:
3534:
3531:
3528:
3525:
3522:
3519:
3518:5 tetrahedral
3516:
3512:
3511:
3504:
3497:
3490:
3483:
3476:
3469:
3465:
3464:
3372:
3280:
3188:
3096:
3004:
2912:
2908:
2907:
2868:
2829:
2790:
2751:
2712:
2673:
2667:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2642:
2641:
2627:
2613:
2603:
2589:
2575:
2561:
2557:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2522:Symmetry group
2518:
2517:
2503:
2502:
2497:
2492:
2487:
2482:
2477:
2472:
2467:
2462:
2457:
2451:
2450:
2443:
2436:
2429:
2422:
2415:
2408:
2401:
2394:
2387:
2373:Petrie polygon
2361:
2358:
2343:dihedral angle
2315:
2312:
2309:
2308:
2292:
2291:
2274:
2273:
2254:
2225:
2224:
2207:
2183:
2182:
2169:
2142:
2140:
2139:
2130:
2129:
2121:
2120:
2119:
2118:
2117:
2114:
2113:
2111:
2109:
2107:
2103:
2102:
2095:
2088:
2081:
2077:
2076:
2073:
2069:
2066:
2063:
2059:
2055:
2052:
2049:
2048:Coxeter plane
2045:
2044:
2042:
2040:
2038:
2032:
2031:
2024:
2017:
2010:
2006:
2005:
2002:
1999:
1996:
1992:
1989:
1986:
1983:
1965:
1849:
1846:
1832:
1822:
1819:
1817:
1814:
1812:
1809:
1807:
1804:
1803:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1781:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1759:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1737:
1734:
1733:
1731:
1704:(16,32,24,8).
1692:
1689:
1688:
1687:
1676:
1673:
1670:
1664:
1659:
1647:
1636:
1631:
1626:
1620:
1615:
1600:
1589:
1584:
1579:
1573:
1568:
1553:
1540:
1536:
1532:
1529:
1526:
1523:
1512:
1499:
1495:
1491:
1488:
1458:
1451:
1421:
1418:
1406:
1401:
1393:
1390:
1384:
1381:
1375:
1372:
1366:
1363:
1357:
1354:
1348:
1345:
1338:
1335:
1329:
1323:
1309:edge lengths.
1298:
1295:
1290:
1268:
1263:
1241:
1236:
1214:
1205:
1198:
1176:
1139:
1136:
1121:
1116:symmetry group
1062:(a particular
1027:demitesseracts
1013:
1012:
1008:4-dimensional:
1005:
1002:3-dimensional:
999:
996:2-dimensional:
993:
990:1-dimensional:
967:
964:
935:
932:
919:in each axis.
906:
898:
895:
889:
882:
879:
873:
867:
845:
840:
832:
829:
823:
820:
814:
811:
805:
802:
796:
793:
787:
784:
777:
774:
768:
762:
748:in each axis.
744:of the closed
714:unit tesseract
709:
706:
624:
621:
553:
548:labels it the
528:
456:of a tesseract
440:
439:
436:
430:
429:
408:
404:
403:
398:
392:
391:
387:
384:
378:
377:
372:
370:Petrie polygon
366:
365:
354:
348:
347:
344:
338:
337:
334:
328:
327:
321:
315:
314:
303:
297:
296:
105:
99:
98:
94:
88:
82:
76:
71:
65:
64:
59:
55:
54:
46:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7433:
7422:
7419:
7417:
7414:
7413:
7411:
7400:
7396:
7392:
7387:
7384:
7380:
7377:
7375:
7368:
7361:
7355:
7353:
7349:
7346:
7344:
7340:
7336:
7332:
7329:
7327:
7323:
7320:
7318:
7314:
7310:
7309:
7306:
7304:
7302:
7299:
7297:
7293:
7290:
7288:
7285:
7283:
7280:
7279:
7276:
7274:
7272:
7269:
7267:
7263:
7260:
7258:
7255:
7253:
7250:
7249:
7246:
7244:
7237:
7230:
7224:
7222:
7219:
7217:
7213:
7210:
7208:
7205:
7203:
7200:
7199:
7196:
7194:
7187:
7180:
7174:
7172:
7169:
7167:
7163:
7160:
7158:
7155:
7153:
7150:
7149:
7146:
7144:
7137:
7131:
7129:
7126:
7124:
7120:
7117:
7115:
7112:
7110:
7107:
7106:
7103:
7101:
7099:
7096:
7094:
7090:
7087:
7085:
7082:
7080:
7077:
7076:
7073:
7069:
7066:
7064:
7061:
7059:
7058:Demitesseract
7056:
7054:
7050:
7047:
7045:
7042:
7040:
7037:
7036:
7033:
7029:
7026:
7024:
7022:
7019:
7017:
7013:
7010:
7008:
7005:
7003:
7000:
6999:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6976:
6973:
6971:
6968:
6967:
6964:
6958:
6955:
6951:
6944:
6940:
6933:
6929:
6924:
6920:
6915:
6911:
6906:
6904:
6902:
6898:
6888:
6884:
6882:
6880:
6876:
6872:
6870:
6868:
6864:
6860:
6858:
6855:
6854:
6849:
6845:
6837:
6832:
6830:
6825:
6823:
6818:
6817:
6814:
6789:
6770:
6769:
6767:
6761:
6742:
6741:
6739:
6733:
6714:
6713:
6711:
6705:
6670:
6669:
6667:
6661:
6642:
6641:
6639:
6633:
6614:
6613:
6611:
6605:
6586:
6585:
6583:
6577:
6558:
6557:
6555:
6549:
6530:
6529:
6527:
6521:
6502:
6501:
6499:
6498:
6495:
6490:
6488:
6481:
6479:
6474:
6472:
6465:
6463:
6456:
6454:
6449:
6447:
6442:
6440:
6433:
6431:
6426:
6424:
6419:
6418:
6413:
6409:
6402:
6398:
6378:
6375:
6372:
6371:
6369:
6363:
6360:
6357:
6356:
6354:
6348:
6345:
6342:
6341:
6339:
6333:
6330:
6327:
6326:
6324:
6318:
6315:
6312:
6311:
6309:
6303:
6300:
6297:
6296:
6294:
6293:
6290:
6287:
6285:
6282:
6280:
6277:
6275:
6272:
6270:
6267:
6265:
6262:
6261:
6256:
6252:
6245:
6241:
6237:
6230:
6225:
6223:
6218:
6216:
6211:
6210:
6207:
6201:
6198:
6196:
6192:
6189:
6187:
6183:
6180:
6176:
6171:
6170:
6166:
6161:
6157:
6154:
6149:
6145:
6144:
6142:
6139:
6136:
6132:
6128:
6124:
6120:
6119:
6114:
6110:
6107:
6103:
6099:
6096:
6089:
6085:
6081:
6077:
6074:
6070:
6066:
6062:
6058:
6053:
6049:
6046:
6042:
6039:
6035:
6031:
6030:
6029:
6027:
6023:
6019:
6015:
6011:
6007:
6006:
6001:
5997:
5996:
5992:
5976:
5972:
5966:
5963:
5959:
5957:9781522504818
5953:
5949:
5945:
5938:
5935:
5931:
5927:
5922:
5921:10.1038/34063
5917:
5913:
5909:
5905:
5901:
5900:
5892:
5889:
5885:
5881:
5877:
5873:
5869:
5865:
5861:
5857:
5850:
5847:
5843:
5837:
5834:
5830:
5825:
5822:
5818:
5813:
5810:
5806:
5802:
5797:
5792:
5788:
5784:
5783:
5775:
5772:
5768:
5764:
5760:
5756:
5752:
5748:
5743:
5738:
5734:
5730:
5729:
5721:
5718:
5715:, p. 18.
5714:
5709:
5706:
5694:
5690:
5684:
5681:
5676:
5668:
5664:
5663:
5657:
5650:
5647:
5639:
5634:
5631:
5626:
5624:1-4181-7968-X
5620:
5616:
5612:
5606:
5603:
5592:
5588:
5582:
5579:
5572:
5568:
5565:
5564:
5560:
5558:
5556:
5552:
5544:
5543:
5539:
5536:
5532:
5528:
5525:
5521:
5520:
5516:
5513:
5509:
5505:
5501:
5497:
5493:
5492:
5491:
5485:
5483:
5433:
5415:
5375:
5357:
5353:
5349:
5346:
5342:
5341:
5337:
5334:
5333:
5330:
5328:
5325:
5321:
5317:
5313:
5310:
5306:
5302:
5298:
5294:
5289:
5287:
5283:
5279:
5276:As a uniform
5257:
5254:
5248:
5239:
5222:
5207:
5204:
5198:
5189:
5172:
5157:
5149:
5134:
5126:
5111:
5108:
5102:
5099:
5088:
5073:
5070:
5065:
5060:
5054:
5050:
5043:
5035:
5031:
5019:
5016:
5015:
4999:
4996:
4992:
4983:
4976:
4972:
4968:
4961:
4956:
4953:
4950:
4943:
4939:
4931:
4916:
4913:
4909:
4900:
4896:
4892:
4887:
4880:
4876:
4868:
4853:
4850:
4846:
4840:
4836:
4829:
4825:
4817:
4802:
4794:
4779:
4776:
4772:
4766:
4763:
4757:
4753:
4745:
4730:
4727:
4723:
4717:
4711:
4706:
4699:
4695:
4687:
4684:
4683:
4667:
4664:
4660:
4651:
4647:
4643:
4636:
4631:
4628:
4625:
4618:
4614:
4606:
4591:
4588:
4584:
4575:
4571:
4567:
4562:
4555:
4551:
4543:
4528:
4525:
4521:
4514:
4511:
4504:
4500:
4492:
4477:
4469:
4454:
4451:
4447:
4440:
4437:
4430:
4426:
4418:
4403:
4400:
4396:
4390:
4384:
4379:
4372:
4368:
4360:
4357:
4356:
4340:
4337:
4330:
4325:
4321:
4308:
4293:
4290:
4283:
4278:
4274:
4261:
4246:
4243:
4236:
4233:
4221:
4203:
4200:
4189:
4171:
4168:
4157:
4139:
4136:
4125:
4123:Short radius
4122:
4121:
4105:
4102:
4093:
4086:
4082:
4077:
4066:
4051:
4048:
4042:
4039:
4028:
4013:
4005:
3990:
3982:
3967:
3964:
3959:
3949:
3934:
3931:
3924:
3921:
3909:
3906:
3905:
3889:
3881:
3866:
3858:
3843:
3835:
3820:
3812:
3797:
3789:
3774:
3766:
3763:
3762:
3758:
3754:
3751:
3747:
3744:
3740:
3737:
3733:
3730:
3726:
3723:
3719:
3717:
3714:
3713:
3709:
3705:
3702:
3698:
3695:
3691:
3688:
3685:
3681:
3679:
3677:
3674:
3673:
3670:10 600-cells
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3650:
3647:
3643:
3641:
3637:
3635:
3631:
3629:
3625:
3623:
3622:8-tetrahedron
3619:
3617:
3616:5-tetrahedron
3613:
3611:
3608:
3607:
3603:
3600:
3598:24 octahedra
3597:
3594:
3591:
3589:5 tetrahedra
3588:
3585:
3584:
3580:
3577:
3575:96 triangles
3574:
3571:
3569:32 triangles
3568:
3566:10 triangles
3565:
3562:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3540:
3537:
3536:
3532:
3529:
3526:
3523:
3521:8 octahedral
3520:
3517:
3514:
3513:
3509:
3505:
3502:
3498:
3495:
3491:
3488:
3484:
3481:
3477:
3474:
3470:
3467:
3466:
3373:
3281:
3189:
3097:
3005:
2913:
2910:
2909:
2869:
2830:
2791:
2752:
2713:
2674:
2672:
2669:
2668:
2664:
2661:
2658:
2655:
2652:
2649:
2647:
2644:
2643:
2640:
2637:
2631:
2628:
2626:
2623:
2617:
2614:
2612:
2607:
2604:
2602:
2599:
2593:
2590:
2588:
2585:
2579:
2576:
2574:
2571:
2565:
2562:
2559:
2558:
2555:
2547:
2541:
2539:
2531:
2525:
2523:
2520:
2519:
2516:
2512:
2509:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2448:
2444:
2441:
2437:
2434:
2430:
2427:
2423:
2420:
2416:
2413:
2409:
2406:
2402:
2399:
2395:
2392:
2388:
2385:
2381:
2377:
2374:
2369:
2367:
2359:
2357:
2355:
2351:
2346:
2344:
2340:
2336:
2332:
2328:
2325:
2321:
2313:
2307:
2303:
2298:
2294:
2293:
2289:
2284:
2280:
2272:
2270:
2264:
2259:
2255:
2253:
2251:
2247:
2243:
2235:
2231:
2222:
2218:
2212:
2208:
2189:
2180:
2174:
2170:
2167:
2161:
2148:
2134:
2125:
2112:
2110:
2108:
2105:
2100:
2096:
2093:
2089:
2086:
2082:
2079:
2067:
2053:
2050:
2047:
2043:
2041:
2039:
2037:
2034:
2029:
2025:
2022:
2018:
2015:
2011:
2008:
2000:
1990:
1984:
1982:
1981:Coxeter plane
1979:
1975:
1962:
1958:
1955:= (1,β1,β1,1)
1954:
1948:= (β1,1,β1,1)
1947:
1941:= (1,1,β1,β1)
1940:
1935:
1931:
1927:
1923:
1919:
1914:
1912:
1908:
1903:
1901:
1897:
1892:
1890:
1886:
1882:
1874:
1870:
1865:
1857:
1853:
1847:
1845:
1830:
1820:
1815:
1810:
1805:
1798:
1793:
1788:
1783:
1776:
1771:
1766:
1761:
1754:
1749:
1744:
1739:
1729:
1719:
1717:
1716:vertex figure
1712:
1709:
1705:
1703:
1698:
1690:
1674:
1671:
1668:
1662:
1657:
1648:
1634:
1629:
1624:
1618:
1613:
1604:
1603:Cell diagonal
1601:
1587:
1582:
1577:
1571:
1566:
1557:
1556:Face diagonal
1554:
1538:
1534:
1530:
1527:
1524:
1521:
1513:
1497:
1493:
1489:
1486:
1478:
1475:
1474:
1473:
1464:
1457:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1419:
1417:
1404:
1391:
1388:
1382:
1379:
1373:
1370:
1364:
1361:
1355:
1352:
1346:
1343:
1336:
1333:
1327:
1310:
1296:
1293:
1288:
1266:
1261:
1239:
1234:
1212:
1203:
1196:
1174:
1166:
1162:
1158:
1157:cuboctahedron
1154:
1150:
1145:
1137:
1135:
1133:
1129:
1125:
1117:
1113:
1109:
1108:
1065:
1061:
1056:
1044:
1040:
1036:
1032:
1028:
1023:
1020:
1018:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
987:
986:
984:
977:
972:
965:
963:
961:
957:
953:
949:
945:
941:
933:
931:
920:
896:
893:
887:
880:
877:
871:
843:
830:
827:
821:
818:
812:
809:
803:
800:
794:
791:
785:
782:
775:
772:
766:
749:
747:
746:unit interval
743:
740:s. It is the
731:
727:
723:
715:
707:
705:
703:
698:
696:
692:
687:
685:
681:
677:
676:dual polytope
673:
669:
668:vertex figure
664:
662:
658:
654:
650:
646:
642:
639:{4,3,3} with
638:
634:
630:
622:
620:
618:
613:
607:
601:
595:
591:
587:
586:
582:'s 1888 book
581:
577:
573:
572:
566:
564:
560:
552:
547:
543:
539:
535:
531:
524:
519:
517:
513:
510:, meeting at
509:
505:
501:
497:
493:
490:
486:
483:
479:
475:
471:
462:
455:
451:
446:
437:
435:
434:Uniform index
431:
428:
424:
420:
416:
412:
409:
405:
402:
399:
397:
393:
385:
383:
382:Coxeter group
379:
376:
373:
371:
367:
364:
359:
355:
353:
352:Vertex figure
349:
345:
343:
339:
335:
333:
329:
326:
322:
320:
316:
312:
308:
304:
302:
298:
106:
104:
100:
72:
70:
66:
63:
60:
56:
52:
47:
38:
33:
19:
7378:
7347:
7338:
7330:
7321:
7312:
7292:10-orthoplex
7052:
7028:Dodecahedron
6949:
6938:
6927:
6918:
6909:
6900:
6896:
6886:
6878:
6874:
6866:
6862:
6268:
6159:
6147:
6140:
6122:
6116:
6108:, Macmillan.
6101:
6079:
6068:
6064:
6051:
6044:
6033:
6017:
6004:
5978:. Retrieved
5974:
5965:
5947:
5937:
5903:
5897:
5891:
5862:(3): 48β52,
5859:
5855:
5849:
5841:
5836:
5829:Coxeter 1973
5824:
5817:Coxeter 1973
5812:
5786:
5780:
5774:
5732:
5726:
5720:
5713:Coxeter 1970
5708:
5696:. Retrieved
5692:
5683:
5660:
5649:
5638:Coxeter 1973
5633:
5614:
5605:
5594:. Retrieved
5590:
5581:
5550:
5548:
5540:
5535:Erik Reitzel
5531:Grande Arche
5517:
5512:Klein bottle
5508:Moebius band
5489:
5371:
5338:Perspective
5319:
5304:
5290:
5285:
5275:
3907:Edge length
3764:Long radius
3667:25 24-cells
3634:6-octahedron
2636:dodecahedron
2633:
2619:
2609:
2595:
2581:
2567:
2506:
2455:Line segment
2363:
2347:
2338:
2317:
2314:Tessellation
2304:3D Disarmed
2302:Stereoscopic
2288:Stereoscopic
2266:
2239:
2216:
2147:hidden lines
1952:
1945:
1938:
1918:vertex-first
1917:
1915:
1906:
1904:
1895:
1893:
1880:
1878:
1851:
1720:
1713:
1710:
1706:
1694:
1467:
1455:
1448:
1441:Kuratowski's
1311:
1164:
1147:few uniform
1141:
1132:mirror walls
1131:
1127:
1105:
1057:
1035:triangulated
1024:
1021:
1014:
1007:
1001:
995:
989:
980:
966:Construction
948:paired trees
947:
942:is called a
937:
921:
750:
725:
713:
711:
699:
688:
665:
626:
616:
603:'four') and
583:
575:
569:
567:
558:
550:
541:
537:
533:
532:, (regular)
526:
522:
520:
512:right angles
504:hypersurface
477:
473:
467:
18:4-4 duoprism
7301:10-demicube
7262:9-orthoplex
7212:8-orthoplex
7162:7-orthoplex
7119:6-orthoplex
7089:5-orthoplex
7044:Pentachoron
7032:Icosahedron
7007:Tetrahedron
6485:icosahedral
6476:great grand
6428:great grand
6421:icosahedral
6334:4-orthoplex
6301:pentachoron
6125:: 179β189.
5656:"tesseract"
5611:Elte, E. L.
5524:Latin cross
5309:tetrahedral
5307:,3,3} with
3661:2 16-cells
3572:24 squares
3527:24 cubical
2622:icosahedron
2570:tetrahedron
2324:tessellates
2246:convex hull
2242:tetrahedron
2145:graph with
1930:rhombohedra
1848:Projections
1477:Hypervolume
1420:Properties
1144:hypersphere
1064:orthoscheme
722:hypervolume
708:Coordinates
702:hyperplanes
672:tetrahedron
631:with three
538:cubic prism
489:dimensional
363:Tetrahedron
7410:Categories
7287:10-simplex
7271:9-demicube
7221:8-demicube
7171:7-demicube
7128:6-demicube
7098:5-demicube
7012:Octahedron
6364:dodecaplex
6186:Ken Perlin
5993:References
5980:21 January
5975:Giant Bomb
5906:(27): 27,
5698:21 January
5677:required.)
5596:2020-11-09
5335:Orthogonal
5017:4-Content
3743:dodecagons
3664:3 8-cells
3652:Inscribed
3546:24 square
2665:{5, 3, 3}
2662:{3, 3, 5}
2659:{3, 4, 3}
2656:{4, 3, 3}
2653:{3, 3, 4}
2650:{3, 3, 3}
2639:600-point
2625:120-point
2584:octahedron
2320:hypercubes
2244:forms the
1926:dissecting
1907:edge-first
1896:face-first
1885:projection
1881:cell-first
1437:non-planar
983:hypercubes
976:dimensions
960:complement
645:hyperprism
534:octachoron
450:DalΓ cross
407:Properties
7335:orthoplex
7257:9-simplex
7207:8-simplex
7157:7-simplex
7114:6-simplex
7084:5-simplex
7053:Tesseract
6522:icosaplex
6469:stellated
6460:stellated
6437:stellated
6379:tetraplex
6316:tesseract
6304:4-simplex
6098:T. Gosset
5884:115769478
5789:: 25β29,
5742:1201.6543
5669:. 199669.
5551:tesseract
5549:The word
5255:≈
5240:×
5205:≈
5190:×
5109:≈
5071:≈
4997:≈
4973:ϕ
4914:≈
4897:ϕ
4851:≈
4777:≈
4728:≈
4665:≈
4648:ϕ
4589:≈
4572:ϕ
4526:≈
4452:≈
4401:≈
4338:≈
4322:ϕ
4291:≈
4275:ϕ
4244:≈
4103:≈
4083:ϕ
4049:≈
4043:ϕ
3965:≈
3932:≈
3515:Vertices
2611:24-point
2601:16-point
2366:hypercube
2339:{4,3,3,4}
2306:Hypercube
1883:parallel
1463:subgraphs
1461:(bottom)
1454:(top) or
1383:±
1365:±
1347:±
1328:±
1149:polytopes
1128:generates
1124:polytopes
1039:simplices
872:−
822:±
804:±
786:±
767:±
661:orthotope
617:tessaract
576:tesseract
559:hypercube
485:hypercube
474:tesseract
423:isohedral
40:Tesseract
7389:Topics:
7352:demicube
7317:polytope
7311:Uniform
7072:600-cell
7068:120-cell
7021:Demicube
6995:Pentagon
6975:Triangle
6494:600-cell
6487:120-cell
6478:120-cell
6471:120-cell
6462:120-cell
6453:120-cell
6446:120-cell
6439:120-cell
6423:120-cell
6349:octaplex
6289:600-cell
6284:120-cell
6162:, Waren.
6002:(1973).
5876:27871086
5767:30946324
5613:(1912).
5561:See also
5432:duoprism
5278:duoprism
3736:octagons
3722:pentagon
3701:decagons
3694:hexagons
3595:8 cubes
2630:120-cell
2616:600-cell
2587:8-point
2573:5-point
2345:of 90Β°.
2335:SchlΓ€fli
2269:3-sphere
2250:infinity
1995:--> A
1900:cuboidal
1702:f-vector
1031:16-cells
940:polytope
649:duoprism
623:Geometry
563:polytope
470:geometry
419:isotoxal
415:isogonal
342:Vertices
44:(4-cube)
7326:simplex
7296:10-cube
7063:24-cell
7049:16-cell
6990:Hexagon
6844:regular
6785:
6773:
6757:
6745:
6729:
6717:
6701:
6689:
6685:
6673:
6657:
6645:
6629:
6617:
6601:
6589:
6573:
6561:
6545:
6533:
6517:
6505:
6373:{3,3,5}
6358:{5,3,3}
6343:{3,4,3}
6328:{3,3,4}
6313:{4,3,3}
6298:{3,3,3}
6279:24-cell
6274:16-cell
6158:(1886)
6131:2369565
6100:(1900)
5930:5317132
5908:Bibcode
5805:0676709
5759:3038527
5322:} with
5318:, {4,3,
5293:16-cell
5288:}Γ{4}.
4685:Volume
3757:30-gons
3750:30-gons
3729:octagon
3684:squares
3462:
3450:
3447:
3435:
3432:
3420:
3417:
3405:
3402:
3390:
3387:
3375:
3370:
3358:
3355:
3343:
3340:
3328:
3325:
3313:
3310:
3298:
3295:
3283:
3278:
3266:
3263:
3251:
3248:
3236:
3233:
3221:
3218:
3206:
3203:
3191:
3186:
3174:
3171:
3159:
3156:
3144:
3141:
3129:
3126:
3114:
3111:
3099:
3094:
3082:
3079:
3067:
3064:
3052:
3049:
3037:
3034:
3022:
3019:
3007:
3002:
2990:
2987:
2975:
2972:
2960:
2957:
2945:
2942:
2930:
2927:
2915:
2606:24-cell
2578:16-cell
2500:10-cube
2337:symbol
1889:cubical
1431:that a
1161:hexagon
1153:24-cell
958:in its
680:16-cell
657:squares
655:of two
600:ΟΞΟΟΞ±ΟΞ±
594:tΓ©ssara
546:Coxeter
401:16-cell
375:octagon
95:0,1,2,3
73:{4,3,3}
7266:9-cube
7216:8-cube
7166:7-cube
7123:6-cube
7093:5-cube
6980:Square
6857:Family
6319:4-cube
6269:8-cell
6264:5-cell
6248:Convex
6129:
6086:
6024:
5954:
5928:
5899:Nature
5882:
5874:
5803:
5765:
5757:
5621:
5510:, the
5000:18.118
4917:16.693
4854:11.314
4668:90.366
4592:198.48
4529:41.569
4455:27.713
4404:10.825
3628:4-cube
3586:Cells
3563:Faces
3468:Graph
2634:Hyper-
2620:Hyper-
2596:Hyper-
2592:8-cell
2582:Hyper-
2568:Hyper-
2564:5-cell
2495:9-cube
2490:8-cube
2485:7-cube
2480:6-cube
2475:5-cube
2470:4-cube
2460:Square
2080:Graph
2051:Other
2009:Graph
1479:(4D):
1439:using
928:2 = 16
736:s and
674:. The
523:8-cell
502:, the
492:square
478:4-cube
411:convex
42:8-cell
6985:p-gon
6771:{3,3,
6659:,3,5}
6615:{5,3,
6575:,5,3}
6547:,3,3}
6503:{3,5,
6492:grand
6483:great
6467:grand
6458:great
6451:grand
6444:great
6435:small
6127:JSTOR
6012:β123.
5926:S2CID
5880:S2CID
5872:JSTOR
5763:S2CID
5737:arXiv
5671:
5573:Notes
5467:, or
5401:, in
5327:cells
5324:cubic
5258:4.193
5236:Short
5208:3.863
5186:Short
5112:0.667
5074:0.146
4780:5.333
4731:2.329
4358:Area
4341:0.926
4294:0.926
4247:0.707
4106:0.270
4052:0.618
3968:1.414
3935:1.581
3539:Edges
2560:Name
1695:This
1110:is a
633:cubes
627:As a
612:αΌΞΊΟΞ―Ο
606:aktΓs
590:Greek
536:, or
508:cells
500:faces
480:is a
332:Edges
319:Faces
307:{4,3}
301:Cells
89:0,2,3
7343:cube
7016:Cube
6846:and
6405:Star
6084:ISBN
6022:ISBN
5982:2018
5952:ISBN
5700:2018
5619:ISBN
5529:The
5372:The
4552:1200
3759:x 4
3752:x 6
3745:x 4
3738:x 4
3731:x 3
3724:x 2
3710:x 4
3706:100
3703:x 6
3696:x 4
3686:x 3
3610:Tori
2598:cube
2465:Cube
2240:The
1934:cube
1916:The
1905:The
1894:The
1879:The
1867:The
952:tree
651:, a
568:The
496:cube
472:, a
452:, a
448:The
396:Dual
58:Type
6892:(p)
6759:,5}
6743:{3,
6731:,3}
6715:{5,
6687:,5,
6603:,5}
6587:{5,
6010:122
5916:doi
5904:391
5864:doi
5791:doi
5747:doi
5542:Fez
5498:",
5471:{}Γ
5438:{4}
5379:{4}
5362:{4}
5303:, {
5284:: {
5244:Vol
5194:Vol
4940:120
4877:600
4615:720
3755:20
3699:12
3644:12
3638:20
2072:/ D
2062:/ A
2058:/ D
1459:3,3
1443:or
1435:is
1134:).
1052:256
1049:487
962:).
950:(a
944:net
934:Net
726:The
578:to
476:or
468:In
454:net
325:{4}
323:24
83:0,2
77:0,3
7412::
7397:β’
7393:β’
7373:21
7369:β’
7366:k1
7362:β’
7359:k2
7337:β’
7294:β’
7264:β’
7242:21
7238:β’
7235:41
7231:β’
7228:42
7214:β’
7192:21
7188:β’
7185:31
7181:β’
7178:32
7164:β’
7142:21
7138:β’
7135:22
7121:β’
7091:β’
7070:β’
7051:β’
7030:β’
7014:β’
6946:/
6935:/
6925:/
6916:/
6894:/
6123:15
6121:.
6104:,
6092:n1
6082:,
6067:,
6054:,
6047:,
6036:,
5973:.
5946:,
5924:,
5914:,
5902:,
5878:,
5870:,
5860:84
5858:,
5801:MR
5799:,
5787:40
5785:,
5761:,
5755:MR
5753:,
5745:,
5733:49
5731:,
5691:.
5659:.
5589:.
5557:.
5383:,
5329:.
5036:24
4951:15
4893:12
4826:24
4754:16
4718:24
4632:10
4626:25
4515:16
4501:96
4478:24
4427:32
4369:10
3748:4
3741:2
3734:2
3727:1
3720:1
3692:4
3682:2
3632:4
3626:2
3620:2
3614:1
3453:π
3438:π
3423:π
3408:π
3393:π
3378:π
3361:π
3346:π
3331:π
3316:π
3301:π
3286:π
3269:π
3254:π
3239:π
3224:π
3209:π
3194:π
3177:π
3162:π
3147:π
3132:π
3117:π
3102:π
3085:π
3070:π
3055:π
3040:π
3025:π
3010:π
2993:π
2978:π
2963:π
2948:π
2933:π
2918:π
2368::
2322:,
2271:)
2221:4D
1957:.
1950:,
1943:,
1811:12
1794:24
1767:32
1740:16
1605::
1558::
1472::
1047:92
930:.
712:A
686:.
619:.
565:.
525:,
518:.
438:10
425:,
421:,
417:,
413:,
390:,
346:16
336:32
305:8
7381:-
7379:n
7371:k
7364:2
7357:1
7350:-
7348:n
7341:-
7339:n
7333:-
7331:n
7324:-
7322:n
7315:-
7313:n
7240:4
7233:2
7226:1
7190:3
7183:2
7176:1
7140:2
7133:1
6962:n
6960:H
6953:2
6950:G
6942:4
6939:F
6931:8
6928:E
6922:7
6919:E
6913:6
6910:E
6901:n
6897:D
6890:2
6887:I
6879:n
6875:B
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