608:
44:
643:
629:
636:
4211:
722:
704:
4521:
615:
749:
731:
4191:
4495:
622:
4549:
740:
713:
4178:
4454:
4202:
4480:
653:
577:
31:
584:
563:
556:
570:
4254:
4847:
4838:
4816:
4807:
4798:
4776:
4767:
4758:
4732:
4240:
4231:
4222:
4706:
4680:
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4641:
4628:
4606:
4584:
4575:
5135:
591:
3810:
4372:
Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, and so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral
1767:
1583:
1142:
2409:
2076:
4160:
between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three-dimensional convex polyhedron is combinatorially equivalent to one of the two opposite faces of a four-dimensional antiprism, constructed from its
3557:
4378:
Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, and so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be
1343:
883:
1226:
3336:
3918:
3479:
1594:
1004:
233:
observed the existence of the infinite family of antiprisms. This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the
981:
1447:
1436:
2087:
1778:
2636:
3805:{\displaystyle V=n\int _{0}^{h}dz={\frac {nh}{3}}R(0)^{2}\sin {\frac {\pi }{n}}(1+2\cos {\frac {\pi }{n}})={\frac {nh}{12}}l^{2}{\frac {1+2\cos {\frac {\pi }{n}}}{\sin {\frac {\pi }{n}}}}.}
1249:
5196:
2799:
264:
used to cancel the effects of a primary optimal element, the first use of "antiprism" in
English in its geometric sense appears to be in the early 20th century in the works of
3181:
3022:
2872:
2671:
1149:
3086:
2936:
3122:
3838:
2963:
2694:
3549:
3042:
2892:
2819:
2734:
2714:
2472:
2452:
2432:
1370:
5066:
Dobbins, Michael Gene (2017). "Antiprismlessness, or: reducing combinatorial equivalence to projective equivalence in realizability problems for polytopes".
910:
791:
492:
4368:
In the retrograde forms, but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:
3189:
5189:
3347:
1772:
Via linear interpolation, points on the outer triangular edges of the antiprism that connect vertices at the bottom with vertices at the top are at
1762:{\displaystyle \left({\begin{array}{c}R(0)\cos {\frac {2\pi (m+1/2)}{n}}\\R(0)\sin {\frac {2\pi (m+1/2)}{n}}\\h\end{array}}\right),\quad m=0..n-1.}
4165:
and its polar dual. However, there exist four-dimensional polychora that cannot be combined with their duals to form five-dimensional antiprisms.
5182:
5068:
65:
1578:{\displaystyle \left({\begin{array}{c}R(0)\cos {\frac {2\pi m}{n}}\\R(0)\sin {\frac {2\pi m}{n}}\\0\end{array}}\right),\quad m=0..n-1;}
5116:
4426:
87:
1137:{\displaystyle V={\frac {n{\sqrt {4\cos ^{2}{\frac {\pi }{2n}}-1}}\sin {\frac {3\pi }{2n}}}{12\sin ^{2}{\frac {\pi }{n}}}}~a^{3},}
2404:{\displaystyle \left({\begin{array}{c}{\frac {R(0)}{h}}\\{\frac {R(0)}{h}}\\\\z\end{array}}\right),\quad 0\leq z\leq h,m=0..n-1.}
2071:{\displaystyle \left({\begin{array}{c}{\frac {R(0)}{h}}\\{\frac {R(0)}{h}}\\\\z\end{array}}\right),\quad 0\leq z\leq h,m=0..n-1}
1378:
5033:
485:
249:
The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to
4527:
5603:
4647:
2480:
4501:
4262:
4182:
391:
5395:
5336:
4931:; Sternath, Maria Luise (July 2008). "New light on the rediscovery of the Archimedean solids during the Renaissance".
4555:
607:
58:
52:
4712:
4128:-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform
363:
bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a
5598:
5425:
5385:
478:
155:
5169:
4686:
4673:
5420:
5415:
182:, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are
69:
4975:
Smyth, Piazzi (1881). "XVII. On the
Constitution of the Lines forming the Low-Temperature Spectrum of Oxygen".
4460:
2874:. (These are derived from the length of the difference of the previous two vectors.) It can be dissected into
5526:
5521:
5400:
5306:
5390:
5331:
5321:
5266:
4486:
5139:
5410:
5326:
5281:
4738:
4634:
4320:
764:
544:
400:
324:
2739:
250:
5370:
5296:
5244:
4162:
4046:
4033:
3127:
2968:
2454:
coordinates in one of the previous two vectors, the squared circumradius of this section at altitude
660:
536:
526:
420:
430:
Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For
5536:
5405:
5380:
5365:
5301:
5249:
2824:
2644:
1338:{\displaystyle V={\frac {nhl^{2}}{12}}\left(\csc {\frac {\pi }{n}}+2\cot {\frac {\pi }{n}}\right).}
531:
239:
128:
4520:
4210:
721:
703:
642:
628:
5551:
5516:
5375:
5270:
5219:
4940:
4351:
4257:
This shows all the non-star and star antiprisms up to 15 sides, together with those of a 29-agon.
995:
465:
375:
243:
5029:
5002:
Coxeter, H. S. M. (January 1928). "The pure
Archimedean polytopes in six and seven dimensions".
4401:
have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.
748:
730:
635:
4494:
4190:
3047:
2897:
614:
5531:
5341:
5316:
5260:
5147:
5112:
4914:
4902:
4548:
3091:
4959:
4283:
and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting
5470:
5077:
5011:
4984:
4906:
3824:
739:
712:
693:
621:
521:
265:
225:
179:
5089:
5052:
2941:
255:
5085:
5048:
4898:
4871:
4453:
4329:
4326:
4177:
4157:
403:
282:
230:
200:
172:
131:
5164:
2676:
289:
bases, one usually considers the case where these two copies are twisted by an angle of
5291:
5214:
4928:
4479:
4201:
4153:
4036:
3929:
3487:
3027:
2877:
2804:
2719:
2699:
2457:
2437:
2417:
1355:
261:
235:
878:{\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}
5592:
5496:
5352:
5286:
4284:
1221:{\displaystyle A={\frac {n}{2}}\left(\cot {\frac {\pi }{n}}+{\sqrt {3}}\right)a^{2}.}
317:
212:
193:
576:
4877:
4337:
4268:
583:
30:
3331:{\displaystyle Q_{1}(z)={\frac {R(0)^{2}}{h^{2}}}(h-z)\left\sin {\frac {\pi }{n}}}
652:
562:
17:
5229:
4365:
star antiprism (by translating and/or twisting one of its bases, if necessary).
3974:
555:
442:
360:
5150:
3913:{\displaystyle V_{\mathrm {prism} }={\frac {nhl^{2}}{4}}\cot {\frac {\pi }{n}}}
3474:{\displaystyle Q_{2}(z)={\frac {R(0)^{2}}{h^{2}}}z\left\sin {\frac {\pi }{n}}.}
569:
5561:
5449:
5239:
5206:
5081:
5015:
4988:
4007:
514:
507:
453:
168:
124:
5556:
5546:
5491:
5311:
5155:
4846:
4837:
4815:
4806:
4797:
4775:
4766:
4757:
4731:
4705:
4679:
4666:
4640:
4627:
4605:
4583:
4574:
4239:
4230:
4221:
5134:
785:
isosceles triangle side faces, circumradius of the bases equal to 1) are:
353:
degrees, more regularity is obtained if the bases have the same axis: are
5442:
151:
101:
4944:
590:
5566:
5541:
4253:
355:
141:
976:{\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}
907:-antiprism is uniform (i.e. if the triangles are equilateral), then:
5174:
441:(degenerate antiprism), which is visually identical to the regular
4172:
29:
3939:-antiprism (i.e. with regular bases and isosceles side faces) is
3183:). According to Heron's formula the areas of these triangles are
260:. Although the English "anti-prism" had been used earlier for an
5234:
5178:
5004:
Mathematical
Proceedings of the Cambridge Philosophical Society
4880:, a three-dimensional polygon whose convex hull is an antiprism
4075:: the regular tetrahedron, which has the larger rotation group
1352:
The circumradius of the horizontal circumcircle of the regular
4101:: the regular octahedron, which has the larger rotation group
37:
1431:{\displaystyle R(0)={\frac {l}{2\sin {\frac {\pi }{n}}}}.}
4152:
Four-dimensional antiprisms can be defined as having two
5123:
Chapter 2: Archimedean polyhedra, prisms and antiprisms
5111:. California: University of California Press Berkeley.
2096:
1787:
1603:
1456:
320:
to the polygon plane and lying in the polygon centre.
3841:
3560:
3490:
3350:
3192:
3130:
3094:
3050:
3030:
2971:
2944:
2900:
2880:
2827:
2807:
2742:
2722:
2702:
2679:
2647:
2483:
2460:
2440:
2420:
2090:
1781:
1597:
1450:
1381:
1358:
1252:
1152:
1007:
913:
794:
27:
Polyhedron with parallel bases connected by triangles
5509:
5484:
5459:
5434:
5350:
5258:
5213:
5034:"Are prisms and antiprisms really boring? (Part 3)"
2631:{\displaystyle R(z)^{2}={\frac {R(0)^{2}}{h^{2}}}.}
3912:
3804:
3543:
3473:
3330:
3175:
3116:
3080:
3036:
3016:
2957:
2930:
2886:
2866:
2813:
2793:
2728:
2708:
2688:
2665:
2630:
2466:
2446:
2426:
2403:
2070:
1761:
1577:
1430:
1364:
1337:
1220:
1136:
975:
877:
4383:Also, star antiprism compounds with regular star
696:of these semiregular antiprisms are as follows:
4977:Transactions of the Royal Society of Edinburgh
5190:
4373:triangle: it is a degenerate star polyhedron.
486:
8:
3920:which is smaller than that of an antiprism.
4361:convex or star polygon bases can be made a
4287:, and are denoted by "inverted" fractions:
4267:Uniform star antiprisms are named by their
2414:By building the sums of the squares of the
5463:
5197:
5183:
5175:
4403:
493:
479:
462:
4958:Heinze, Karl (1886). Lucke, Franz (ed.).
4156:as parallel opposite faces, so that each
3900:
3882:
3869:
3847:
3846:
3840:
3786:
3768:
3750:
3744:
3725:
3709:
3681:
3669:
3641:
3614:
3592:
3579:
3574:
3559:
3523:
3501:
3489:
3458:
3425:
3400:
3389:
3373:
3355:
3349:
3318:
3291:
3242:
3231:
3215:
3197:
3191:
3165:
3150:
3129:
3099:
3093:
3049:
3029:
3006:
2991:
2970:
2949:
2943:
2899:
2879:
2856:
2832:
2826:
2806:
2780:
2747:
2741:
2721:
2701:
2678:
2646:
2612:
2576:
2548:
2533:
2522:
2506:
2497:
2482:
2459:
2439:
2419:
2310:
2268:
2223:
2186:
2144:
2099:
2095:
2089:
1977:
1947:
1902:
1865:
1835:
1790:
1786:
1780:
1705:
1684:
1645:
1624:
1602:
1596:
1517:
1477:
1455:
1449:
1412:
1397:
1380:
1357:
1317:
1295:
1272:
1259:
1251:
1209:
1193:
1180:
1159:
1151:
1125:
1105:
1096:
1067:
1038:
1029:
1020:
1014:
1006:
955:
936:
921:
912:
861:
830:
806:
793:
88:Learn how and when to remove this message
4252:
698:
51:This article includes a list of general
4964:(in German). B. G. Teubner. p. 14.
4890:
476:
4010:, which has the larger symmetry group
3977:, which has the larger symmetry group
1001:-gonal antiprism; then the volume is:
144:, connected by an alternating band of
5170:Paper models of prisms and antiprisms
5069:Discrete & Computational Geometry
4933:Archive for History of Exact Sciences
1230:Furthermore, the volume of a regular
7:
4323:
3933:
1231:
768:
4124:-antiprisms have congruent regular
2641:The horizontal section at altitude
333:-gon bases, twisted by an angle of
3860:
3857:
3854:
3851:
3848:
57:it lacks sufficient corresponding
25:
4393:-gon bases can be constructed if
2794:{\displaystyle l_{1}(z)=l(1-z/h)}
316:of a regular polygon is the line
171:, and are a (degenerate) type of
134:copies (not mirror images) of an
5133:
4845:
4836:
4814:
4805:
4796:
4774:
4765:
4756:
4730:
4704:
4678:
4665:
4639:
4626:
4604:
4582:
4573:
4547:
4519:
4493:
4478:
4452:
4238:
4229:
4220:
4209:
4200:
4189:
4176:
3819:Note that the volume of a right
1441:The vertices at the base are at
747:
738:
729:
720:
711:
702:
651:
641:
634:
627:
620:
613:
606:
589:
582:
575:
568:
561:
554:
42:
5165:Nonconvex Prisms and Antiprisms
4312:; example: 5/3 instead of 5/2.
3176:{\displaystyle R(z)+l_{2}(z)/2}
3017:{\displaystyle R(z)+l_{1}(z)/2}
2364:
2031:
1740:
1588:the vertices at the top are at
1553:
4713:Enneagrammic crossed-antiprism
4528:Pentagrammic crossed-antiprism
4083:, which has three versions of
3988:, which has three versions of
3719:
3691:
3666:
3659:
3629:
3626:
3620:
3604:
3598:
3585:
3538:
3535:
3529:
3513:
3507:
3494:
3386:
3379:
3367:
3361:
3282:
3270:
3262:
3250:
3228:
3221:
3209:
3203:
3162:
3156:
3140:
3134:
3111:
3105:
3075:
3069:
3060:
3054:
3003:
2997:
2981:
2975:
2925:
2919:
2910:
2904:
2844:
2838:
2788:
2768:
2759:
2753:
2622:
2603:
2591:
2541:
2519:
2512:
2494:
2487:
2340:
2331:
2316:
2289:
2277:
2259:
2247:
2244:
2235:
2229:
2216:
2207:
2192:
2165:
2153:
2135:
2123:
2120:
2111:
2105:
2007:
1998:
1983:
1938:
1926:
1923:
1914:
1908:
1895:
1886:
1871:
1826:
1814:
1811:
1802:
1796:
1713:
1693:
1675:
1669:
1653:
1633:
1615:
1609:
1508:
1502:
1468:
1462:
1391:
1385:
1238:with side length of its bases
858:
848:
774:-antiprism (i.e. with regular
154:. They are represented by the
1:
4874:, a four-dimensional polytope
4648:Octagrammic crossed-antiprism
4415:-antiprisms by symmetry, for
4109:, which has four versions of
4021:, which has four versions of
3044:isoceless triangles of edges
2894:isoceless triangles of edges
2867:{\displaystyle l_{2}(z)=lz/h}
2666:{\displaystyle 0\leq z\leq h}
167:Antiprisms are a subclass of
5577:Degenerate polyhedra are in
5109:Polyhedra: A visual approach
4917:, of a heptagonal antiprism.
4854:
4823:
4783:
4743:
4717:
4691:
4687:Enneagrammic antiprism (9/4)
4674:Enneagrammic antiprism (9/2)
4652:
4613:
4591:
4560:
4532:
4506:
4502:crossed pentagonal antiprism
4465:
4437:
4263:Prismatic uniform polyhedron
4218:
4174:
4032:The symmetry group contains
700:
460:(non-degenerate antiprism).
5396:pentagonal icositetrahedron
5337:truncated icosidodecahedron
4556:crossed hexagonal antiprism
3484:The area of the section is
5620:
5426:pentagonal hexecontahedron
5386:deltoidal icositetrahedron
4260:
4065:, except in the cases of:
3963:, except in the cases of:
178:Antiprisms are similar to
5575:
5466:
5421:disdyakis triacontahedron
5416:deltoidal hexecontahedron
5082:10.1007/s00454-017-9874-y
5016:10.1017/s0305004100011786
4989:10.1017/s0080456800029112
4544:
4490:
4449:
4430:
3081:{\displaystyle R(z),R(z)}
2931:{\displaystyle R(z),R(z)}
1146:and the surface area is:
427:triangles as side faces.
413:-gons as base faces, and
4461:Crossed square antiprism
4357:Any star antiprism with
4344:polygon base faces, and
3117:{\displaystyle l_{2}(z)}
994:be the edge-length of a
5527:gyroelongated bipyramid
5401:rhombic triacontahedron
5307:truncated cuboctahedron
4961:Genetische Stereometrie
4911:(in Latin). p. 49.
4903:"Book II, Definition X"
4169:Self-crossing polyhedra
986:Volume and surface area
603:Spherical tiling image
242:has been attributed to
207:-gonal antiprism is an
189:triangles, rather than
72:more precise citations.
5522:truncated trapezohedra
5391:disdyakis dodecahedron
5357:(duals of Archimedean)
5332:rhombicosidodecahedron
5322:truncated dodecahedron
4487:Pentagrammic antiprism
4258:
4158:three-dimensional face
3914:
3806:
3545:
3475:
3332:
3177:
3118:
3082:
3038:
3018:
2959:
2932:
2888:
2868:
2815:
2795:
2730:
2710:
2690:
2667:
2632:
2468:
2448:
2428:
2405:
2072:
1763:
1579:
1432:
1366:
1339:
1222:
1138:
977:
879:
767:for the vertices of a
323:For an antiprism with
281:For an antiprism with
35:
5411:pentakis dodecahedron
5327:truncated icosahedron
5282:truncated tetrahedron
5107:Anthony Pugh (1976).
4739:Decagrammic antiprism
4635:Octagrammic antiprism
4261:Further information:
4256:
3915:
3807:
3546:
3476:
3333:
3178:
3119:
3083:
3039:
3019:
2960:
2958:{\displaystyle l_{1}}
2933:
2889:
2869:
2816:
2796:
2731:
2711:
2691:
2668:
2633:
2469:
2449:
2429:
2406:
2073:
1764:
1580:
1433:
1367:
1340:
1223:
1139:
978:
880:
765:Cartesian coordinates
760:Cartesian coordinates
545:Apeirogonal antiprism
33:
5604:Prismatoid polyhedra
5371:rhombic dodecahedron
5297:truncated octahedron
5142:at Wikimedia Commons
4317:right star antiprism
4163:canonical polyhedron
4148:In higher dimensions
3839:
3558:
3551:, and the volume is
3488:
3348:
3190:
3128:
3092:
3048:
3028:
2969:
2942:
2898:
2878:
2825:
2805:
2740:
2720:
2700:
2677:
2673:above the base is a
2645:
2481:
2458:
2438:
2418:
2088:
1779:
1595:
1448:
1379:
1372:-gon at the base is
1356:
1250:
1150:
1005:
911:
792:
537:Heptagonal antiprism
527:Pentagonal antiprism
515:Triangular antiprism
458:triangular antiprism
246:, who died in 1556.
5406:triakis icosahedron
5381:tetrakis hexahedron
5366:triakis tetrahedron
5302:rhombicuboctahedron
4422:
3584:
648:Plane tiling image
532:Hexagonal antiprism
500:
240:hexagonal antiprism
34:Octagonal antiprism
5376:triakis octahedron
5261:Archimedean solids
5148:Weisstein, Eric W.
4927:Schreiber, Peter;
4404:
4352:isosceles triangle
4259:
3910:
3802:
3570:
3541:
3471:
3328:
3173:
3114:
3078:
3034:
3014:
2955:
2928:
2884:
2864:
2811:
2791:
2726:
2706:
2689:{\displaystyle 2n}
2686:
2663:
2628:
2464:
2444:
2424:
2401:
2355:
2068:
2022:
1759:
1731:
1575:
1544:
1428:
1362:
1335:
1218:
1134:
973:
875:
463:
244:Hieronymus Andreae
36:
5599:Uniform polyhedra
5586:
5585:
5505:
5504:
5342:snub dodecahedron
5317:icosidodecahedron
5138:Media related to
4863:
4862:
4248:
4247:
3908:
3892:
3797:
3794:
3776:
3738:
3717:
3689:
3654:
3544:{\displaystyle n}
3466:
3433:
3406:
3326:
3299:
3248:
3037:{\displaystyle n}
2887:{\displaystyle n}
2814:{\displaystyle n}
2801:alternating with
2729:{\displaystyle n}
2709:{\displaystyle n}
2620:
2539:
2467:{\displaystyle z}
2447:{\displaystyle y}
2427:{\displaystyle x}
2338:
2296:
2242:
2214:
2172:
2118:
2005:
1963:
1921:
1893:
1851:
1809:
1720:
1660:
1533:
1493:
1423:
1420:
1365:{\displaystyle n}
1325:
1303:
1282:
1198:
1188:
1167:
1120:
1116:
1113:
1085:
1059:
1051:
968:
944:
843:
819:
757:
756:
694:Schlegel diagrams
690:
689:
551:Polyhedron image
508:Digonal antiprism
439:digonal antiprism
385:Uniform antiprism
251:Theodor Wittstein
223:In his 1619 book
98:
97:
90:
18:Crossed antiprism
16:(Redirected from
5611:
5464:
5460:Dihedral uniform
5435:Dihedral regular
5358:
5274:
5223:
5199:
5192:
5185:
5176:
5161:
5160:
5137:
5122:
5094:
5093:
5063:
5057:
5056:
5038:
5030:Grünbaum, Branko
5026:
5020:
5019:
4999:
4993:
4992:
4972:
4966:
4965:
4955:
4949:
4948:
4924:
4918:
4912:
4908:Harmonices Mundi
4899:Kepler, Johannes
4895:
4849:
4840:
4833:
4818:
4809:
4800:
4793:
4778:
4769:
4760:
4753:
4734:
4727:
4708:
4701:
4682:
4669:
4662:
4643:
4630:
4623:
4608:
4601:
4586:
4577:
4570:
4551:
4542:
4523:
4516:
4497:
4482:
4475:
4456:
4447:
4423:
4421:
4414:
4400:
4396:
4392:
4350:
4311:
4301:
4282:
4242:
4233:
4224:
4213:
4204:
4193:
4180:
4173:
4138:
4132:-antiprism, for
4131:
4127:
4123:
4120:Note: The right
4115:
4108:
4107:24 = 4 × (2 × 3)
4104:
4100:
4089:
4082:
4081:12 = 3 × (2 × 2)
4078:
4074:
4064:
4057:
4041:
4027:
4020:
4019:48 = 4 × (4 × 3)
4016:
4005:
3994:
3987:
3986:24 = 3 × (4 × 2)
3983:
3972:
3962:
3955:
3938:
3919:
3917:
3916:
3911:
3909:
3901:
3893:
3888:
3887:
3886:
3870:
3865:
3864:
3863:
3834:
3830:
3822:
3811:
3809:
3808:
3803:
3798:
3796:
3795:
3787:
3778:
3777:
3769:
3751:
3749:
3748:
3739:
3734:
3726:
3718:
3710:
3690:
3682:
3674:
3673:
3655:
3650:
3642:
3619:
3618:
3597:
3596:
3583:
3578:
3550:
3548:
3547:
3542:
3528:
3527:
3506:
3505:
3480:
3478:
3477:
3472:
3467:
3459:
3451:
3447:
3434:
3426:
3407:
3405:
3404:
3395:
3394:
3393:
3374:
3360:
3359:
3337:
3335:
3334:
3329:
3327:
3319:
3311:
3307:
3300:
3292:
3249:
3247:
3246:
3237:
3236:
3235:
3216:
3202:
3201:
3182:
3180:
3179:
3174:
3169:
3155:
3154:
3123:
3121:
3120:
3115:
3104:
3103:
3087:
3085:
3084:
3079:
3043:
3041:
3040:
3035:
3023:
3021:
3020:
3015:
3010:
2996:
2995:
2964:
2962:
2961:
2956:
2954:
2953:
2937:
2935:
2934:
2929:
2893:
2891:
2890:
2885:
2873:
2871:
2870:
2865:
2860:
2837:
2836:
2821:sides of length
2820:
2818:
2817:
2812:
2800:
2798:
2797:
2792:
2784:
2752:
2751:
2736:sides of length
2735:
2733:
2732:
2727:
2715:
2713:
2712:
2707:
2696:-gon (truncated
2695:
2693:
2692:
2687:
2672:
2670:
2669:
2664:
2637:
2635:
2634:
2629:
2621:
2613:
2581:
2580:
2553:
2552:
2540:
2538:
2537:
2528:
2527:
2526:
2507:
2502:
2501:
2473:
2471:
2470:
2465:
2453:
2451:
2450:
2445:
2433:
2431:
2430:
2425:
2410:
2408:
2407:
2402:
2360:
2356:
2346:
2339:
2334:
2311:
2297:
2292:
2269:
2243:
2238:
2224:
2215:
2210:
2187:
2173:
2168:
2145:
2119:
2114:
2100:
2077:
2075:
2074:
2069:
2027:
2023:
2013:
2006:
2001:
1978:
1964:
1959:
1948:
1922:
1917:
1903:
1894:
1889:
1866:
1852:
1847:
1836:
1810:
1805:
1791:
1768:
1766:
1765:
1760:
1736:
1732:
1721:
1716:
1709:
1685:
1661:
1656:
1649:
1625:
1584:
1582:
1581:
1576:
1549:
1545:
1534:
1529:
1518:
1494:
1489:
1478:
1437:
1435:
1434:
1429:
1424:
1422:
1421:
1413:
1398:
1371:
1369:
1368:
1363:
1344:
1342:
1341:
1336:
1331:
1327:
1326:
1318:
1304:
1296:
1283:
1278:
1277:
1276:
1260:
1245:
1241:
1236:-gonal antiprism
1235:
1227:
1225:
1224:
1219:
1214:
1213:
1204:
1200:
1199:
1194:
1189:
1181:
1168:
1160:
1143:
1141:
1140:
1135:
1130:
1129:
1118:
1117:
1115:
1114:
1106:
1101:
1100:
1087:
1086:
1084:
1076:
1068:
1060:
1052:
1050:
1039:
1034:
1033:
1021:
1015:
1000:
993:
982:
980:
979:
974:
969:
964:
956:
945:
937:
926:
925:
906:
899:
884:
882:
881:
876:
874:
870:
866:
865:
844:
839:
831:
820:
815:
807:
784:
777:
773:
751:
742:
733:
724:
715:
706:
699:
655:
645:
638:
631:
624:
617:
610:
593:
586:
579:
572:
565:
558:
522:Square antiprism
501:
495:
488:
481:
451:
436:
419:
412:
396:
373:
359:; i.e. (for non-
352:
351:
349:
348:
343:
340:
332:
308:
307:
305:
304:
299:
296:
286:
266:H. S. M. Coxeter
259:
226:Harmonices Mundi
210:
206:
192:
188:
163:
150:
140:
138:
127:composed of two
122:
119:
111:
109:
93:
86:
82:
79:
73:
68:this article by
59:inline citations
46:
45:
38:
21:
5619:
5618:
5614:
5613:
5612:
5610:
5609:
5608:
5589:
5588:
5587:
5582:
5571:
5510:Dihedral others
5501:
5480:
5455:
5430:
5359:
5356:
5355:
5346:
5275:
5264:
5263:
5254:
5217:
5215:Platonic solids
5209:
5203:
5146:
5145:
5130:
5119:
5106:
5103:
5101:Further reading
5098:
5097:
5065:
5064:
5060:
5036:
5028:
5027:
5023:
5001:
5000:
4996:
4974:
4973:
4969:
4957:
4956:
4952:
4929:Fischer, Gisela
4926:
4925:
4921:
4897:
4896:
4892:
4887:
4872:Grand antiprism
4868:
4850:
4841:
4831:
4830:
4829:
4825:
4819:
4810:
4801:
4791:
4790:
4789:
4785:
4779:
4770:
4761:
4751:
4750:
4749:
4745:
4737:
4735:
4725:
4724:
4723:
4719:
4711:
4709:
4699:
4698:
4697:
4693:
4685:
4683:
4672:
4670:
4660:
4659:
4658:
4654:
4646:
4644:
4633:
4631:
4621:
4620:
4619:
4615:
4609:
4599:
4598:
4597:
4593:
4587:
4578:
4568:
4567:
4566:
4562:
4554:
4552:
4540:
4539:
4538:
4534:
4526:
4524:
4514:
4513:
4512:
4508:
4500:
4498:
4485:
4483:
4473:
4472:
4471:
4467:
4459:
4457:
4445:
4444:
4443:
4439:
4416:
4406:
4398:
4394:
4384:
4345:
4303:
4288:
4272:
4265:
4250:
4243:
4234:
4225:
4214:
4205:
4196:
4194:
4185:
4181:
4171:
4150:
4145:
4143:Generalizations
4133:
4129:
4125:
4121:
4114:
4110:
4106:
4102:
4095:
4088:
4084:
4080:
4076:
4069:
4059:
4056:
4050:
4039:
4026:
4022:
4018:
4015:
4011:
4000:
3993:
3989:
3985:
3982:
3978:
3967:
3957:
3954:
3947:
3940:
3936:
3926:
3878:
3871:
3842:
3837:
3836:
3832:
3828:
3820:
3818:
3815:
3779:
3752:
3740:
3727:
3665:
3643:
3610:
3588:
3556:
3555:
3519:
3497:
3486:
3485:
3415:
3411:
3396:
3385:
3375:
3351:
3346:
3345:
3269:
3265:
3238:
3227:
3217:
3193:
3188:
3187:
3146:
3126:
3125:
3124:(semiperimeter
3095:
3090:
3089:
3046:
3045:
3026:
3025:
2987:
2967:
2966:
2965:(semiperimeter
2945:
2940:
2939:
2896:
2895:
2876:
2875:
2828:
2823:
2822:
2803:
2802:
2743:
2738:
2737:
2718:
2717:
2698:
2697:
2675:
2674:
2643:
2642:
2572:
2544:
2529:
2518:
2508:
2493:
2479:
2478:
2456:
2455:
2436:
2435:
2416:
2415:
2354:
2353:
2347:
2344:
2343:
2312:
2270:
2225:
2220:
2219:
2188:
2146:
2101:
2091:
2086:
2085:
2021:
2020:
2014:
2011:
2010:
1979:
1949:
1904:
1899:
1898:
1867:
1837:
1792:
1782:
1777:
1776:
1730:
1729:
1723:
1722:
1686:
1663:
1662:
1626:
1598:
1593:
1592:
1543:
1542:
1536:
1535:
1519:
1496:
1495:
1479:
1451:
1446:
1445:
1402:
1377:
1376:
1354:
1353:
1350:
1288:
1284:
1268:
1261:
1248:
1247:
1243:
1239:
1233:
1205:
1173:
1169:
1148:
1147:
1121:
1092:
1088:
1077:
1069:
1043:
1025:
1016:
1003:
1002:
998:
991:
988:
957:
917:
909:
908:
904:
889:
857:
832:
808:
799:
795:
790:
789:
779:
778:-gon bases and
775:
771:
762:
752:
743:
734:
725:
716:
707:
520:
513:
504:Antiprism name
499:
446:
431:
414:
410:
394:
387:
374:side faces are
368:
365:right antiprism
344:
341:
338:
337:
335:
334:
330:
300:
297:
294:
293:
291:
290:
284:
279:
277:Right antiprism
274:
253:
231:Johannes Kepler
221:
208:
204:
201:dual polyhedron
190:
183:
173:snub polyhedron
158:
156:Conway notation
145:
136:
135:
117:
115:
107:
106:
94:
83:
77:
74:
64:Please help to
63:
47:
43:
28:
23:
22:
15:
12:
11:
5:
5617:
5615:
5607:
5606:
5601:
5591:
5590:
5584:
5583:
5576:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5534:
5529:
5524:
5519:
5513:
5511:
5507:
5506:
5503:
5502:
5500:
5499:
5494:
5488:
5486:
5482:
5481:
5479:
5478:
5473:
5467:
5461:
5457:
5456:
5454:
5453:
5446:
5438:
5436:
5432:
5431:
5429:
5428:
5423:
5418:
5413:
5408:
5403:
5398:
5393:
5388:
5383:
5378:
5373:
5368:
5362:
5360:
5353:Catalan solids
5351:
5348:
5347:
5345:
5344:
5339:
5334:
5329:
5324:
5319:
5314:
5309:
5304:
5299:
5294:
5292:truncated cube
5289:
5284:
5278:
5276:
5259:
5256:
5255:
5253:
5252:
5247:
5242:
5237:
5232:
5226:
5224:
5211:
5210:
5204:
5202:
5201:
5194:
5187:
5179:
5173:
5172:
5167:
5162:
5143:
5129:
5128:External links
5126:
5125:
5124:
5117:
5102:
5099:
5096:
5095:
5076:(4): 966–984.
5058:
5041:Geombinatorics
5021:
4994:
4983:(1): 419–425.
4967:
4950:
4939:(4): 457–467.
4919:
4915:illustration A
4889:
4888:
4886:
4883:
4882:
4881:
4875:
4867:
4864:
4861:
4860:
4857:
4853:
4852:
4843:
4834:
4827:
4822:
4821:
4812:
4803:
4794:
4787:
4782:
4781:
4772:
4763:
4754:
4747:
4742:
4741:
4728:
4721:
4716:
4715:
4702:
4695:
4690:
4689:
4676:
4663:
4656:
4651:
4650:
4637:
4624:
4617:
4612:
4611:
4602:
4595:
4590:
4589:
4580:
4571:
4564:
4559:
4558:
4545:
4543:
4536:
4531:
4530:
4517:
4510:
4505:
4504:
4491:
4489:
4476:
4469:
4464:
4463:
4450:
4448:
4441:
4436:
4435:
4432:
4431:Uniform stars
4429:
4427:Symmetry group
4381:
4380:
4375:
4374:
4285:vertex figures
4246:
4245:
4244:9/5-antiprism
4236:
4235:9/4-antiprism
4227:
4226:9/2-antiprism
4217:
4216:
4215:5/3-antiprism
4207:
4206:5/2-antiprism
4198:
4187:
4170:
4167:
4154:dual polyhedra
4149:
4146:
4144:
4141:
4118:
4117:
4112:
4092:
4091:
4086:
4052:
4047:rotation group
4037:if and only if
4030:
4029:
4024:
4013:
4006:: the regular
3997:
3996:
3991:
3980:
3973:: the regular
3949:
3942:
3930:symmetry group
3925:
3922:
3907:
3904:
3899:
3896:
3891:
3885:
3881:
3877:
3874:
3868:
3862:
3859:
3856:
3853:
3850:
3845:
3827:with the same
3813:
3812:
3801:
3793:
3790:
3785:
3782:
3775:
3772:
3767:
3764:
3761:
3758:
3755:
3747:
3743:
3737:
3733:
3730:
3724:
3721:
3716:
3713:
3708:
3705:
3702:
3699:
3696:
3693:
3688:
3685:
3680:
3677:
3672:
3668:
3664:
3661:
3658:
3653:
3649:
3646:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3617:
3613:
3609:
3606:
3603:
3600:
3595:
3591:
3587:
3582:
3577:
3573:
3569:
3566:
3563:
3540:
3537:
3534:
3531:
3526:
3522:
3518:
3515:
3512:
3509:
3504:
3500:
3496:
3493:
3482:
3481:
3470:
3465:
3462:
3457:
3454:
3450:
3446:
3443:
3440:
3437:
3432:
3429:
3424:
3421:
3418:
3414:
3410:
3403:
3399:
3392:
3388:
3384:
3381:
3378:
3372:
3369:
3366:
3363:
3358:
3354:
3339:
3338:
3325:
3322:
3317:
3314:
3310:
3306:
3303:
3298:
3295:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3268:
3264:
3261:
3258:
3255:
3252:
3245:
3241:
3234:
3230:
3226:
3223:
3220:
3214:
3211:
3208:
3205:
3200:
3196:
3172:
3168:
3164:
3161:
3158:
3153:
3149:
3145:
3142:
3139:
3136:
3133:
3113:
3110:
3107:
3102:
3098:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3033:
3013:
3009:
3005:
3002:
2999:
2994:
2990:
2986:
2983:
2980:
2977:
2974:
2952:
2948:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2883:
2863:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2835:
2831:
2810:
2790:
2787:
2783:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2750:
2746:
2725:
2705:
2685:
2682:
2662:
2659:
2656:
2653:
2650:
2639:
2638:
2627:
2624:
2619:
2616:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2579:
2575:
2571:
2568:
2565:
2562:
2559:
2556:
2551:
2547:
2543:
2536:
2532:
2525:
2521:
2517:
2514:
2511:
2505:
2500:
2496:
2492:
2489:
2486:
2463:
2443:
2423:
2412:
2411:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2363:
2359:
2352:
2349:
2348:
2345:
2342:
2337:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2309:
2306:
2303:
2300:
2295:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2241:
2237:
2234:
2231:
2228:
2222:
2221:
2218:
2213:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2185:
2182:
2179:
2176:
2171:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2117:
2113:
2110:
2107:
2104:
2098:
2097:
2094:
2079:
2078:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2030:
2026:
2019:
2016:
2015:
2012:
2009:
2004:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1976:
1973:
1970:
1967:
1962:
1958:
1955:
1952:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1920:
1916:
1913:
1910:
1907:
1901:
1900:
1897:
1892:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1864:
1861:
1858:
1855:
1850:
1846:
1843:
1840:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1808:
1804:
1801:
1798:
1795:
1789:
1788:
1785:
1770:
1769:
1758:
1755:
1752:
1749:
1746:
1743:
1739:
1735:
1728:
1725:
1724:
1719:
1715:
1712:
1708:
1704:
1701:
1698:
1695:
1692:
1689:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1664:
1659:
1655:
1652:
1648:
1644:
1641:
1638:
1635:
1632:
1629:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1604:
1601:
1586:
1585:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1552:
1548:
1541:
1538:
1537:
1532:
1528:
1525:
1522:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1497:
1492:
1488:
1485:
1482:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1457:
1454:
1439:
1438:
1427:
1419:
1416:
1411:
1408:
1405:
1401:
1396:
1393:
1390:
1387:
1384:
1361:
1349:
1346:
1334:
1330:
1324:
1321:
1316:
1313:
1310:
1307:
1302:
1299:
1294:
1291:
1287:
1281:
1275:
1271:
1267:
1264:
1258:
1255:
1217:
1212:
1208:
1203:
1197:
1192:
1187:
1184:
1179:
1176:
1172:
1166:
1163:
1158:
1155:
1133:
1128:
1124:
1112:
1109:
1104:
1099:
1095:
1091:
1083:
1080:
1075:
1072:
1066:
1063:
1058:
1055:
1049:
1046:
1042:
1037:
1032:
1028:
1024:
1019:
1013:
1010:
987:
984:
972:
967:
963:
960:
954:
951:
948:
943:
940:
935:
932:
929:
924:
920:
916:
886:
885:
873:
869:
864:
860:
856:
853:
850:
847:
842:
838:
835:
829:
826:
823:
818:
814:
811:
805:
802:
798:
761:
758:
755:
754:
745:
736:
727:
718:
709:
688:
687:
684:
681:
678:
675:
672:
669:
666:
663:
661:Vertex config.
657:
656:
649:
646:
639:
632:
625:
618:
611:
604:
600:
599:
597:
594:
587:
580:
573:
566:
559:
552:
548:
547:
542:
539:
534:
529:
524:
517:
510:
505:
498:
497:
490:
483:
475:
452:, the regular
437:, we have the
386:
383:
278:
275:
273:
270:
220:
217:
194:quadrilaterals
96:
95:
50:
48:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5616:
5605:
5602:
5600:
5597:
5596:
5594:
5580:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5514:
5512:
5508:
5498:
5495:
5493:
5490:
5489:
5487:
5483:
5477:
5474:
5472:
5469:
5468:
5465:
5462:
5458:
5452:
5451:
5447:
5445:
5444:
5440:
5439:
5437:
5433:
5427:
5424:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5392:
5389:
5387:
5384:
5382:
5379:
5377:
5374:
5372:
5369:
5367:
5364:
5363:
5361:
5354:
5349:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5287:cuboctahedron
5285:
5283:
5280:
5279:
5277:
5272:
5268:
5262:
5257:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5227:
5225:
5221:
5216:
5212:
5208:
5200:
5195:
5193:
5188:
5186:
5181:
5180:
5177:
5171:
5168:
5166:
5163:
5158:
5157:
5152:
5149:
5144:
5141:
5136:
5132:
5131:
5127:
5120:
5118:0-520-03056-7
5114:
5110:
5105:
5104:
5100:
5091:
5087:
5083:
5079:
5075:
5071:
5070:
5062:
5059:
5054:
5050:
5046:
5042:
5035:
5031:
5025:
5022:
5017:
5013:
5009:
5005:
4998:
4995:
4990:
4986:
4982:
4978:
4971:
4968:
4963:
4962:
4954:
4951:
4946:
4942:
4938:
4934:
4930:
4923:
4920:
4916:
4910:
4909:
4904:
4900:
4894:
4891:
4884:
4879:
4876:
4873:
4870:
4869:
4865:
4858:
4855:
4848:
4844:
4839:
4835:
4824:
4817:
4813:
4808:
4804:
4799:
4795:
4784:
4777:
4773:
4768:
4764:
4759:
4755:
4744:
4740:
4733:
4729:
4718:
4714:
4707:
4703:
4692:
4688:
4681:
4677:
4675:
4668:
4664:
4653:
4649:
4642:
4638:
4636:
4629:
4625:
4614:
4607:
4603:
4592:
4585:
4581:
4576:
4572:
4561:
4557:
4550:
4546:
4533:
4529:
4522:
4518:
4507:
4503:
4496:
4492:
4488:
4481:
4477:
4466:
4462:
4455:
4451:
4438:
4433:
4428:
4425:
4424:
4419:
4413:
4409:
4402:
4391:
4387:
4377:
4376:
4371:
4370:
4369:
4366:
4364:
4360:
4355:
4353:
4349:
4343:
4342:
4341:
4335:
4334:
4333:
4328:
4325:
4322:
4318:
4313:
4310:
4306:
4299:
4295:
4291:
4286:
4280:
4276:
4270:
4264:
4255:
4251:
4241:
4237:
4232:
4228:
4223:
4219:
4212:
4208:
4203:
4199:
4195:5/4-antiprism
4192:
4188:
4184:
4183:3/2-antiprism
4179:
4175:
4168:
4166:
4164:
4159:
4155:
4147:
4142:
4140:
4136:
4116:as subgroups.
4098:
4094:
4093:
4090:as subgroups;
4072:
4068:
4067:
4066:
4063:
4055:
4048:
4043:
4038:
4035:
4028:as subgroups.
4009:
4003:
3999:
3998:
3995:as subgroups;
3976:
3970:
3966:
3965:
3964:
3961:
3952:
3945:
3935:
3931:
3923:
3921:
3905:
3902:
3897:
3894:
3889:
3883:
3879:
3875:
3872:
3866:
3843:
3826:
3816:
3799:
3791:
3788:
3783:
3780:
3773:
3770:
3765:
3762:
3759:
3756:
3753:
3745:
3741:
3735:
3731:
3728:
3722:
3714:
3711:
3706:
3703:
3700:
3697:
3694:
3686:
3683:
3678:
3675:
3670:
3662:
3656:
3651:
3647:
3644:
3638:
3635:
3632:
3623:
3615:
3611:
3607:
3601:
3593:
3589:
3580:
3575:
3571:
3567:
3564:
3561:
3554:
3553:
3552:
3532:
3524:
3520:
3516:
3510:
3502:
3498:
3491:
3468:
3463:
3460:
3455:
3452:
3448:
3444:
3441:
3438:
3435:
3430:
3427:
3422:
3419:
3416:
3412:
3408:
3401:
3397:
3390:
3382:
3376:
3370:
3364:
3356:
3352:
3344:
3343:
3342:
3323:
3320:
3315:
3312:
3308:
3304:
3301:
3296:
3293:
3288:
3285:
3279:
3276:
3273:
3266:
3259:
3256:
3253:
3243:
3239:
3232:
3224:
3218:
3212:
3206:
3198:
3194:
3186:
3185:
3184:
3170:
3166:
3159:
3151:
3147:
3143:
3137:
3131:
3108:
3100:
3096:
3072:
3066:
3063:
3057:
3051:
3031:
3011:
3007:
3000:
2992:
2988:
2984:
2978:
2972:
2950:
2946:
2922:
2916:
2913:
2907:
2901:
2881:
2861:
2857:
2853:
2850:
2847:
2841:
2833:
2829:
2808:
2785:
2781:
2777:
2774:
2771:
2765:
2762:
2756:
2748:
2744:
2723:
2703:
2683:
2680:
2660:
2657:
2654:
2651:
2648:
2625:
2617:
2614:
2609:
2606:
2600:
2597:
2594:
2588:
2585:
2582:
2577:
2573:
2569:
2566:
2563:
2560:
2557:
2554:
2549:
2545:
2534:
2530:
2523:
2515:
2509:
2503:
2498:
2490:
2484:
2477:
2476:
2475:
2461:
2441:
2421:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2361:
2357:
2350:
2335:
2328:
2325:
2322:
2319:
2313:
2307:
2304:
2301:
2298:
2293:
2286:
2283:
2280:
2274:
2271:
2265:
2262:
2256:
2253:
2250:
2239:
2232:
2226:
2211:
2204:
2201:
2198:
2195:
2189:
2183:
2180:
2177:
2174:
2169:
2162:
2159:
2156:
2150:
2147:
2141:
2138:
2132:
2129:
2126:
2115:
2108:
2102:
2092:
2084:
2083:
2082:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2028:
2024:
2017:
2002:
1995:
1992:
1989:
1986:
1980:
1974:
1971:
1968:
1965:
1960:
1956:
1953:
1950:
1944:
1941:
1935:
1932:
1929:
1918:
1911:
1905:
1890:
1883:
1880:
1877:
1874:
1868:
1862:
1859:
1856:
1853:
1848:
1844:
1841:
1838:
1832:
1829:
1823:
1820:
1817:
1806:
1799:
1793:
1783:
1775:
1774:
1773:
1756:
1753:
1750:
1747:
1744:
1741:
1737:
1733:
1726:
1717:
1710:
1706:
1702:
1699:
1696:
1690:
1687:
1681:
1678:
1672:
1666:
1657:
1650:
1646:
1642:
1639:
1636:
1630:
1627:
1621:
1618:
1612:
1606:
1599:
1591:
1590:
1589:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1550:
1546:
1539:
1530:
1526:
1523:
1520:
1514:
1511:
1505:
1499:
1490:
1486:
1483:
1480:
1474:
1471:
1465:
1459:
1452:
1444:
1443:
1442:
1425:
1417:
1414:
1409:
1406:
1403:
1399:
1394:
1388:
1382:
1375:
1374:
1373:
1359:
1347:
1345:
1332:
1328:
1322:
1319:
1314:
1311:
1308:
1305:
1300:
1297:
1292:
1289:
1285:
1279:
1273:
1269:
1265:
1262:
1256:
1253:
1246:is given by:
1237:
1228:
1215:
1210:
1206:
1201:
1195:
1190:
1185:
1182:
1177:
1174:
1170:
1164:
1161:
1156:
1153:
1144:
1131:
1126:
1122:
1110:
1107:
1102:
1097:
1093:
1089:
1081:
1078:
1073:
1070:
1064:
1061:
1056:
1053:
1047:
1044:
1040:
1035:
1030:
1026:
1022:
1017:
1011:
1008:
997:
985:
983:
970:
965:
961:
958:
952:
949:
946:
941:
938:
933:
930:
927:
922:
918:
914:
901:
897:
893:
871:
867:
862:
854:
851:
845:
840:
836:
833:
827:
824:
821:
816:
812:
809:
803:
800:
796:
788:
787:
786:
783:
770:
766:
759:
750:
746:
741:
737:
732:
728:
723:
719:
714:
710:
705:
701:
697:
695:
685:
682:
679:
676:
673:
670:
667:
664:
662:
659:
658:
654:
650:
647:
644:
640:
637:
633:
630:
626:
623:
619:
616:
612:
609:
605:
602:
601:
598:
595:
592:
588:
585:
581:
578:
574:
571:
567:
564:
560:
557:
553:
550:
549:
546:
543:
540:
538:
535:
533:
530:
528:
525:
523:
518:
516:
511:
509:
506:
503:
502:
496:
491:
489:
484:
482:
477:
474:
470:
467:
461:
459:
455:
449:
444:
440:
434:
428:
426:
425:
424:
418:
409:
408:
407:
402:
398:
393:
384:
382:
380:
378:
372:
366:
362:
358:
357:
347:
329:
326:
321:
319:
318:perpendicular
315:
310:
303:
288:
276:
272:Special cases
271:
269:
267:
263:
262:optical prism
257:
252:
247:
245:
241:
237:
232:
228:
227:
218:
216:
214:
213:trapezohedron
202:
197:
195:
187:
181:
176:
174:
170:
165:
162:
157:
153:
149:
143:
133:
130:
126:
121:
113:
103:
92:
89:
81:
71:
67:
61:
60:
54:
49:
40:
39:
32:
19:
5578:
5497:trapezohedra
5475:
5448:
5441:
5245:dodecahedron
5154:
5108:
5073:
5067:
5061:
5047:(2): 69–78.
5044:
5040:
5024:
5007:
5003:
4997:
4980:
4976:
4970:
4960:
4953:
4936:
4932:
4922:
4907:
4893:
4878:Skew polygon
4434:Right stars
4417:
4411:
4407:
4389:
4385:
4382:
4367:
4362:
4358:
4356:
4354:side faces.
4347:
4339:
4338:
4331:
4330:
4316:
4314:
4308:
4304:
4297:
4293:
4289:
4278:
4274:
4269:star polygon
4266:
4249:
4151:
4134:
4119:
4096:
4070:
4061:
4053:
4044:
4031:
4001:
3968:
3959:
3950:
3943:
3927:
3817:
3814:
3483:
3340:
2640:
2413:
2080:
1771:
1587:
1440:
1351:
1229:
1145:
989:
902:
895:
891:
887:
781:
763:
691:
519:(Tetragonal)
472:
468:
457:
447:
438:
432:
429:
422:
421:
416:
405:
404:
390:
388:
376:
370:
364:
354:
345:
327:
322:
313:
311:
301:
280:
248:
224:
222:
198:
185:
177:
166:
160:
147:
116:
105:
99:
84:
78:January 2013
75:
56:
5267:semiregular
5250:icosahedron
5230:tetrahedron
5151:"Antiprism"
4851:3.3.3.12/7
4842:3.3.3.12/5
4820:3.3.3.11/7
4811:3.3.3.11/5
4802:3.3.3.11/3
4780:3.3.3.11/6
4771:3.3.3.11/4
4762:3.3.3.11/2
4302:instead of
4197:nonuniform
4186:nonuniform
3975:tetrahedron
2716:-gon) with
1242:and height
443:tetrahedron
423:equilateral
254: [
169:prismatoids
70:introducing
5593:Categories
5562:prismatoid
5492:bipyramids
5476:antiprisms
5450:hosohedron
5240:octahedron
5140:Antiprisms
5010:(1): 1–9.
4885:References
4736:3.3.3.10/3
4610:3.3.3.7/3
4588:3.3.3.7/4
4579:3.3.3.7/2
4008:octahedron
1348:Derivation
512:(Trigonal)
473:antiprisms
464:Family of
454:octahedron
397:-antiprism
367:, and its
125:polyhedron
120:-antiprism
53:references
5557:birotunda
5547:bifrustum
5312:snub cube
5207:polyhedra
5156:MathWorld
4913:See also
4752:(*2.2.11)
4710:3.3.3.9/5
4684:3.3.3.9/4
4671:3.3.3.9/2
4645:3.3.3.8/5
4632:3.3.3.8/3
4553:3.3/2.3.6
4525:3.3.3.5/3
4499:3.3/2.3.5
4484:3.3.3.5/2
4458:3.3/2.3.4
4321:congruent
4105:of order
4079:of order
4058:of order
4034:inversion
4017:of order
3984:of order
3956:of order
3903:π
3898:
3789:π
3784:
3771:π
3766:
3712:π
3707:
3684:π
3679:
3572:∫
3461:π
3456:
3442:−
3428:π
3423:
3321:π
3316:
3294:π
3289:
3277:−
3257:−
2775:−
2658:≤
2652:≤
2615:π
2610:
2598:−
2555:−
2396:−
2375:≤
2369:≤
2314:π
2308:
2275:π
2266:
2254:−
2190:π
2184:
2151:π
2142:
2130:−
2063:−
2042:≤
2036:≤
1981:π
1975:
1954:π
1945:
1933:−
1869:π
1863:
1842:π
1833:
1821:−
1754:−
1691:π
1682:
1631:π
1622:
1567:−
1524:π
1515:
1484:π
1475:
1415:π
1410:
1320:π
1315:
1298:π
1293:
1183:π
1178:
1108:π
1103:
1074:π
1065:
1054:−
1041:π
1036:
962:π
953:
947:−
939:π
934:
852:−
837:π
828:
813:π
804:
401:congruent
379:triangles
377:isosceles
325:congruent
309:degrees.
152:triangles
112:antiprism
5537:bicupola
5517:pyramids
5443:dihedron
5032:(2005).
4945:41134285
4901:(1619).
4866:See also
4379:uniform.
4319:has two
4042:is odd.
3924:Symmetry
686:∞.3.3.3
680:7.3.3.3
677:6.3.3.3
674:5.3.3.3
671:4.3.3.3
668:3.3.3.3
665:2.3.3.3
399:has two
361:coplanar
283:regular
129:parallel
102:geometry
5579:italics
5567:scutoid
5552:rotunda
5542:frustum
5271:uniform
5220:regular
5205:Convex
5090:3639611
5053:2298896
4359:regular
4327:regular
4324:coaxial
4271:bases,
3823:-gonal
3024:) plus
2081:and at
996:uniform
903:if the
471:-gonal
466:uniform
406:regular
392:uniform
356:coaxial
350:
336:
328:regular
306:
292:
219:History
211:-gonal
142:polygon
66:improve
5532:cupola
5485:duals:
5471:prisms
5115:
5088:
5051:
4943:
4832:(2*12)
4792:(2*11)
4726:(2*10)
4661:(*229)
4569:(*227)
4474:(*225)
4332:convex
1232:right
1119:
888:where
445:; for
203:of an
180:prisms
139:-sided
132:direct
110:-gonal
55:, but
5037:(PDF)
4941:JSTOR
4700:(2*9)
4622:(2*8)
4600:(2*7)
4541:(2*6)
4515:(2*5)
4446:(2*4)
4405:Star
4363:right
3934:right
3932:of a
3825:prism
769:right
456:as a
258:]
238:of a
123:is a
104:, an
5235:cube
5113:ISBN
4859:...
4856:...
4420:≤ 12
4397:and
4340:star
4045:The
3928:The
3835:is:
3831:and
3341:and
3088:and
2938:and
2434:and
990:Let
890:0 ≤
692:The
683:...
596:...
541:...
314:axis
312:The
287:-gon
199:The
5269:or
5078:doi
5012:doi
4985:doi
4828:12d
4788:11d
4748:11h
4722:10d
4336:or
4137:≥ 4
4099:= 3
4073:= 2
4049:is
4004:= 3
3971:= 2
3948:= D
3895:cot
3781:sin
3763:cos
3704:cos
3676:sin
3453:sin
3420:cos
3313:sin
3286:cos
2607:cos
2474:is
2390:0..
2305:sin
2263:sin
2181:cos
2139:cos
2057:0..
1972:sin
1942:sin
1860:cos
1830:cos
1748:0..
1679:sin
1619:cos
1561:0..
1512:sin
1472:cos
1407:sin
1312:cot
1290:csc
1175:cot
1094:sin
1062:sin
1027:cos
950:cos
931:cos
898:– 1
894:≤ 2
825:sin
801:cos
753:A8
744:A7
735:A6
726:A5
717:A4
708:A3
450:= 3
435:= 2
339:180
295:180
236:net
175:.
114:or
100:In
5595::
5153:.
5086:MR
5084:.
5074:57
5072:.
5049:MR
5045:15
5043:.
5039:.
5008:24
5006:.
4981:30
4979:.
4937:62
4935:.
4905:.
4696:9d
4657:9h
4618:8d
4596:7d
4565:7h
4537:6d
4511:5d
4470:5h
4442:4d
4315:A
4296:–
4292:/(
4281:},
4139:.
4025:3d
3992:2d
3736:12
2399:1.
1757:1.
1280:12
1090:12
900:;
389:A
381:.
268:.
256:de
229:,
215:.
196:.
164:.
5581:.
5273:)
5265:(
5222:)
5218:(
5198:e
5191:t
5184:v
5159:.
5121:.
5092:.
5080::
5055:.
5018:.
5014::
4991:.
4987::
4947:.
4826:D
4786:D
4746:D
4720:D
4694:D
4655:D
4616:D
4594:D
4563:D
4535:D
4509:D
4468:D
4440:D
4418:p
4412:q
4410:/
4408:p
4399:q
4395:p
4390:q
4388:/
4386:p
4348:n
4346:2
4309:q
4307:/
4305:p
4300:)
4298:q
4294:p
4290:p
4279:q
4277:/
4275:p
4273:{
4135:n
4130:n
4126:n
4122:n
4113:3
4111:D
4103:O
4097:n
4087:2
4085:D
4077:T
4071:n
4062:n
4060:2
4054:n
4051:D
4040:n
4023:D
4014:h
4012:O
4002:n
3990:D
3981:d
3979:T
3969:n
3960:n
3958:4
3953:v
3951:n
3946:d
3944:n
3941:D
3937:n
3906:n
3890:4
3884:2
3880:l
3876:h
3873:n
3867:=
3861:m
3858:s
3855:i
3852:r
3849:p
3844:V
3833:h
3829:l
3821:n
3800:.
3792:n
3774:n
3760:2
3757:+
3754:1
3746:2
3742:l
3732:h
3729:n
3723:=
3720:)
3715:n
3701:2
3698:+
3695:1
3692:(
3687:n
3671:2
3667:)
3663:0
3660:(
3657:R
3652:3
3648:h
3645:n
3639:=
3636:z
3633:d
3630:]
3627:)
3624:z
3621:(
3616:2
3612:Q
3608:+
3605:)
3602:z
3599:(
3594:1
3590:Q
3586:[
3581:h
3576:0
3568:n
3565:=
3562:V
3539:]
3536:)
3533:z
3530:(
3525:2
3521:Q
3517:+
3514:)
3511:z
3508:(
3503:1
3499:Q
3495:[
3492:n
3469:.
3464:n
3449:]
3445:z
3439:h
3436:+
3431:n
3417:z
3413:[
3409:z
3402:2
3398:h
3391:2
3387:)
3383:0
3380:(
3377:R
3371:=
3368:)
3365:z
3362:(
3357:2
3353:Q
3324:n
3309:]
3305:z
3302:+
3297:n
3283:)
3280:z
3274:h
3271:(
3267:[
3263:)
3260:z
3254:h
3251:(
3244:2
3240:h
3233:2
3229:)
3225:0
3222:(
3219:R
3213:=
3210:)
3207:z
3204:(
3199:1
3195:Q
3171:2
3167:/
3163:)
3160:z
3157:(
3152:2
3148:l
3144:+
3141:)
3138:z
3135:(
3132:R
3112:)
3109:z
3106:(
3101:2
3097:l
3076:)
3073:z
3070:(
3067:R
3064:,
3061:)
3058:z
3055:(
3052:R
3032:n
3012:2
3008:/
3004:)
3001:z
2998:(
2993:1
2989:l
2985:+
2982:)
2979:z
2976:(
2973:R
2951:1
2947:l
2926:)
2923:z
2920:(
2917:R
2914:,
2911:)
2908:z
2905:(
2902:R
2882:n
2862:h
2858:/
2854:z
2851:l
2848:=
2845:)
2842:z
2839:(
2834:2
2830:l
2809:n
2789:)
2786:h
2782:/
2778:z
2772:1
2769:(
2766:l
2763:=
2760:)
2757:z
2754:(
2749:1
2745:l
2724:n
2704:n
2684:n
2681:2
2661:h
2655:z
2649:0
2626:.
2623:]
2618:n
2604:)
2601:z
2595:h
2592:(
2589:z
2586:2
2583:+
2578:2
2574:z
2570:2
2567:+
2564:z
2561:h
2558:2
2550:2
2546:h
2542:[
2535:2
2531:h
2524:2
2520:)
2516:0
2513:(
2510:R
2504:=
2499:2
2495:)
2491:z
2488:(
2485:R
2462:z
2442:y
2422:x
2393:n
2387:=
2384:m
2381:,
2378:h
2372:z
2366:0
2362:,
2358:)
2351:z
2341:]
2336:n
2332:)
2329:1
2326:+
2323:m
2320:2
2317:(
2302:z
2299:+
2294:n
2290:)
2287:1
2284:+
2281:m
2278:(
2272:2
2260:)
2257:z
2251:h
2248:(
2245:[
2240:h
2236:)
2233:0
2230:(
2227:R
2217:]
2212:n
2208:)
2205:1
2202:+
2199:m
2196:2
2193:(
2178:z
2175:+
2170:n
2166:)
2163:1
2160:+
2157:m
2154:(
2148:2
2136:)
2133:z
2127:h
2124:(
2121:[
2116:h
2112:)
2109:0
2106:(
2103:R
2093:(
2066:1
2060:n
2054:=
2051:m
2048:,
2045:h
2039:z
2033:0
2029:,
2025:)
2018:z
2008:]
2003:n
1999:)
1996:1
1993:+
1990:m
1987:2
1984:(
1969:z
1966:+
1961:n
1957:m
1951:2
1939:)
1936:z
1930:h
1927:(
1924:[
1919:h
1915:)
1912:0
1909:(
1906:R
1896:]
1891:n
1887:)
1884:1
1881:+
1878:m
1875:2
1872:(
1857:z
1854:+
1849:n
1845:m
1839:2
1827:)
1824:z
1818:h
1815:(
1812:[
1807:h
1803:)
1800:0
1797:(
1794:R
1784:(
1751:n
1745:=
1742:m
1738:,
1734:)
1727:h
1718:n
1714:)
1711:2
1707:/
1703:1
1700:+
1697:m
1694:(
1688:2
1676:)
1673:0
1670:(
1667:R
1658:n
1654:)
1651:2
1647:/
1643:1
1640:+
1637:m
1634:(
1628:2
1616:)
1613:0
1610:(
1607:R
1600:(
1573:;
1570:1
1564:n
1558:=
1555:m
1551:,
1547:)
1540:0
1531:n
1527:m
1521:2
1509:)
1506:0
1503:(
1500:R
1491:n
1487:m
1481:2
1469:)
1466:0
1463:(
1460:R
1453:(
1426:.
1418:n
1404:2
1400:l
1395:=
1392:)
1389:0
1386:(
1383:R
1360:n
1333:.
1329:)
1323:n
1309:2
1306:+
1301:n
1286:(
1274:2
1270:l
1266:h
1263:n
1257:=
1254:V
1244:h
1240:l
1234:n
1216:.
1211:2
1207:a
1202:)
1196:3
1191:+
1186:n
1171:(
1165:2
1162:n
1157:=
1154:A
1132:,
1127:3
1123:a
1111:n
1098:2
1082:n
1079:2
1071:3
1057:1
1048:n
1045:2
1031:2
1023:4
1018:n
1012:=
1009:V
999:n
992:a
971:.
966:n
959:2
942:n
928:=
923:2
919:h
915:2
905:n
896:n
892:k
872:)
868:h
863:k
859:)
855:1
849:(
846:,
841:n
834:k
822:,
817:n
810:k
797:(
782:n
780:2
776:n
772:n
494:e
487:t
480:v
469:n
448:n
433:n
417:n
415:2
411:n
395:n
371:n
369:2
346:n
342:/
331:n
302:n
298:/
285:n
209:n
205:n
191:n
186:n
184:2
161:n
159:A
148:n
146:2
137:n
118:n
108:n
91:)
85:(
80:)
76:(
62:.
20:)
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