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Like the duoantiprisms as alternated duoprisms, there is a set of p-gonal double antiprismoids created by alternating the 2p-gonal ditetragoltriates, creating p-gonal antiprisms and tetrahedra while reinterpreting the non-corealmic triangular bipyramidal spaces as two tetrahedra. The resulting figure
2522:; then the p-gonal ditetragoltriate is the convex hull of two p-p duoprisms (where the p-gons are similar but not congruent, having different sizes) in perpendicular orientations. The resulting polychoron is isogonal and has 2p p-gonal prisms and p rectangular trapezoprisms (a
3173:
701:
2550:
and its conjugate, the pentagrammic double antiprismoid (with p = 5 and 5/3 respectively), represented as the alternation of a decagonal or decagrammic ditetragoltriate. The vertex figure is a variant of the
1871:
Vertex-centered orthogonal projections of p-p duoprisms project into symmetry for odd degrees, and for even degrees. There are n vertices projected into the center. For 4,4, it represents the A
766:
An alternative, more concise way of specifying a particular duoprism is by prefixing with numbers denoting the base polygons, for example: 3,5-duoprism for the triangular-pentagonal duoprism.
2348:{p,2,q}, the "duoantiprisms", which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the
2751:
2792:
2628:
969:-gonal faces, and form a second loop perpendicular to the first. These two loops are attached to each other via their square faces, and are mutually perpendicular.
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The p-q duoprisms are identical to the q-p duoprisms, but look different in these projections because they are projected in the center of different cells.
2158:
910:-gonal prisms. For example, the Cartesian product of a triangle and a hexagon is a duoprism bounded by 6 triangular prisms and 3 hexagonal prisms.
566:
3272:
824:, a Cartesian product of two or more polytopes of dimension at least two. The duoprisms are proprisms formed from exactly two polytopes.
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3172:, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online:
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operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the
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2188:-gonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a n-m duoprism, but these are not
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930:-gonal prisms. For example, the Cartesian product of two triangles is a duoprism bounded by 6 triangular prisms.
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Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
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of a p-gon can be clearly defined if one assumes that the octagon is the convex hull of two perpendicular
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A close up inside the 23-29 duoprism projected onto a 3-sphere, and perspective projected to 3-space. As
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Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two
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3248:(Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
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813:
265:
47:
3176:—contains a description of duoprisms (double prisms) and duocylinders (double cylinders).
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759:-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: a
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2169:, divided into a checkerboard surface of squares from the {4,4|n} skew polyhedron
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The two sets of 16-gonal prisms are shown. The top and bottom faces of the vertical
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are the sets of the points contained in the respective polygons. Such a duoprism is
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of two polytopes, each of two dimensions or higher. The
Cartesian product of an
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Also related are the ditetragoltriates or octagoltriates, formed by taking the
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696:{\displaystyle P_{1}\times P_{2}=\{(x,y,z,w)|(x,y)\in P_{1},(z,w)\in P_{2}\}}
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are identically 4, the resulting duoprism is bounded by 8 square prisms (
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985:
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2571:, is first in a dimensional series of uniform polytopes, expressed by
2514:(considered to be a ditetragon or a truncated square) to a p-gon. The
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1417:
A cell-centered perspective projection makes a duoprism look like a
2579:
series. The 3-3 duoprism is the vertex figure for the second, the
2214:
2199:
2156:
1467:
1418:
29:
1421:, with two sets of orthogonal cells, p-gonal and q-gonal prisms.
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symmetry) but cannot be made uniform. The vertex figure is a
552:
being the
Cartesian product of two polygons in 2-dimensional
1466:. Similarly a 6-6 duoprism projected into 3D approximates a
980:
approach infinity, the corresponding duoprisms approach the
892:-sided polygon with the same edge length. It is bounded by
254:
32:
3230:, Dover Publications, Inc., 1973, New York, p. 124.
2590:, and the final is a paracompact hyperbolic honeycomb, 3
2177:, {4,4|n}, exists in 4-space as the n square faces of a
2261:. The 16-cell is the only convex uniform duoantiprism.
763:
is the
Cartesian product of a triangle and a pentagon.
2764:
2723:
2600:
2438:
The only nonconvex uniform solution is p=5, q=5/3, ht
961:-gonal faces, and form a closed loop. Similarly, the
922:
are identical, the resulting duoprism is bounded by 2
569:
3253:
Regular Skew
Polyhedra in Three and Four Dimensions.
1423:
831:
469:
become large, a duoprism approaches the geometry of
2546:is generally not uniform except for two cases: the
965:-gonal prisms are attached to each other via their
957:-gonal prisms are attached to each other via their
2786:
2745:
2622:
1887:Orthogonal projection wireframes of p-p duoprisms
695:
3285:, Ph.D. Dissertation, University of Toronto, 1966
3189:Jonathan Bowers - Miscellaneous Uniform Polychora
2241:, there is a set of 4-dimensional duoantiprisms:
3283:The Theory of Uniform Polytopes and Honeycombs
2583:. The fourth figure is a Euclidean honeycomb,
808:is coined by George Olshevsky, shortened from
8:
1879:. The 5,5 projection is identical to the 3D
728:if both bases are convex, and is bounded by
690:
596:
3201:http://www.polychora.com/12GudapsMovie.gif
2257:) which creates the uniform (and regular)
2181:, using all 2n edges and n vertices. The 2
862:are connected when folded together in 4D.
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2767:
2766:
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2726:
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2722:
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1470:, hexagonal both in plan and in section.
684:
653:
626:
587:
574:
568:
3263:, Heidi Burgiel, Chaim Goodman-Strauss,
3255:Proc. London Math. Soc. 43, 33-62, 1937.
2640:
2634:is constructed from the previous as its
1885:
1478:
995:
456:
3162:
984:. As such, duoprisms are useful as non-
880:is created by the product of a regular
3238:The Beauty of Geometry: Twelve Essays
3174:The Fourth Dimension Simply Explained
3170:The Fourth Dimension Simply Explained
2393:{2,2,2}, with its alternation as the
7:
988:approximations of the duocylinder.
869:Geometry of 4-dimensional duoprisms
2498:, and 50 tetrahedra, known as the
27:Cartesian product of two polytopes
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2746:{\displaystyle {\tilde {E}}_{6}}
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2496:pentagrammic crossed-antiprisms
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2787:{\displaystyle {\bar {T}}_{7}}
2772:
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2623:{\displaystyle {\bar {T}}_{7}}
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2228:pentagrammic crossed-antiprism
761:triangular-pentagonal duoprism
674:
662:
643:
631:
627:
623:
599:
1:
744:of the same edge length is a
489:of 4 dimensions or higher, a
48:Prismatic uniform 4-polytopes
3240:, Dover Publications, 1999,
2131:
1999:
1799:
1743:
1679:
1617:
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1010:
997:
833:
556:. More precisely, it is the
266:Prismatic uniform 4-polytope
256:Set of uniform p-p duoprisms
3210:Animation of cross sections
945:), and is identical to the
3315:
2245:that can be created by an
2018:
1891:
477:-gonal prism approaches a
3141:Convex regular 4-polytope
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2012:
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769:Other alternative names:
249:
3265:The Symmetries of Things
2646:figures in n dimensions
2224:stereographic projection
2163:stereographic projection
816:proposed a similar name
2594:, with Coxeter group ,
2175:regular skew polyhedron
1881:rhombic triacontahedron
1413:Perspective projections
540:The lowest-dimensional
2788:
2747:
2624:
2490:, constructed from 10
2230:
2212:
2170:
1867:Orthogonal projections
828:Example 16-16 duoprism
697:
482:
293:Coxeter-Dynkin diagram
75:Coxeter-Dynkin diagram
2789:
2748:
2625:
2581:birectified 5-simplex
2492:pentagonal antiprisms
2218:
2203:
2160:
1875:Coxeter plane of the
698:
533:are dimensions of 2 (
460:
2762:
2721:
2598:
2541:Double antiprismoids
2535:triangular bipyramid
567:
3299:Uniform 4-polytopes
2647:
2630:. Each progressive
1888:
1483:
1428:
790:-gonal double prism
751:A duoprism made of
546:4-dimensional space
501:resulting from the
3251:Coxeter, H. S. M.
3206:2014-02-22 at the
2784:
2743:
2641:
2620:
2500:great duoantiprism
2231:
2220:Great duoantiprism
2213:
2171:
1886:
1479:
1424:
814:John Horton Conway
693:
483:
3273:978-1-56881-220-5
3224:Regular Polytopes
3123:
3122:
2775:
2734:
2611:
2506:Ditetragoltriates
2226:, centred on one
2204:p-q duoantiprism
2153:Related polytopes
2150:
2149:
1864:
1863:
1481:Schlegel diagrams
1474:
1473:
1426:Schlegel diagrams
1410:
1409:
866:
865:
800:-gonal hyperprism
525:-polytope, where
509:-polytope and an
503:Cartesian product
455:
454:
16:(Redirected from
3306:
3228:H. S. M. Coxeter
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1154:
1143:
1132:
1121:
1108:
1097:
1086:
1075:
1062:
1051:
1040:
1027:
1016:
1003:
996:
873:A 4-dimensional
856:
843:
836:Schlegel diagram
832:
746:uniform duoprism
742:regular polygons
723:
714:
702:
700:
699:
694:
689:
688:
658:
657:
630:
592:
591:
579:
578:
532:
528:
524:
513:-polytope is an
512:
508:
476:
468:
464:
450:Facet-transitive
430:
418:
411:
399:
387:
373:
362:
346:
333:
332:
331:
327:
326:
322:
321:
317:
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307:
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301:
287:
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216:
196:
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125:
115:
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113:
109:
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84:
83:
69:
37:
30:
21:
3314:
3313:
3309:
3308:
3307:
3305:
3304:
3303:
3289:
3288:
3220:
3215:
3214:
3208:Wayback Machine
3199:
3195:
3187:
3183:
3168:
3164:
3159:
3128:
3118:
3110:
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2705:
2697:
2645:
2601:
2596:
2595:
2593:
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2578:
2570:
2561:
2548:grand antiprism
2543:
2531:
2508:
2485:
2480:
2475:
2470:
2465:
2460:
2455:
2450:
2445:
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2329:
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2314:
2309:
2307:
2305:
2297:
2292:
2287:
2282:
2277:
2272:
2267:
2265:
2210:gyrobifastigium
2198:
2155:
1874:
1869:
1857:
1848:
1837:
1828:
1817:
1806:
1795:
1786:
1777:
1768:
1759:
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1737:
1728:
1717:
1708:
1697:
1686:
1675:
1666:
1657:
1646:
1635:
1624:
1611:
1602:
1591:
1580:
1569:
1558:
1545:
1536:
1525:
1514:
1503:
1492:
1460:hexagonal prism
1415:
1403:
1392:
1381:
1370:
1359:
1348:
1337:
1326:
1313:
1302:
1291:
1280:
1269:
1258:
1247:
1234:
1223:
1212:
1201:
1190:
1179:
1166:
1155:
1144:
1133:
1122:
1109:
1098:
1087:
1076:
1063:
1052:
1041:
1028:
1017:
1004:
994:
871:
857:
851:
844:
838:
830:
738:
730:prismatic cells
722:
716:
713:
707:
680:
649:
583:
570:
565:
564:
554:Euclidean space
530:
526:
514:
510:
506:
474:
466:
462:
428:
413:
409:
395:
382:
368:
367:
358:
341:
329:
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314:
309:
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299:
297:
277:
273:Schläfli symbol
226:
211:
197:
180:
167:
156:
155:
151:
150:
143:
132:
131:
123:
111:
106:
101:
96:
91:
86:
81:
79:
59:
55:Schläfli symbol
35:
34:Set of uniform
28:
23:
22:
15:
12:
11:
5:
3312:
3310:
3302:
3301:
3291:
3290:
3287:
3286:
3276:
3261:John H. Conway
3258:
3257:
3256:
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3216:
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3193:
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2655:
2652:
2643:
2617:
2610:
2607:
2591:
2586:
2576:
2568:
2560:
2559:k_22 polytopes
2557:
2542:
2539:
2529:
2507:
2504:
2439:
2390:
2345:
2303:
2264:The duoprisms
2237:as alternated
2197:
2194:
2165:of a rotating
2154:
2151:
2148:
2147:
2145:
2143:
2141:
2139:
2137:
2135:
2133:
2130:
2129:
2122:
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2108:
2101:
2094:
2087:
2080:
2073:
2066:
2059:
2052:
2044:
2043:
2038:
2033:
2028:
2021:
2020:
2016:
2015:
2013:
2011:
2009:
2007:
2005:
2003:
2001:
1998:
1997:
1990:
1983:
1976:
1969:
1962:
1955:
1948:
1941:
1934:
1927:
1920:
1912:
1911:
1908:
1905:
1900:
1894:
1893:
1872:
1868:
1865:
1862:
1861:
1850:
1841:
1830:
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1798:
1797:
1788:
1779:
1770:
1761:
1752:
1742:
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1730:
1721:
1710:
1701:
1690:
1678:
1677:
1668:
1659:
1650:
1639:
1628:
1616:
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1584:
1573:
1562:
1550:
1549:
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1472:
1471:
1455:
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1363:
1352:
1341:
1330:
1318:
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1273:
1262:
1251:
1239:
1238:
1227:
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1126:
1114:
1113:
1102:
1091:
1080:
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1056:
1045:
1033:
1032:
1021:
1009:
1008:
993:
990:
951:
950:
931:
888:and a regular
870:
867:
864:
863:
846:
829:
826:
802:
801:
791:
781:
755:-polygons and
737:
734:
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704:
703:
692:
687:
683:
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664:
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619:
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446:vertex-uniform
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393:
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244:vertex-uniform
237:
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135:-gonal prisms
121:
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77:
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57:
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26:
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14:
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3262:
3259:
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3246:0-486-40919-8
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2636:vertex figure
2633:
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2202:
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2017:
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1981:
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1142:
1138:
1136:
1131:
1127:
1125:
1120:
1116:
1115:
1112:
1107:
1103:
1101:
1096:
1092:
1090:
1085:
1081:
1079:
1074:
1070:
1069:
1066:
1061:
1057:
1055:
1050:
1046:
1044:
1039:
1035:
1034:
1031:
1026:
1022:
1020:
1015:
1011:
1007:
1002:
998:
991:
989:
987:
983:
979:
975:
970:
968:
964:
960:
956:
948:
944:
940:
936:
932:
929:
925:
921:
917:
913:
912:
911:
909:
906:
902:
898:
895:
891:
887:
883:
879:
877:
868:
861:
855:
850:
847:
842:
837:
834:
827:
825:
823:
822:product prism
819:
815:
811:
807:
799:
795:
792:
789:
785:
782:
779:
775:
772:
771:
770:
767:
764:
762:
758:
754:
749:
747:
743:
735:
733:
731:
727:
719:
710:
685:
681:
677:
671:
668:
665:
659:
654:
650:
646:
640:
637:
634:
620:
617:
614:
611:
608:
605:
602:
593:
588:
584:
580:
575:
571:
563:
562:
561:
559:
555:
551:
547:
543:
538:
537:) or higher.
536:
522:
518:
504:
500:
496:
492:
488:
480:
472:
459:
451:
447:
443:
440:
436:
433:
427:
425:
421:
417:
408:
406:
402:
398:
394:
390:
386:
381:
377:
372:
365:
361:
357:
353:
350:
345:
340:
336:
296:
294:
290:
285:
281:
276:
274:
270:
267:
264:
260:
257:
253:
248:
245:
241:
238:
234:
231:
225:
223:
219:
215:
209:
207:
203:
200:
195:
191:
189:
188:Vertex figure
185:
179:
175:
171:
166:
162:
148:
142:
138:
129:
122:
118:
78:
76:
72:
67:
63:
58:
56:
52:
49:
46:
42:
39:
31:
19:
3282:
3279:N.W. Johnson
3275:(Chapter 26)
3264:
3252:
3237:
3223:
3196:
3184:
3169:
3165:
2565:3-3 duoprism
2562:
2553:sphenocorona
2544:
2527:
2515:
2509:
2437:
2435:, s{2}s{2}.
2263:
2251:4-4 duoprism
2250:
2232:
2196:Duoantiprism
2189:
2185:
2182:
2179:n-n duoprism
2178:
2172:
2027:(tesseract)
1870:
1475:
1463:
1452:6-6 duoprism
1416:
977:
973:
971:
966:
962:
958:
954:
952:
938:
934:
927:
923:
919:
915:
907:
904:
896:
893:
889:
881:
874:
872:
859:
821:
810:double prism
809:
805:
803:
797:
793:
787:
783:
780:-gonal prism
777:
773:
768:
765:
760:
756:
752:
750:
745:
739:
736:Nomenclature
717:
708:
705:
541:
539:
520:
516:
494:
491:double prism
490:
484:
473:just like a
415:
396:
384:
370:
359:
343:
283:
279:
255:
213:
169:
65:
61:
33:
18:Duoantiprism
3146:Duocylinder
2660:Hyperbolic
2442:{5,2,5/3},
2247:alternation
2243:4-polytopes
2167:duocylinder
982:duocylinder
560:of points:
550:4-polytopes
471:duocylinder
3218:References
3191:965. Gudap
3178:Googlebook
3136:4-polytope
2657:Euclidean
2520:rectangles
2235:antiprisms
926:identical
438:Properties
432:duopyramid
236:Properties
230:duopyramid
199:disphenoid
3151:Tesseract
2773:¯
2732:~
2609:¯
2502:(gudap).
2350:tesseract
2255:tesseract
2233:Like the
1877:tesseract
947:tesseract
804:The term
678:∈
647:∈
581:×
544:exist in
542:duoprisms
38:duoprisms
3293:Category
3204:Archived
3132:Polytope
3126:See also
3083:−1
3074:∞
3071:∞
3042:∞
3039:103,680
3007:Symmetry
1448:6-prism
878:duoprism
860:cylinder
818:proprism
806:duoprism
499:polytope
495:duoprism
487:geometry
479:cylinder
405:Symmetry
392:Vertices
210:, order
206:Symmetry
177:Vertices
3234:Coxeter
2805:diagram
2803:Coxeter
2696:Coxeter
2654:Finite
2573:Coxeter
2516:octagon
2512:octagon
2440:0,1,2,3
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3271:
3267:2008,
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3047:Graph
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3157:Notes
3079:Name
3036:1440
3029:Order
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2041:10-10
2019:Even
1468:torus
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933:When
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3269:ISBN
3242:ISBN
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2563:The
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2208:, a
2173:The
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1907:7-7
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1787:7-7
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1729:6-7
1709:6-5
1676:5-8
1667:5-7
1658:5-6
1603:4-7
1537:3-7
1394:9-10
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1819:8-4
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1124:3-7
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1043:3-5
1030:4-4
1019:3-4
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3281::
3236:,
3226:,
3117:22
3109:22
3101:22
3093:22
3085:22
3023:]
3020:]
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3014:]
3011:]
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2587:22
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