458:
1434:
2112:
2119:
3052:
2098:
2091:
2077:
2070:
2056:
1994:
1987:
1973:
1966:
1952:
1945:
1931:
1924:
1917:
2126:
2105:
2084:
1980:
1938:
1959:
2216:
1400:
854:
1389:
1378:
1367:
1356:
1345:
1334:
1323:
1310:
1299:
1288:
1277:
1266:
1255:
1244:
1231:
1220:
1209:
1198:
1187:
1176:
1163:
1152:
1141:
1130:
1119:
1106:
1095:
1084:
1073:
1060:
1049:
1038:
1014:
1001:
194:
3066:
841:
3059:
2049:
1441:
1854:
1845:
1834:
1825:
1814:
1803:
1792:
1783:
1774:
1765:
1756:
1747:
1734:
1725:
1714:
1705:
1694:
1683:
1672:
1663:
1654:
1643:
1632:
1621:
1608:
1599:
1588:
1577:
1566:
1555:
1542:
1533:
1522:
1511:
1500:
1489:
2063:
1025:
2201:
2545:
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.
1476:
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.
3245:
3172:, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online:
2822:
2998:
2963:
2939:
2914:
2890:
2875:
2851:
2846:
2486:
2466:
2456:
2446:
2431:
2421:
2411:
2401:
2385:
2375:
2365:
2355:
2340:
2330:
2320:
2310:
2298:
2288:
2278:
2268:
320:
300:
102:
82:
2958:
2948:
2909:
2899:
2870:
2860:
2841:
2831:
2812:
2988:
2978:
2968:
2929:
2919:
2880:
2249:
operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the
330:
310:
112:
92:
2953:
2904:
2865:
2836:
2817:
2476:
1462:, projected into the plane by perspective, centered on a hexagonal face, looks like a double hexagon connected by (distorted)
3278:
3227:
3177:
2495:
2481:
2461:
2426:
2416:
2406:
2325:
2227:
2993:
2983:
2973:
2934:
2924:
2885:
2471:
2451:
2380:
2370:
2360:
2335:
2315:
2293:
2283:
2273:
325:
315:
305:
107:
97:
87:
3298:
3203:
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
2802:
3140:
930:-gonal prisms. For example, the Cartesian product of two triangles is a duoprism bounded by 6 triangular prisms.
2223:
2162:
845:
Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
2688:
2683:
2174:
1880:
2673:
2668:
2518:
of a p-gon can be clearly defined if one assumes that the octagon is the convex hull of two perpendicular
2246:
1433:
457:
292:
74:
461:
A close up inside the 23-29 duoprism projected onto a 3-sphere, and perspective projected to 3-space. As
3090:
2678:
2580:
740:
Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two
2720:
2534:
2491:
2761:
2597:
1393:
1382:
1371:
1360:
1349:
1338:
1327:
545:
478:
1314:
1303:
1292:
1281:
1270:
1259:
1248:
1224:
1202:
1167:
1156:
1145:
1134:
3260:
3248:(Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
2499:
2219:
875:
813:
265:
47:
3176:—contains a description of duoprisms (double prisms) and duocylinders (double cylinders).
272:
54:
759:-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: a
3268:
3241:
557:
502:
363:
146:
3006:
2631:
2238:
2157:
2111:
1480:
1425:
900:
835:
729:
449:
404:
348:
205:
127:
3207:
2547:
2209:
1459:
942:
741:
725:
553:
441:
239:
3051:
2118:
2090:
2069:
1993:
1972:
1951:
1930:
1916:
3114:
3106:
2584:
2125:
2040:
1404:
848:
445:
243:
2169:, divided into a checkerboard surface of squares from the {4,4|n} skew polyhedron
2097:
2076:
2055:
1986:
1965:
1944:
1923:
3292:
3098:
2695:
2635:
2205:
423:
221:
187:
2104:
2083:
1979:
1937:
858:
The two sets of 16-gonal prisms are shown. The top and bottom faces of the vertical
724:
are the sets of the points contained in the respective polygons. Such a duoprism is
3082:
2564:
2552:
2035:
2030:
2024:
1958:
1902:
1897:
1858:
1838:
1818:
1807:
1738:
1718:
1698:
1687:
1647:
1636:
1625:
1612:
1592:
1581:
1570:
1559:
1546:
1526:
1515:
1504:
1493:
1451:
1235:
1213:
1191:
1180:
1123:
1110:
1099:
1088:
1077:
1064:
1053:
1042:
1029:
1018:
1005:
2215:
1399:
853:
3145:
3028:
2242:
2166:
1388:
1377:
1366:
1355:
1344:
1333:
1322:
981:
505:
of two polytopes, each of two dimensions or higher. The
Cartesian product of an
470:
2510:
Also related are the ditetragoltriates or octagoltriates, formed by taking the
1309:
1298:
1287:
1276:
1265:
1254:
1243:
1230:
1219:
1208:
1197:
1186:
1175:
1162:
1151:
1140:
1129:
1118:
1105:
1094:
1083:
1072:
1059:
1048:
1037:
1013:
1000:
193:
3188:
3135:
549:
431:
229:
198:
3200:
696:{\displaystyle P_{1}\times P_{2}=\{(x,y,z,w)|(x,y)\in P_{1},(z,w)\in P_{2}\}}
3150:
2519:
2349:
2254:
2234:
1876:
946:
840:
17:
3065:
2200:
3058:
2048:
1853:
1844:
1833:
1824:
1813:
1802:
1791:
1782:
1773:
1764:
1755:
1746:
1733:
1724:
1713:
1704:
1693:
1682:
1671:
1662:
1653:
1642:
1631:
1620:
1607:
1598:
1587:
1576:
1565:
1554:
1541:
1532:
1521:
1510:
1499:
1488:
1440:
3131:
941:
are identically 4, the resulting duoprism is bounded by 8 square prisms (
817:
498:
486:
3233:
2572:
2511:
2394:
2258:
1024:
985:
885:
534:
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
2062:
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.
2523:
2533:
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.
2778:
2767:
2766:
2763:
2737:
2726:
2725:
2722:
2614:
2603:
2602:
2599:
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
25:
3064:
3057:
3050:
2996:
2991:
2986:
2981:
2976:
2971:
2966:
2961:
2956:
2951:
2946:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2902:
2897:
2888:
2883:
2878:
2873:
2868:
2863:
2858:
2849:
2844:
2839:
2834:
2829:
2820:
2815:
2810:
2746:{\displaystyle {\tilde {E}}_{6}}
2484:
2479:
2474:
2469:
2464:
2459:
2454:
2449:
2444:
2429:
2424:
2419:
2414:
2409:
2404:
2399:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2306:{p,2,q}, can be alternated into
2296:
2291:
2286:
2281:
2276:
2271:
2266:
2124:
2117:
2110:
2103:
2096:
2089:
2082:
2075:
2068:
2061:
2054:
2047:
1992:
1985:
1978:
1971:
1964:
1957:
1950:
1943:
1936:
1929:
1922:
1915:
1852:
1843:
1832:
1823:
1812:
1801:
1790:
1781:
1772:
1763:
1754:
1745:
1732:
1723:
1712:
1703:
1692:
1681:
1670:
1661:
1652:
1641:
1630:
1619:
1606:
1597:
1586:
1575:
1564:
1553:
1540:
1531:
1520:
1509:
1498:
1487:
1439:
1432:
1398:
1387:
1376:
1365:
1354:
1343:
1332:
1321:
1308:
1297:
1286:
1275:
1264:
1253:
1242:
1229:
1218:
1207:
1196:
1185:
1174:
1161:
1150:
1139:
1128:
1117:
1104:
1093:
1082:
1071:
1058:
1047:
1036:
1023:
1012:
999:
852:
839:
328:
323:
318:
313:
308:
303:
298:
192:
110:
105:
100:
95:
90:
85:
80:
2496:pentagrammic crossed-antiprisms
437:
422:
403:
391:
378:
354:
337:
291:
271:
261:
235:
220:
204:
186:
176:
163:
139:
119:
73:
53:
43:
2787:{\displaystyle {\bar {T}}_{7}}
2772:
2731:
2623:{\displaystyle {\bar {T}}_{7}}
2608:
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:
1551:
1485:
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
3041:
2653:
2144:
2140:
2136:
2132:
2039:
2034:
2029:
2023:
2012:
2008:
2004:
2000:
1909:
1906:
1901:
1896:
1457:
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
3211:
3198:
3192:
3186:
3180:
3167:
3068:
3061:
3054:
3001:
3000:
2999:
2995:
2994:
2990:
2989:
2985:
2984:
2980:
2979:
2975:
2974:
2970:
2969:
2965:
2964:
2960:
2959:
2955:
2954:
2950:
2949:
2942:
2941:
2940:
2936:
2935:
2931:
2930:
2926:
2925:
2921:
2920:
2916:
2915:
2911:
2910:
2906:
2905:
2901:
2900:
2893:
2892:
2891:
2887:
2886:
2882:
2881:
2877:
2876:
2872:
2871:
2867:
2866:
2862:
2861:
2854:
2853:
2852:
2848:
2847:
2843:
2842:
2838:
2837:
2833:
2832:
2825:
2824:
2823:
2819:
2818:
2814:
2813:
2793:
2791:
2790:
2785:
2783:
2782:
2777:
2776:
2768:
2752:
2750:
2749:
2744:
2742:
2741:
2736:
2735:
2727:
2648:
2632:uniform polytope
2629:
2627:
2626:
2621:
2619:
2618:
2613:
2612:
2604:
2489:
2488:
2487:
2483:
2482:
2478:
2477:
2473:
2472:
2468:
2467:
2463:
2462:
2458:
2457:
2453:
2452:
2448:
2447:
2434:
2433:
2432:
2428:
2427:
2423:
2422:
2418:
2417:
2413:
2412:
2408:
2407:
2403:
2402:
2388:
2387:
2386:
2382:
2381:
2377:
2376:
2372:
2371:
2367:
2366:
2362:
2361:
2357:
2356:
2343:
2342:
2341:
2337:
2336:
2332:
2331:
2327:
2326:
2322:
2321:
2317:
2316:
2312:
2311:
2301:
2300:
2299:
2295:
2294:
2290:
2289:
2285:
2284:
2280:
2279:
2275:
2274:
2270:
2269:
2128:
2121:
2114:
2107:
2100:
2093:
2086:
2079:
2072:
2065:
2058:
2051:
1996:
1989:
1982:
1975:
1968:
1961:
1954:
1947:
1940:
1933:
1926:
1919:
1889:
1856:
1847:
1836:
1827:
1816:
1805:
1794:
1785:
1776:
1767:
1758:
1749:
1736:
1727:
1716:
1707:
1696:
1685:
1674:
1665:
1656:
1645:
1634:
1623:
1610:
1601:
1590:
1579:
1568:
1557:
1544:
1535:
1524:
1513:
1502:
1491:
1484:
1443:
1436:
1429:
1402:
1391:
1380:
1369:
1358:
1347:
1336:
1325:
1312:
1301:
1290:
1279:
1268:
1257:
1246:
1233:
1222:
1211:
1200:
1189:
1178:
1165:
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:
316:
312:
311:
307:
306:
302:
301:
287:
228:
216:
196:
182:
172:
158:
153:
145:
134:
125:
115:
114:
113:
109:
108:
104:
103:
99:
98:
94:
93:
89:
88:
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:
3102:
3094:
3086:
2997:
2992:
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2945:
2938:
2933:
2928:
2923:
2918:
2913:
2908:
2903:
2898:
2896:
2889:
2884:
2879:
2874:
2869:
2864:
2859:
2857:
2850:
2845:
2840:
2835:
2830:
2828:
2821:
2816:
2811:
2809:
2804:
2797:
2765:
2760:
2759:
2756:
2724:
2719:
2718:
2715:
2709:
2705:
2697:
2645:
2601:
2596:
2595:
2593:
2588:
2578:
2570:
2561:
2548:grand antiprism
2543:
2531:
2508:
2485:
2480:
2475:
2470:
2465:
2460:
2455:
2450:
2445:
2443:
2441:
2430:
2425:
2420:
2415:
2410:
2405:
2400:
2398:
2392:
2384:
2379:
2374:
2369:
2364:
2359:
2354:
2352:
2347:
2339:
2334:
2329:
2324:
2319:
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:
1750:
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:
324:
319:
314:
309:
304:
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:
3231:
3219:
3216:
3213:
3212:
3193:
3181:
3161:
3160:
3158:
3155:
3154:
3153:
3148:
3143:
3138:
3127:
3124:
3121:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3080:
3076:
3075:
3072:
3069:
3062:
3055:
3048:
3044:
3043:
3040:
3037:
3034:
3031:
3025:
3024:
3021:
3018:
3015:
3012:
3009:
3003:
3002:
2943:
2894:
2855:
2826:
2807:
2799:
2798:
2795:
2781:
2774:
2771:
2757:
2754:
2740:
2733:
2730:
2716:
2713:
2710:
2707:
2703:
2700:
2692:
2691:
2686:
2681:
2676:
2671:
2666:
2662:
2661:
2658:
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:
2115:
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:
1821:
1810:
1798:
1797:
1788:
1779:
1770:
1761:
1752:
1742:
1741:
1730:
1721:
1710:
1701:
1690:
1678:
1677:
1668:
1659:
1650:
1639:
1628:
1616:
1615:
1604:
1595:
1584:
1573:
1562:
1550:
1549:
1538:
1529:
1518:
1507:
1496:
1472:
1471:
1455:
1454:
1449:
1445:
1444:
1437:
1414:
1411:
1408:
1407:
1396:
1385:
1374:
1363:
1352:
1341:
1330:
1318:
1317:
1306:
1295:
1284:
1273:
1262:
1251:
1239:
1238:
1227:
1216:
1205:
1194:
1183:
1171:
1170:
1159:
1148:
1137:
1126:
1114:
1113:
1102:
1091:
1080:
1068:
1067:
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:
720:
711:
704:
703:
692:
687:
683:
679:
676:
673:
670:
667:
664:
661:
656:
652:
648:
645:
642:
639:
636:
633:
629:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
590:
586:
582:
577:
573:
453:
452:
446:vertex-uniform
439:
435:
434:
426:
420:
419:
407:
401:
400:
393:
389:
388:
380:
376:
375:
356:
352:
351:
339:
335:
334:
295:
289:
288:
275:
269:
268:
263:
259:
258:
252:
251:
247:
246:
244:vertex-uniform
237:
233:
232:
224:
218:
217:
208:
202:
201:
190:
184:
183:
178:
174:
173:
165:
161:
160:
141:
137:
136:
135:-gonal prisms
121:
117:
116:
77:
71:
70:
57:
51:
50:
45:
41:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3311:
3300:
3297:
3296:
3294:
3284:
3280:
3277:
3274:
3270:
3266:
3262:
3259:
3254:
3250:
3249:
3247:
3246:0-486-40919-8
3243:
3239:
3235:
3232:
3229:
3225:
3222:
3221:
3217:
3209:
3205:
3202:
3197:
3194:
3190:
3185:
3182:
3179:
3175:
3171:
3166:
3163:
3156:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3133:
3130:
3129:
3125:
3119:
3113:
3111:
3105:
3103:
3097:
3095:
3089:
3087:
3081:
3078:
3077:
3073:
3070:
3067:
3063:
3060:
3056:
3053:
3049:
3046:
3045:
3038:
3035:
3032:
3030:
3027:
3026:
3022:
3019:
3016:
3013:
3010:
3008:
3005:
3004:
2944:
2895:
2856:
2827:
2808:
2806:
2801:
2800:
2779:
2769:
2758:
2738:
2728:
2717:
2711:
2701:
2699:
2694:
2693:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2664:
2663:
2659:
2656:
2650:
2649:
2639:
2637:
2636:vertex figure
2633:
2615:
2605:
2589:
2582:
2574:
2566:
2558:
2556:
2554:
2549:
2540:
2538:
2536:
2532:
2525:
2521:
2517:
2513:
2505:
2503:
2501:
2497:
2493:
2436:
2396:
2351:
2262:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2229:
2225:
2221:
2217:
2211:
2207:
2206:vertex figure
2202:
2195:
2193:
2191:
2187:
2184:
2180:
2176:
2168:
2164:
2159:
2152:
2146:
2142:
2138:
2134:
2127:
2123:
2120:
2116:
2113:
2109:
2106:
2102:
2099:
2095:
2092:
2088:
2085:
2081:
2078:
2074:
2071:
2067:
2064:
2060:
2057:
2053:
2050:
2046:
2045:
2042:
2037:
2032:
2026:
2022:
2017:
2014:
2010:
2006:
2002:
1995:
1991:
1988:
1984:
1981:
1977:
1974:
1970:
1967:
1963:
1960:
1956:
1953:
1949:
1946:
1942:
1939:
1935:
1932:
1928:
1925:
1921:
1918:
1914:
1913:
1904:
1899:
1895:
1890:
1884:
1882:
1878:
1866:
1860:
1855:
1851:
1846:
1842:
1840:
1835:
1831:
1826:
1822:
1820:
1815:
1811:
1809:
1804:
1800:
1793:
1789:
1784:
1780:
1775:
1771:
1766:
1762:
1757:
1753:
1748:
1744:
1740:
1735:
1731:
1726:
1722:
1720:
1715:
1711:
1706:
1702:
1700:
1695:
1691:
1689:
1684:
1680:
1673:
1669:
1664:
1660:
1655:
1651:
1649:
1644:
1640:
1638:
1633:
1629:
1627:
1622:
1618:
1614:
1609:
1605:
1600:
1596:
1594:
1589:
1585:
1583:
1578:
1574:
1572:
1567:
1563:
1561:
1556:
1552:
1548:
1543:
1539:
1534:
1530:
1528:
1523:
1519:
1517:
1512:
1508:
1506:
1501:
1497:
1495:
1490:
1486:
1482:
1477:
1469:
1465:
1461:
1456:
1453:
1450:
1447:
1446:
1442:
1438:
1435:
1431:
1430:
1427:
1422:
1420:
1412:
1406:
1401:
1397:
1395:
1390:
1386:
1384:
1379:
1375:
1373:
1368:
1364:
1362:
1357:
1353:
1351:
1346:
1342:
1340:
1335:
1331:
1329:
1324:
1320:
1319:
1316:
1311:
1307:
1305:
1300:
1296:
1294:
1289:
1285:
1283:
1278:
1274:
1272:
1267:
1263:
1261:
1256:
1252:
1250:
1245:
1241:
1240:
1237:
1232:
1228:
1226:
1221:
1217:
1215:
1210:
1206:
1204:
1199:
1195:
1193:
1188:
1184:
1182:
1177:
1173:
1172:
1169:
1164:
1160:
1158:
1153:
1149:
1147:
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:
18:3-8 duoprism
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
2395:16-cell
2391:0,1,2,3
2346:0,1,2,3
2304:0,1,2,3
2259:16-cell
2190:regular
1464:squares
986:quadric
899:-gonal
886:polygon
884:-sided
876:uniform
796:-gonal-
786:-gonal-
776:-gonal-
535:polygon
364:squares
347:-gonal
250:
147:squares
126:-gonal
3271:
3267:2008,
3244:
3047:Graph
2651:Space
2239:prisms
901:prisms
726:convex
706:where
442:convex
412:order
374:-gons
349:prisms
240:convex
159:-gons
154:-gons,
128:prisms
3157:Notes
3079:Name
3036:1440
3029:Order
2698:group
2526:with
2494:, 10
2041:10-10
2019:Even
1468:torus
1419:torus
1405:10-10
943:cubes
933:When
914:When
497:is a
379:Edges
355:Faces
338:Cells
164:Edges
140:Faces
120:Cells
3269:ISBN
3242:ISBN
3134:and
2575:as k
2567:, -1
2563:The
2524:cube
2344:, ht
2208:, a
2173:The
1910:9-9
1907:7-7
1892:Odd
1849:8-7
1829:8-5
1796:7-8
1787:7-7
1778:7-6
1769:7-5
1760:7-4
1751:7-3
1729:6-7
1709:6-5
1676:5-8
1667:5-7
1658:5-6
1603:4-7
1537:3-7
1394:9-10
1383:8-10
1372:7-10
1361:6-10
1350:5-10
1339:4-10
1328:3-10
992:Nets
976:and
953:The
937:and
918:and
903:and
820:for
715:and
529:and
465:and
424:Dual
262:Type
222:Dual
44:Type
3033:72
2389:, t
2302:, t
2192:.)
2036:8-8
2031:6-6
2025:4-4
1903:5-5
1898:3-3
1883:.
1859:8-8
1839:8-6
1819:8-4
1808:8-3
1739:6-8
1719:6-6
1699:6-4
1688:6-3
1648:5-5
1637:5-4
1626:5-3
1613:4-8
1593:4-6
1582:4-5
1571:4-4
1560:4-3
1547:3-8
1527:3-6
1516:3-5
1505:3-4
1494:3-3
1315:9-9
1304:8-9
1293:7-9
1282:6-9
1271:5-9
1260:4-9
1249:3-9
1236:8-8
1225:7-8
1214:6-8
1203:5-8
1192:4-8
1181:3-8
1168:7-7
1157:6-7
1146:5-7
1135:4-7
1124:3-7
1111:6-6
1100:5-6
1089:4-6
1078:3-6
1065:5-5
1054:4-5
1043:3-5
1030:4-4
1019:3-4
1006:3-3
972:As
849:net
558:set
548:as
493:or
485:In
429:p-p
410:= ,
371:p p
344:p p
282:}Ă—{
227:p-q
157:q p
152:p q
133:q p
124:p q
64:}Ă—{
36:p-q
3295::
3281::
3236:,
3226:,
3117:22
3109:22
3101:22
3093:22
3085:22
3023:]
3020:]
3017:]
3014:]
3011:]
2794:=E
2753:=E
2665:n
2644:22
2638:.
2592:22
2587:22
2577:22
2569:22
2555:.
2537:.
2530:2d
2397:,
2222:,
2161:A
1458:A
812:.
748:.
732:.
448:,
444:,
286:}
242:,
214:pq
181:pq
170:pq
144:pq
68:}
3115:3
3107:2
3099:1
3091:0
2796:6
2780:7
2770:T
2755:6
2739:6
2729:E
2714:6
2712:E
2708:2
2706:A
2704:2
2702:A
2689:8
2684:7
2679:6
2674:5
2669:4
2642:k
2616:7
2606:T
2585:2
2528:D
2253:(
2186:n
2183:n
1873:3
978:n
974:m
967:n
963:n
959:m
955:m
949:.
939:n
935:m
928:n
924:n
920:n
916:m
908:n
905:m
897:m
894:n
890:m
882:n
798:p
794:q
788:p
784:q
778:p
774:q
757:m
753:n
721:2
718:P
712:1
709:P
691:}
686:2
682:P
675:)
672:w
669:,
666:z
663:(
660:,
655:1
651:P
644:)
641:y
638:,
635:x
632:(
628:|
624:)
621:w
618:,
615:z
612:,
609:y
606:,
603:x
600:(
597:{
594:=
589:2
585:P
576:1
572:P
531:m
527:n
523:)
521:m
519:+
517:n
515:(
511:m
507:n
481:.
475:p
467:n
463:m
416:p
414:8
397:p
385:p
383:2
369:2
366:,
360:p
342:2
284:p
280:p
278:{
212:4
168:2
149:,
130:,
66:q
62:p
60:{
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
