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The volume of any prism is the product of the area of the base and the distance between the two bases. In the case of a triangular prism, its base is a triangle, so its volume can be calculated by multiplying the area of a triangle and the length of the prism:
542:
its part at an oblique angle. As a result, the two bases are not parallel and every height has a different edge length. If the edges connecting bases are perpendicular to one of its bases, the prism is called a
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is another polyhedron constructed from a triangular prism with equilateral triangle bases. This way, one of its bases rotates around the prism's centerline and breaks the square faces into
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onto the base of a triangular prism. The augmented triangular prism, biaugmented triangular prism, and triaugmented triangular prism are constructed by attaching
370:
228:. A semiregular prism means that the number of its polygonal base's edges equals the number of its square faces. More generally, the triangular prism is
671:. Each square face can be re-triangulated with two triangles to form a non-convex dihedral angle. As a result, the Schönhardt polyhedron cannot be
771:
775:
675:
by a partition into tetrahedra. It is also that the Schönhardt polyhedron has no internal diagonals. It is named after German mathematician
583:
523:
onto the square face of the prism. The gyrobifastigium is constructed by attaching two triangular prisms along one of its square faces.
271:. The triangular bipyramid has the same symmetry as the triangular prism. The dihedral angle between two adjacent square faces is the
779:
763:
743:
767:
187:. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3
751:
355:
is the distance between the triangular faces. In the case of a right triangular prism, where all its edges are equal in length
3912:
3885:
3804:
3775:
3746:
3644:
3616:
3492:
Berman, Leah Wrenn; Williams, Gordon (2009). "Exploring
Polyhedra and Discovering Euler's Formula". In Hopkin, Brian (ed.).
259:
of order 12: the appearance is unchanged if the triangular prism is rotated one- and two- thirds of a full angle around its
747:
728:
148:
with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a
3497:
720:
716:
304:
3279:
3171:
475:
Beyond the triangular bipyramid as its dual polyhedron, many other polyhedrons are related to the triangular prism. A
724:
99:
3937:
241:
3663:
755:
479:
is a convex polyhedron with regular faces, and this definition is sometimes omitted uniform polyhedrons such as
759:
496:
3155:
3932:
3443:
512:
2024:
1935:
508:
492:
3098:
2196:
515:. The elongated triangular pyramid and the gyroelongated triangular pyramid are constructed by attaching
179:
A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its
1075:
520:
504:
988:
2067:
20:
1034:
2579:
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672:
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225:
153:
50:
3494:
Resources for
Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles
3308:
3250:
3200:
3181:
3130:
3111:
163:
The triangular prism can be used in constructing another polyhedron. Examples are some of the
3823:
3599:
King, Robert B. (1994). "Polyhedral
Dynamics". In Bonchev, Danail D.; Mekenyan, O.G. (eds.).
3317:
3289:
3259:
3165:
2450:
2407:
2321:
2110:
909:
889:
840:
694:
539:
3868:
Todesco, Gian Marco (2020). "Hyperbolic
Honeycomb". In Emmer, Michele; Abate, Marco (eds.).
361:, its volume can be calculated as the product of the equilateral triangle's area and length
3861:
3686:
3555:
3526:
2854:
2536:
819:
791:
268:
217:
124:
3710:
Messer, Peter W. (2002). "Closed-Form
Expressions for Uniform Polyhedra and Their Duals".
3702:
191:
as other faces. If the prism's edges are perpendicular to the base, the lateral faces are
8:
3636:
3626:
2940:
2682:
1806:
348:
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3738:
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3742:
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2364:
1849:
1720:
491:. There are 6 Johnson solids with their construction involving the triangular prism:
480:
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89:
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3763:
3719:
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679:, who described it in 1928, although the related structure was exhibited by artist
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145:
46:
3857:
3786:
3757:
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264:
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passing through the center's base, and reflecting across a horizontal plane. The
237:
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79:
61:
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2016:
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3591:
688:
668:
526:
3534:
Bezdek, Andras; Carrigan, Braxton (2016). "On nontriangulable polyhedra".
2263:
2009:
1549:
516:
431:
2256:
3832:
3788:
Convex
Polyhedra with Regularity Conditions and Hilbert's Third Problem
3464:
1995:
1705:
The triangular prism exists as cells of a number of four-dimensional
815:
739:
There are 9 uniform honeycombs that include triangular prism cells:
488:
192:
3456:
2002:
1981:
421:{\displaystyle {\frac {\sqrt {3}}{2}}l^{2}\cdot l\approx 0.433l^{3}}
3078:
3064:
1988:
1581:
704:
207:
184:
133:
67:
35:
3071:
3057:
3050:
3036:
553:
is the area of the triangular prism's base, and the three heights
3840:
Skilling, John (1976), "Uniform
Compounds of Uniform Polyhedra",
3043:
3029:
2646:
2632:
827:
811:
2639:
2625:
3842:
Mathematical
Proceedings of the Cambridge Philosophical Society
3818:
3472:
Bansod, Yogesh Deepak; Nandanwar, Deepesh; Burša, Jiřà (2014).
1588:
823:
72:
2674:
2660:
2667:
2653:
830:'s notation the triangular prism is given the symbol −1
715:
There are 4 uniform compounds of triangular prisms. They are
3791:. Texts and Readings in Mathematics. Hindustan Book Agency.
1616:
1609:
1602:
3505:
Berman, Martin (1971). "Regular-faced convex polyhedra".
790:
The triangular prism is first in a dimensional series of
580:, its volume can be determined in the following formula:
199:. This prism may also be considered a special case of a
3819:"Ăśber die Zerlegung von Dreieckspolyedern in Tetraeder"
3347:
3225:
3143:
1037:
991:
586:
373:
307:
3625:
3471:
3415:
3407:
3381:
3246:
3601:
3894:
3629:; Moore, Teresa E.; Prassidis, Efstratios (2011).
3313:
3267:
3213:
3103:
3101:
1715:Four dimensional polytopes with triangular prisms
1059:
1013:
647:{\displaystyle {\frac {A(h_{1}+h_{2}+h_{3})}{3}}.}
646:
420:
331:
3755:
3285:
806:identified this series in 1900 as containing all
3924:
3730:
3177:
470:
3870:Imagine Math 7: Between Culture and Mathematics
430:The triangular prism can be represented as the
3533:
3491:
3474:"Overview of tensegrity – I: Basic structures"
3373:
3196:
772:rhombitriangular-hexagonal prismatic honeycomb
286:, and that between a square and a triangle is
3756:Pisanski, TomaĹľ; Servatius, Brigitte (2013).
776:snub triangular-hexagonal prismatic honeycomb
167:, the truncated right triangular prism, and
343:is the length of one side of the triangle,
3895:Williams, Kim; Monteleone, Cosino (2021).
3813:
3731:O'Keeffe, Michael; Hyde, Bruce G. (2020).
3582:Kern, William F.; Bland, James R. (1938).
3402:
3368:
183:, and the bases of a triangular prism are
34:
3734:Crystal Structures: Patterns and Symmetry
3676:
3581:
3353:
3231:
3149:
1700:
826:in the case of the triangular prism). In
232:. This means that a triangular prism has
3839:
3759:Configuration from a Graphical Viewpoint
3437:
3421:
3387:
780:elongated triangular prismatic honeycomb
764:triangular-hexagonal prismatic honeycomb
744:Gyroelongated alternated cubic honeycomb
687:
655:
525:
212:3D model of a (uniform) triangular prism
205:
3867:
3784:
3653:
3441:(1948). "On indecomposable polyhedra".
3334:
3304:
3219:
768:truncated hexagonal prismatic honeycomb
152:. A right triangular prism may be both
3925:
3709:
3504:
3339:
3273:
3205:
3135:
752:gyrated triangular prismatic honeycomb
3897:Daniele Barbaro's Perspective of 1568
3659:"Convex polyhedra with regular faces"
538:is a triangular prism constructed by
465:
224:, then the right triangular prism is
3598:
3570:
3408:Bansod, Nandanwar & Burša (2014)
3255:
3247:Kinsey, Moore & Prassidis (2011)
3161:
3126:
3107:
785:
748:elongated alternated cubic honeycomb
729:compound of twenty triangular prisms
3498:Mathematical Association of America
721:compound of eight triangular prisms
13:
3536:Beiträge zur Algebra und Geometrie
717:compound of four triangular prisms
442:. More generally, the prism graph
14:
3949:
3507:Journal of the Franklin Institute
725:compound of ten triangular prisms
332:{\displaystyle {\frac {bhl}{2}},}
3314:Williams & Monteleone (2021)
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1103:
1098:
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1088:
1083:
1014:{\displaystyle {\tilde {E}}_{8}}
545:truncated right triangular prism
530:Truncated right triangular prism
242:three-dimensional symmetry group
3664:Canadian Journal of Mathematics
3430:
3393:
3359:
3325:
3295:
3286:Pisanski & Servatius (2013)
756:snub square prismatic honeycomb
244:of a right triangular prism is
206:
3237:
3187:
3117:
1060:{\displaystyle {\bar {T}}_{8}}
1045:
999:
760:triangular prismatic honeycomb
632:
593:
497:elongated triangular bipyramid
1:
3584:Solid Mensuration with proofs
3444:American Mathematical Monthly
3087:
2025:Rhomb-icosidodecahedral prism
734:
703:with a triangular prism as a
513:triaugmented triangular prism
471:In construction of polyhedron
174:
3519:10.1016/0016-0032(71)90071-8
3374:Bezdek & Carrigan (2016)
3197:Berman & Williams (2009)
3092:
3026:
2680:
2622:
2276:
2239:
2022:
1978:
1936:Truncated dodecahedral prism
1718:
509:biaugmented triangular prism
493:elongated triangular pyramid
195:, and the prism is called a
7:
2197:n-gonal antiprismatic prism
521:equilateral square pyramids
275:of an equilateral triangle
267:of a triangular prism is a
19:For the optical prism, see
10:
3954:
3178:O'Keeffe & Hyde (2020)
2068:Rhombi-cuboctahedral prism
1714:
839:
802:of the previous polytope.
536:truncated triangular prism
505:augmented triangular prism
240:symmetry on vertices. The
220:and the lateral faces are
18:
3905:10.1007/978-3-030-76687-0
3878:10.1007/978-3-030-42653-8
3854:10.1017/S0305004100052440
3797:10.1007/978-93-86279-06-4
3768:10.1007/978-0-8176-8364-1
3724:10.1007/s00454-001-0078-2
3609:10.1007/978-94-011-1202-4
3548:10.1007/s13366-015-0248-4
1571:
854:
118:
98:
88:
78:
60:
42:
33:
28:
21:Triangular prism (optics)
16:Prism with a 3-sided base
3712:Discrete Comput Geometry
2580:Runcitruncated tesseract
2494:Cantitruncated tesseract
351:drawn to that side, and
3938:Space-filling polyhedra
3785:Rajwade, A. R. (2001).
3354:Kern & Bland (1938)
3232:Kern & Bland (1938)
3150:Kern & Bland (1938)
2984:Runcitruncated 120-cell
2898:Cantitruncated 120-cell
2154:Snub dodecahedral prism
1893:Icosidodecahedral prism
810:facets, containing all
3678:10.4153/cjm-1966-021-8
2812:Runcitruncated 24-cell
2726:Cantitruncated 24-cell
1701:Four dimensional space
1061:
1015:
712:
661:
648:
531:
422:
333:
213:
197:right triangular prism
150:right triangular prism
51:Semiregular polyhedron
3824:Mathematische Annalen
3637:John Wiley & Sons
3632:Geometry and Symmetry
3577:. Ginn & Company.
3571:Haul, Wm. S. (1893).
3481:Engineering Mechanics
2451:Cantellated tesseract
2408:Runcitruncated 5-cell
2322:Cantitruncated 5-cell
2111:Truncated cubic prism
1062:
1016:
820:equilateral triangles
792:semiregular polytopes
691:
665:Schönhardt polyhedron
660:Schönhardt polyhedron
659:
649:
529:
423:
334:
211:
169:Schönhardt polyhedron
3933:Prismatoid polyhedra
3627:Kinsey, L. Christine
2855:Cantellated 120-cell
2537:Runcinated tesseract
1035:
989:
697:triangular antiprism
584:
371:
347:is the length of an
305:
269:triangular bipyramid
125:Triangular bipyramid
2941:Runcinated 120-cell
2683:Cantellated 24-cell
1807:Cuboctahedral prism
1707:uniform 4-polytopes
794:. Each progressive
709:isosceles triangles
3833:10.1007/BF01451597
3739:Dover Publications
3655:Johnson, Norman W.
2769:Runcinated 24-cell
2279:Cantellated 5-cell
1057:
1011:
713:
701:vertex arrangement
662:
644:
532:
481:Archimedean solids
466:Related polyhedron
418:
329:
214:
55:Uniform polyhedron
3914:978-3-030-76687-0
3887:978-3-030-42653-8
3806:978-93-86279-06-4
3777:978-0-8176-8363-4
3748:978-0-486-83654-6
3646:978-0-470-49949-8
3618:978-94-011-1202-4
3403:Schönhardt (1928)
3369:Schönhardt (1928)
3085:
3084:
2365:Runcinated 5-cell
1850:Icosahedral prism
1721:Tetrahedral prism
1698:
1697:
1048:
1002:
786:Related polytopes
639:
384:
380:
324:
130:
129:
3945:
3918:
3891:
3864:
3836:
3810:
3781:
3752:
3727:
3706:
3680:
3650:
3622:
3595:
3578:
3567:
3530:
3501:
3488:
3478:
3468:
3425:
3419:
3413:
3397:
3391:
3385:
3379:
3363:
3357:
3351:
3345:
3329:
3323:
3299:
3293:
3283:
3277:
3271:
3265:
3241:
3235:
3229:
3223:
3217:
3211:
3191:
3185:
3175:
3169:
3159:
3153:
3147:
3141:
3121:
3115:
3105:
3081:
3074:
3067:
3060:
3053:
3046:
3039:
3032:
3023:
3022:
3021:
3017:
3016:
3012:
3011:
3007:
3006:
3002:
3001:
2997:
2996:
2992:
2991:
2980:
2979:
2978:
2974:
2973:
2969:
2968:
2964:
2963:
2959:
2958:
2954:
2953:
2949:
2948:
2937:
2936:
2935:
2931:
2930:
2926:
2925:
2921:
2920:
2916:
2915:
2911:
2910:
2906:
2905:
2894:
2893:
2892:
2888:
2887:
2883:
2882:
2878:
2877:
2873:
2872:
2868:
2867:
2863:
2862:
2851:
2850:
2849:
2845:
2844:
2840:
2839:
2835:
2834:
2830:
2829:
2825:
2824:
2820:
2819:
2808:
2807:
2806:
2802:
2801:
2797:
2796:
2792:
2791:
2787:
2786:
2782:
2781:
2777:
2776:
2765:
2764:
2763:
2759:
2758:
2754:
2753:
2749:
2748:
2744:
2743:
2739:
2738:
2734:
2733:
2722:
2721:
2720:
2716:
2715:
2711:
2710:
2706:
2705:
2701:
2700:
2696:
2695:
2691:
2690:
2677:
2670:
2663:
2656:
2649:
2642:
2635:
2628:
2619:
2618:
2617:
2613:
2612:
2608:
2607:
2603:
2602:
2598:
2597:
2593:
2592:
2588:
2587:
2576:
2575:
2574:
2570:
2569:
2565:
2564:
2560:
2559:
2555:
2554:
2550:
2549:
2545:
2544:
2533:
2532:
2531:
2527:
2526:
2522:
2521:
2517:
2516:
2512:
2511:
2507:
2506:
2502:
2501:
2490:
2489:
2488:
2484:
2483:
2479:
2478:
2474:
2473:
2469:
2468:
2464:
2463:
2459:
2458:
2447:
2446:
2445:
2441:
2440:
2436:
2435:
2431:
2430:
2426:
2425:
2421:
2420:
2416:
2415:
2404:
2403:
2402:
2398:
2397:
2393:
2392:
2388:
2387:
2383:
2382:
2378:
2377:
2373:
2372:
2361:
2360:
2359:
2355:
2354:
2350:
2349:
2345:
2344:
2340:
2339:
2335:
2334:
2330:
2329:
2318:
2317:
2316:
2312:
2311:
2307:
2306:
2302:
2301:
2297:
2296:
2292:
2291:
2287:
2286:
2273:
2266:
2259:
2252:
2245:
2236:
2235:
2234:
2230:
2229:
2225:
2224:
2220:
2219:
2215:
2214:
2210:
2209:
2205:
2204:
2193:
2192:
2191:
2187:
2186:
2182:
2181:
2177:
2176:
2172:
2171:
2167:
2166:
2162:
2161:
2150:
2149:
2148:
2144:
2143:
2139:
2138:
2134:
2133:
2129:
2128:
2124:
2123:
2119:
2118:
2107:
2106:
2105:
2101:
2100:
2096:
2095:
2091:
2090:
2086:
2085:
2081:
2080:
2076:
2075:
2064:
2063:
2062:
2058:
2057:
2053:
2052:
2048:
2047:
2043:
2042:
2038:
2037:
2033:
2032:
2019:
2012:
2005:
1998:
1991:
1984:
1975:
1974:
1973:
1969:
1968:
1964:
1963:
1959:
1958:
1954:
1953:
1949:
1948:
1944:
1943:
1932:
1931:
1930:
1926:
1925:
1921:
1920:
1916:
1915:
1911:
1910:
1906:
1905:
1901:
1900:
1889:
1888:
1887:
1883:
1882:
1878:
1877:
1873:
1872:
1868:
1867:
1863:
1862:
1858:
1857:
1846:
1845:
1844:
1840:
1839:
1835:
1834:
1830:
1829:
1825:
1824:
1820:
1819:
1815:
1814:
1803:
1802:
1801:
1797:
1796:
1792:
1791:
1787:
1786:
1782:
1781:
1777:
1776:
1772:
1771:
1764:Octahedral prism
1760:
1759:
1758:
1754:
1753:
1749:
1748:
1744:
1743:
1739:
1738:
1734:
1733:
1729:
1728:
1712:
1711:
1619:
1612:
1605:
1598:
1591:
1584:
1521:
1520:
1519:
1515:
1514:
1510:
1509:
1505:
1504:
1500:
1499:
1495:
1494:
1490:
1489:
1485:
1484:
1480:
1479:
1475:
1474:
1470:
1469:
1465:
1464:
1460:
1459:
1455:
1454:
1450:
1449:
1445:
1444:
1440:
1439:
1432:
1431:
1430:
1426:
1425:
1421:
1420:
1416:
1415:
1411:
1410:
1406:
1405:
1401:
1400:
1396:
1395:
1391:
1390:
1386:
1385:
1381:
1380:
1376:
1375:
1371:
1370:
1366:
1365:
1361:
1360:
1353:
1352:
1351:
1347:
1346:
1342:
1341:
1337:
1336:
1332:
1331:
1327:
1326:
1322:
1321:
1317:
1316:
1312:
1311:
1307:
1306:
1302:
1301:
1297:
1296:
1292:
1291:
1284:
1283:
1282:
1278:
1277:
1273:
1272:
1268:
1267:
1263:
1262:
1258:
1257:
1253:
1252:
1248:
1247:
1243:
1242:
1238:
1237:
1233:
1232:
1225:
1224:
1223:
1219:
1218:
1214:
1213:
1209:
1208:
1204:
1203:
1199:
1198:
1194:
1193:
1189:
1188:
1184:
1183:
1176:
1175:
1174:
1170:
1169:
1165:
1164:
1160:
1159:
1155:
1154:
1150:
1149:
1145:
1144:
1137:
1136:
1135:
1131:
1130:
1126:
1125:
1121:
1120:
1116:
1115:
1108:
1107:
1106:
1102:
1101:
1097:
1096:
1092:
1091:
1087:
1086:
1066:
1064:
1063:
1058:
1056:
1055:
1050:
1049:
1041:
1020:
1018:
1017:
1012:
1010:
1009:
1004:
1003:
995:
847:in n dimensions
837:
836:
808:regular polytope
796:uniform polytope
685:
681:Karlis Johansons
677:Erich Schönhardt
653:
651:
650:
645:
640:
635:
631:
630:
618:
617:
605:
604:
588:
579:
570:
561:
552:
461:
459:
452:
451:
441:
440:
427:
425:
424:
419:
417:
416:
395:
394:
385:
376:
375:
366:
360:
354:
346:
342:
338:
336:
335:
330:
325:
320:
309:
296:
295:
292:
285:
284:
281:
261:axis of symmetry
258:
210:
138:triangular prism
114:
38:
29:Triangular prism
26:
25:
3953:
3952:
3948:
3947:
3946:
3944:
3943:
3942:
3923:
3922:
3921:
3915:
3888:
3807:
3778:
3749:
3647:
3619:
3476:
3457:10.2307/2306130
3433:
3428:
3422:Skilling (1976)
3420:
3416:
3412:
3398:
3394:
3388:Bagemihl (1948)
3386:
3382:
3378:
3364:
3360:
3352:
3348:
3344:
3330:
3326:
3322:
3300:
3296:
3284:
3280:
3272:
3268:
3264:
3242:
3238:
3230:
3226:
3218:
3214:
3210:
3192:
3188:
3176:
3172:
3160:
3156:
3148:
3144:
3140:
3122:
3118:
3106:
3099:
3095:
3090:
3019:
3014:
3009:
3004:
2999:
2994:
2989:
2987:
2986:
2976:
2971:
2966:
2961:
2956:
2951:
2946:
2944:
2943:
2933:
2928:
2923:
2918:
2913:
2908:
2903:
2901:
2900:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2858:
2857:
2847:
2842:
2837:
2832:
2827:
2822:
2817:
2815:
2814:
2804:
2799:
2794:
2789:
2784:
2779:
2774:
2772:
2771:
2761:
2756:
2751:
2746:
2741:
2736:
2731:
2729:
2728:
2718:
2713:
2708:
2703:
2698:
2693:
2688:
2686:
2685:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2583:
2582:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2540:
2539:
2529:
2524:
2519:
2514:
2509:
2504:
2499:
2497:
2496:
2486:
2481:
2476:
2471:
2466:
2461:
2456:
2454:
2453:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2411:
2410:
2400:
2395:
2390:
2385:
2380:
2375:
2370:
2368:
2367:
2357:
2352:
2347:
2342:
2337:
2332:
2327:
2325:
2324:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2282:
2281:
2232:
2227:
2222:
2217:
2212:
2207:
2202:
2200:
2199:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2157:
2156:
2146:
2141:
2136:
2131:
2126:
2121:
2116:
2114:
2113:
2103:
2098:
2093:
2088:
2083:
2078:
2073:
2071:
2070:
2060:
2055:
2050:
2045:
2040:
2035:
2030:
2028:
2027:
1971:
1966:
1961:
1956:
1951:
1946:
1941:
1939:
1938:
1928:
1923:
1918:
1913:
1908:
1903:
1898:
1896:
1895:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1853:
1852:
1842:
1837:
1832:
1827:
1822:
1817:
1812:
1810:
1809:
1799:
1794:
1789:
1784:
1779:
1774:
1769:
1767:
1766:
1756:
1751:
1746:
1741:
1736:
1731:
1726:
1724:
1723:
1703:
1693:
1685:
1677:
1669:
1661:
1653:
1645:
1637:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1452:
1447:
1442:
1437:
1435:
1428:
1423:
1418:
1413:
1408:
1403:
1398:
1393:
1388:
1383:
1378:
1373:
1368:
1363:
1358:
1356:
1349:
1344:
1339:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1287:
1280:
1275:
1270:
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1230:
1228:
1221:
1216:
1211:
1206:
1201:
1196:
1191:
1186:
1181:
1179:
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1140:
1133:
1128:
1123:
1118:
1113:
1111:
1104:
1099:
1094:
1089:
1084:
1082:
1077:
1070:
1051:
1040:
1039:
1038:
1036:
1033:
1032:
1030:
1024:
1005:
994:
993:
992:
990:
987:
986:
984:
977:
969:
961:
954:
950:
944:
940:
934:
930:
926:
918:
870:
844:
833:
798:is constructed
788:
737:
707:, with lateral
683:
626:
622:
613:
609:
600:
596:
589:
587:
585:
582:
581:
578:
572:
569:
563:
560:
554:
548:
501:gyrobifastigium
473:
468:
455:
454:
453:represents the
450:
444:
443:
439:
435:
434:
412:
408:
390:
386:
374:
372:
369:
368:
362:
356:
352:
344:
340:
310:
308:
306:
303:
302:
290:
288:
287:
279:
277:
276:
265:dual polyhedron
257:
248:
216:If the base is
177:
120:Dual polyhedron
113:
104:
70:
53:
49:
24:
17:
12:
11:
5:
3951:
3941:
3940:
3935:
3920:
3919:
3913:
3892:
3886:
3865:
3837:
3815:Schönhardt, E.
3811:
3805:
3782:
3776:
3753:
3747:
3728:
3707:
3651:
3645:
3623:
3617:
3596:
3579:
3568:
3531:
3513:(5): 329–352.
3502:
3489:
3469:
3451:(7): 411–413.
3434:
3432:
3429:
3427:
3426:
3414:
3411:
3410:
3405:
3399:
3392:
3380:
3377:
3376:
3371:
3365:
3358:
3346:
3343:
3342:
3337:
3335:Rajwade (2001)
3331:
3324:
3321:
3320:
3311:
3305:Todesco (2020)
3301:
3294:
3278:
3266:
3263:
3262:
3253:
3243:
3236:
3224:
3220:Johnson (1966)
3212:
3209:
3208:
3203:
3193:
3186:
3170:
3154:
3142:
3139:
3138:
3133:
3123:
3116:
3096:
3094:
3091:
3089:
3086:
3083:
3082:
3075:
3068:
3061:
3054:
3047:
3040:
3033:
3025:
3024:
2981:
2938:
2895:
2852:
2809:
2766:
2723:
2679:
2678:
2671:
2664:
2657:
2650:
2643:
2636:
2629:
2621:
2620:
2577:
2534:
2491:
2448:
2405:
2362:
2319:
2275:
2274:
2267:
2260:
2253:
2246:
2238:
2237:
2194:
2151:
2108:
2065:
2021:
2020:
2013:
2006:
1999:
1992:
1985:
1977:
1976:
1933:
1890:
1847:
1804:
1761:
1717:
1716:
1709:, including:
1702:
1699:
1696:
1695:
1691:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1627:
1626:
1623:
1620:
1613:
1606:
1599:
1592:
1585:
1578:
1574:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1546:
1545:
1543:
1541:
1539:
1537:
1535:
1533:
1531:
1529:
1523:
1522:
1433:
1354:
1285:
1226:
1177:
1138:
1109:
1080:
1072:
1071:
1068:
1054:
1047:
1044:
1028:
1025:
1022:
1008:
1001:
998:
982:
979:
975:
971:
967:
963:
959:
955:
952:
948:
945:
942:
938:
935:
932:
928:
924:
921:
913:
912:
907:
902:
897:
892:
887:
882:
877:
872:
868:
863:
862:
859:
856:
853:
849:
848:
842:
831:
804:Thorold Gosset
787:
784:
783:
782:
736:
733:
643:
638:
634:
629:
625:
621:
616:
612:
608:
603:
599:
595:
592:
576:
567:
558:
485:Catalan solids
472:
469:
467:
464:
446:
437:
415:
411:
407:
404:
401:
398:
393:
389:
383:
379:
328:
323:
319:
316:
313:
273:internal angle
252:
246:dihedral group
189:parallelograms
176:
173:
165:Johnson solids
142:trigonal prism
128:
127:
122:
116:
115:
108:
102:
100:Symmetry group
96:
95:
92:
86:
85:
82:
76:
75:
64:
58:
57:
44:
40:
39:
31:
30:
15:
9:
6:
4:
3:
2:
3950:
3939:
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3808:
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3528:
3524:
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3508:
3503:
3499:
3495:
3490:
3487:(5): 355–367.
3486:
3482:
3475:
3470:
3466:
3462:
3458:
3454:
3450:
3446:
3445:
3440:
3436:
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3404:
3401:
3400:
3396:
3389:
3384:
3375:
3372:
3370:
3367:
3366:
3362:
3356:, p. 81.
3355:
3350:
3341:
3340:Berman (1971)
3338:
3336:
3333:
3332:
3328:
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3315:
3312:
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3298:
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3282:
3275:
3274:Berman (1971)
3270:
3261:
3257:
3254:
3252:
3248:
3245:
3244:
3240:
3234:, p. 26.
3233:
3228:
3221:
3216:
3207:
3206:Messer (2002)
3204:
3202:
3198:
3195:
3194:
3190:
3183:
3179:
3174:
3167:
3163:
3158:
3152:, p. 25.
3151:
3146:
3137:
3136:Berman (1971)
3134:
3132:
3128:
3125:
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2018:
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800:vertex figure
797:
793:
781:
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718:
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674:
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669:skew polygons
666:
658:
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641:
636:
627:
623:
619:
614:
610:
606:
601:
597:
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566:
557:
551:
547:. Given that
546:
541:
537:
528:
524:
522:
518:
514:
510:
506:
502:
498:
494:
490:
487:, prisms and
486:
482:
478:
477:Johnson solid
463:
462:sided prism.
458:
449:
433:
428:
413:
409:
405:
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396:
391:
387:
381:
377:
365:
359:
350:
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311:
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293:
282:
274:
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266:
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256:
251:
247:
243:
239:
235:
234:regular faces
231:
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172:
170:
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159:
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147:
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135:
126:
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117:
112:
107:
103:
101:
97:
93:
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77:
74:
69:
65:
63:
59:
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52:
48:
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41:
37:
32:
27:
22:
3899:. Springer.
3896:
3872:. Springer.
3869:
3845:
3841:
3822:
3787:
3762:. Springer.
3758:
3733:
3715:
3711:
3668:
3662:
3631:
3603:. Springer.
3600:
3583:
3573:
3542:(1): 51–66.
3539:
3535:
3510:
3506:
3493:
3484:
3480:
3448:
3442:
3439:Bagemihl, F.
3431:Bibliography
3417:
3395:
3383:
3361:
3349:
3327:
3297:
3281:
3269:
3239:
3227:
3215:
3189:
3173:
3157:
3145:
3119:
1704:
1633:
1569:696,729,600
789:
738:
714:
693:
673:triangulated
663:
573:
564:
555:
549:
544:
535:
533:
474:
456:
447:
429:
363:
357:
299:
289:
278:
254:
249:
215:
196:
180:
178:
162:
149:
141:
137:
131:
110:
105:
3848:: 447–457,
3827:: 309–312.
3718:: 353–375.
3671:: 169–200.
3574:Mensuration
3256:Haul (1893)
3162:Haul (1893)
3127:King (1994)
3108:King (1994)
861:Hyperbolic
816:orthoplexes
699:shares its
517:tetrahedron
432:prism graph
236:and has an
226:semiregular
218:equilateral
154:semiregular
3927:Categories
3703:0132.14603
3316:, p.
3307:, p.
3288:, p.
3258:, p.
3249:, p.
3199:, p.
3180:, p.
3164:, p.
3129:, p.
3110:, p.
3088:References
1566:2,903,040
858:Euclidean
735:Honeycombs
540:truncating
489:antiprisms
193:rectangles
175:Properties
3695:122006114
3564:118484882
3093:Citations
1046:¯
1000:~
812:simplexes
403:≈
397:⋅
185:triangles
68:triangles
3817:(1928).
3657:(1966).
1634:−1
1572:∞
1527:Symmetry
705:faceting
684:in 1921.
349:altitude
294:/2 = 90°
283:/3 = 60°
238:isogonal
134:geometry
90:Vertices
3862:0397554
3687:0185507
3592:1035479
3556:3457762
3527:0290245
3465:2306130
1563:51,840
1078:diagram
1076:Coxeter
917:Coxeter
855:Finite
845:figures
828:Coxeter
824:squares
695:crossed
230:uniform
158:uniform
73:squares
3911:
3884:
3860:
3803:
3774:
3745:
3701:
3693:
3685:
3643:
3615:
3590:
3562:
3554:
3525:
3463:
1577:Graph
1560:1,920
852:Space
571:, and
511:, and
339:where
222:square
3691:S2CID
3560:S2CID
3477:(PDF)
3461:JSTOR
1630:Name
1550:Order
919:group
406:0.433
201:wedge
181:bases
146:prism
144:is a
80:Edges
62:Faces
47:Prism
3909:ISBN
3882:ISBN
3801:ISBN
3772:ISBN
3743:ISBN
3641:ISBN
3613:ISBN
3588:OCLC
1557:120
822:and
814:and
156:and
136:, a
43:Type
3901:doi
3874:doi
3850:doi
3829:doi
3793:doi
3764:doi
3720:doi
3699:Zbl
3673:doi
3605:doi
3544:doi
3515:doi
3511:291
3453:doi
3309:282
3251:389
3201:100
3182:139
3131:113
3112:113
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1021:= E
140:or
132:In
3929::
3907:.
3880:.
3858:MR
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3846:79
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3100:^
1692:21
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1676:21
1668:21
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951:=D
941:=A
927:=A
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