202:
328:
319:
79:
97:
88:
182:
published a list including ninety-two
Johnson solids—excluding the five Platonic solids, the thirteen Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms—and gave them their names and numbers. He did not prove that there were only ninety-two,
213:
The naming of
Johnson solids follows a flexible and precise descriptive formula that allows many solids to be named in multiple different ways without compromising the accuracy of each name as a description. Most Johnson solids can be constructed from the first few solids
209:. Here, it is constructed from triangular prism by joining three equilateral square pyramids onto each of its squares (tri-). The process of this construction known as "augmentation", making its first name is "triaugmented".
1115:(also called the pseudorhombicuboctahedron) is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle. However, it is not
143:(meaning they are not segments of the same line). Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not
242:; the center of a particular solid's name will reflect these ingredients. From there, a series of prefixes are attached to the word to indicate additions, rotations, and transformations:
385:
solid has three removed pyramids or cupolae. In certain large solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique.
1000:. This means the polyhedron cannot be separated by a plane to create two small convex polyhedrons with regular faces; examples of Johnson solids are the first six Johnson solids—
525:
denoted the first
Johnson solid, the equilateral square pyramid). The following is the list of ninety-two Johnson solids, with the enumeration followed according to the list of
1088:, meaning they do have polyhedrons of the same or larger size that may pass through a hole inside of them. However, the other five Johnson solids do not have this property:
404:
The last few
Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature:
135:. Here, a polyhedron is said to be convex if the shortest path between any two of its vertices lies either within its interior or on its boundary, none of its faces are
309:
indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, as in the difference between ortho- and gyrobicupolae.
1080:
As the definition above, a
Johnson solid is a convex polyhedron with regular polygons as their faces. However, there are several properties possessed by each of them.
523:
476:
496:
57:; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians
249:
indicates that two copies of the solid are joined base-to-base. For cupolae and rotundas, the solids can be joined so that either like faces (
2321:
2316:
1105:
919:
914:
2341:
2326:
939:
924:
1829:
2336:
2331:
2089:
1029:
934:
929:
729:
2281:
2276:
2124:
2094:
1500:
879:
874:
764:
734:
1732:
1677:
1607:
1587:
1362:
1298:
1263:
1160:
2346:
1033:
944:
2301:
2296:
2286:
2237:
2099:
1097:
1093:
899:
894:
884:
849:
739:
2129:
2104:
2079:
2064:
769:
744:
719:
704:
2311:
2119:
2109:
2084:
2069:
1535:
1483:
1228:
909:
759:
749:
724:
709:
2306:
2271:
2014:
1926:
1101:
904:
869:
654:
584:
2256:
2183:
2178:
2009:
1999:
854:
809:
804:
649:
639:
349:
345:
1956:
1951:
1941:
1775:
1381:
614:
609:
599:
171:
163:. A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a
58:
2291:
2158:
2114:
2074:
2049:
1112:
1089:
889:
784:
754:
714:
689:
206:
106:
2405:
2227:
2217:
2212:
2188:
2168:
2153:
2054:
1994:
1921:
1916:
1906:
1822:
1716:
1065:
989:
839:
829:
824:
814:
794:
779:
694:
634:
579:
574:
564:
2004:
1989:
1979:
644:
629:
619:
170:
The
Johnson solid, sometimes known as Johnson–Zalgaller solid, was named after two mathematicians
2266:
2163:
2148:
1946:
1801:
1579:
864:
789:
774:
604:
534:
2232:
2222:
2173:
2059:
1025:
844:
834:
799:
699:
1698:
2261:
2039:
2024:
1911:
859:
679:
664:
569:
262:
2437:
2044:
1984:
1815:
684:
624:
201:
164:
1726:
2417:
2380:
2207:
1119:, as it has different isometry at different vertices, making it a Johnson solid rather than an
964:
819:
196:
1659:
1473:
1245:
2370:
2357:
1768:
1573:
1344:
1319:
1280:
1142:
1041:
954:
139:(meaning they do not share the same plane, and do not "lie flat"), and none of its edges are
1756:
1244:
Todesco, Gian Marco (2020). "Hyperbolic
Honeycomb". In Emmer, Michele; Abate, Marco (eds.).
2390:
2029:
1936:
1931:
1636:
1455:
1069:
1053:
997:
974:
669:
594:
589:
501:
454:
258:
8:
2034:
674:
301:
indicates a pyramid or cupola is removed from one or more faces of the solid in question.
1971:
1898:
1887:
1867:
1854:
1846:
1544:
1193:
1021:
1005:
559:
539:
481:
288:
223:
215:
144:
54:
46:
42:
1603:
1211:
Boissonnat, J. D.; Yvinec, M. (June 1989). "Probing a scene of non convex polyhedra".
2248:
1967:
1882:
1872:
1850:
1739:
1673:
1583:
1479:
1447:
1358:
1294:
1259:
1224:
1156:
1120:
1116:
1017:
1009:
554:
544:
292:
231:
219:
152:
50:
34:
2395:
2385:
2140:
1665:
1622:
1554:
1515:
1469:
1443:
1411:
1393:
1350:
1286:
1251:
1216:
1185:
1148:
1057:
1049:
979:
969:
272:
235:
175:
156:
110:
62:
1742:
1558:
1519:
327:
318:
2400:
2365:
2019:
1779:
1720:
1632:
1451:
1085:
1061:
1037:
984:
949:
659:
389:
indicates the former, that the solid in question has altered parallel faces, and
132:
114:
38:
1176:
Litchenberg, Dorovan R. (1988). "Pyramids, Prisms, Antiprisms, and
Deltahedra".
78:
2199:
1862:
1315:
1147:. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39.
1001:
227:
148:
1669:
1354:
1346:
Introduction to
Computational Origami: The World of New Computational Geometry
1290:
1255:
1152:
2431:
1877:
1797:
1013:
549:
45:. There are ninety-two solids with such a property: the first solids are the
2375:
1397:
1045:
959:
96:
1189:
140:
118:
1762:
1220:
1197:
1320:"Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons"
1072:
because they are constructed by attaching some elementary polyhedrons.
128:
261:
is a solid constructed by attaching two bases of pentagonal pyramids.
1747:
1627:
412:
is a complex of two triangles attached to opposite sides of a square.
280:
239:
160:
117:, as some of its diagonals lie outside the shape. The third presents
1794:
aces), a generalization of the
Johnson solids to 4-dimensional space
1772:
1478:. Undergraduate Texts in Mathematics. Springer-Verlag. p. 464.
377:
indicate a double and triple operation respectively. For example, a
1549:
1213:
Proceedings of the Fifth Annual Symposium on Computational Geometry
428:- indicates a blunt complex of two lunes separated by a third lune.
305:
136:
87:
22:
1807:
1533:
Fredriksson, Albin (2024). "Optimizing for the Rupert property".
16:
92 non-uniform convex polyhedra, with each face a regular polygon
1713:
418:- indicates a wedgelike complex formed by two adjacent lunes.
369:—can be performed multiple times for certain large solids.
1327:
Bridges: Mathematical Connections in Art, Music and Science
265:
is constructed by two triangular cupolas along their bases.
295:, is joined to one or more faces of the solid in question.
275:
is joined to the base of the solid, or between the bases;
1434:
Berman, Martin (1971). "Regular-faced convex polyhedra".
1737:
504:
484:
457:
451:
The enumeration of Johnson solids may be denoted as
1278:
517:
490:
470:
440:is a larger crownlike complex of twelve triangles.
397:solid has had two parallel faces augmented, and a
393:the latter, altered oblique faces. For example, a
1733:Images of all 92 solids, categorized, on one page
996:Some of the Johnson solids may be categorized as
183:but he did conjecture that there were no others.
105:Among these three polyhedra, only the first, the
2429:
1658:Lando, Sergei K.; Zvonkin, Alexander K. (2004).
1210:
1492:
1247:Imagine Math 7: Between Culture and Mathematics
1701:[Convex polyhedra with regular faces]
498:denoted the list's enumeration (an example is
1823:
1699:"Les polyèdres convexes aux faces régulières"
1647:. Princeton University Press. pp. 18–31.
2322:metagyrate diminished rhombicosidodecahedron
2317:paragyrate diminished rhombicosidodecahedron
1657:
1384:(1966). "Convex Solids with Regular Faces".
1106:paragyrate diminished rhombicosidodecahedron
920:Metagyrate diminished rhombicosidodecahedron
915:Paragyrate diminished rhombicosidodecahedron
68:
41:. They are sometimes defined to exclude the
1532:
1175:
147:. This means that a Johnson solid is not a
2342:gyrate bidiminished rhombicosidodecahedron
2327:bigyrate diminished rhombicosidodecahedron
1830:
1816:
1804:applied to them, including Johnson solids.
1798:https://levskaya.github.io/polyhedronisme/
1498:
1468:
1462:
1313:
1307:
1279:Williams, Kim; Monteleone, Cosino (2021).
1169:
940:Gyrate bidiminished rhombicosidodecahedron
925:Bigyrate diminished rhombicosidodecahedron
434:is a crownlike complex of eight triangles.
1661:Graphs on Surfaces and Their Applications
1626:
1548:
1429:
1427:
1425:
1410:
1376:
1374:
401:solid has had two oblique faces gyrated.
190:
187:proved that Johnson's list was complete.
184:
1602:
1596:
1571:
1565:
1404:
1338:
1336:
200:
2337:metabidiminished rhombicosidodecahedron
2332:parabidiminished rhombicosidodecahedron
2090:elongated pentagonal orthocupolarotunda
1380:
1272:
1243:
1030:parabidiminished rhombicosidodecahedron
935:Metabidiminished rhombicosidodecahedron
930:Parabidiminished rhombicosidodecahedron
730:Elongated pentagonal orthocupolarotunda
526:
287:indicates another polyhedron, namely a
179:
2430:
2282:metabiaugmented truncated dodecahedron
2277:parabiaugmented truncated dodecahedron
2125:gyroelongated pentagonal cupolarotunda
2095:elongated pentagonal gyrocupolarotunda
1696:
1642:
1433:
1422:
1371:
1342:
1237:
1140:
880:Metabiaugmented truncated dodecahedron
875:Parabiaugmented truncated dodecahedron
765:Gyroelongated pentagonal cupolarotunda
735:Elongated pentagonal gyrocupolarotunda
109:, is a Johnson solid. The second, the
1811:
1738:
1501:"Junction of Non-composite Polyhedra"
1333:
1282:Daniele Barbaro's Perspective of 1568
1134:
447:indicates a belt of twelve triangles.
381:solid has two rotated cupolae, and a
2347:tridiminished rhombicosidodecahedron
1645:The Best Writing on Mathematics 2010
1526:
1111:From all of the Johnson solids, the
1034:tridiminished rhombicosidodecahedron
945:Tridiminished rhombicosidodecahedron
2302:metabigyrate rhombicosidodecahedron
2297:parabigyrate rhombicosidodecahedron
2287:triaugmented truncated dodecahedron
2238:augmented tridiminished icosahedron
2100:elongated pentagonal orthobirotunda
1786:onvex 4-dimensional polytopes with
1508:St. Petersburg Mathematical Journal
1416:Convex Polyhedra with Regular Faces
1098:metabigyrate rhombicosidodecahedron
1094:parabigyrate rhombicosidodecahedron
900:Metabigyrate rhombicosidodecahedron
895:Parabigyrate rhombicosidodecahedron
885:Triaugmented truncated dodecahedron
850:Augmented tridiminished icosahedron
740:Elongated pentagonal orthobirotunda
257:) meet. Using this nomenclature, a
13:
2130:gyroelongated pentagonal birotunda
2105:elongated pentagonal gyrobirotunda
2080:elongated pentagonal orthobicupola
2065:elongated triangular orthobicupola
1837:
1651:
770:Gyroelongated pentagonal birotunda
745:Elongated pentagonal gyrobirotunda
720:Elongated pentagonal orthobicupola
705:Elongated triangular orthobicupola
14:
2449:
2312:diminished rhombicosidodecahedron
2120:gyroelongated pentagonal bicupola
2110:gyroelongated triangular bicupola
2085:elongated pentagonal gyrobicupola
2070:elongated triangular gyrobicupola
1790:egular polygons as 2-dimensional
1690:
1536:The American Mathematical Monthly
1436:Journal of the Franklin Institute
1204:
910:Diminished rhombicosidodecahedron
760:Gyroelongated pentagonal bicupola
750:Gyroelongated triangular bicupola
725:Elongated pentagonal gyrobicupola
710:Elongated triangular gyrobicupola
2307:trigyrate rhombicosidodecahedron
2272:augmented truncated dodecahedron
2015:gyroelongated pentagonal rotunda
1927:gyroelongated pentagonal pyramid
1102:trigyrate rhombicosidodecahedron
905:Trigyrate rhombicosidodecahedron
870:Augmented truncated dodecahedron
655:Gyroelongated pentagonal rotunda
585:Gyroelongated pentagonal pyramid
326:
317:
95:
86:
77:
2257:augmented truncated tetrahedron
2184:metabiaugmented hexagonal prism
2179:parabiaugmented hexagonal prism
2010:gyroelongated pentagonal cupola
2000:gyroelongated triangular cupola
1800:a generator of polyhedrons and
1769:CRF polychora discovery project
1386:Canadian Journal of Mathematics
1144:Multi-shell Polyhedral Clusters
1068:. The other Johnson solids are
855:Augmented truncated tetrahedron
810:Metabiaugmented hexagonal prism
805:Parabiaugmented hexagonal prism
650:Gyroelongated pentagonal cupola
640:Gyroelongated triangular cupola
350:metabiaugmented hexagonal prism
346:parabiaugmented hexagonal prism
131:polyhedron whose faces are all
1957:gyroelongated square bipyramid
1952:elongated pentagonal bipyramid
1942:elongated triangular bipyramid
615:Gyroelongated square bipyramid
610:Elongated pentagonal bipyramid
600:Elongated triangular bipyramid
1:
2292:gyrate rhombicosidodecahedron
2159:triaugmented triangular prism
2115:gyroelongated square bicupola
2075:elongated square gyrobicupola
2050:pentagonal orthocupolarotunda
1763:VRML models of Johnson Solids
1757:VRML models of Johnson Solids
1559:10.1080/00029890.2023.2285200
1520:10.1090/S1061-0022-10-01105-2
1127:
1113:elongated square gyrobicupola
1090:gyrate rhombicosidodecahedron
1075:
890:Gyrate rhombicosidodecahedron
785:Triaugmented triangular prism
755:Gyroelongated square bicupola
715:Elongated square gyrobicupola
690:Pentagonal orthocupolarotunda
422:indicates two such complexes.
207:triaugmented triangular prism
107:elongated square gyrobicupola
31:Johnson–Zalgaller solid
2406:triangular hebesphenorotunda
2228:metabidiminished icosahedron
2218:metabiaugmented dodecahedron
2213:parabiaugmented dodecahedron
2189:triaugmented hexagonal prism
2169:biaugmented pentagonal prism
2154:biaugmented triangular prism
2055:pentagonal gyrocupolarotunda
1995:elongated pentagonal rotunda
1922:gyroelongated square pyramid
1917:elongated pentagonal pyramid
1907:elongated triangular pyramid
1643:Pitici, Mircea, ed. (2011).
1448:10.1016/0016-0032(71)90071-8
1066:triangular hebesphenorotunda
990:Triangular hebesphenorotunda
840:Metabidiminished icosahedron
830:Metabiaugmented dodecahedron
825:Parabiaugmented dodecahedron
815:Triaugmented hexagonal prism
795:Biaugmented pentagonal prism
780:Biaugmented triangular prism
695:Pentagonal gyrocupolarotunda
635:Elongated pentagonal rotunda
580:Gyroelongated square pyramid
575:Elongated pentagonal pyramid
565:Elongated triangular pyramid
29:, sometimes also known as a
7:
2005:gyroelongated square cupola
1990:elongated pentagonal cupola
1980:elongated triangular cupola
1572:Cromwell, Peter R. (1997).
1475:Geometry: Euclid and Beyond
645:Gyroelongated square cupola
630:Elongated pentagonal cupola
620:Elongated triangular cupola
10:
2454:
2267:biaugmented truncated cube
2164:augmented pentagonal prism
2149:augmented triangular prism
1947:elongated square bipyramid
1580:Cambridge University Press
1499:Timofeenko, A. V. (2010).
865:Biaugmented truncated cube
790:Augmented pentagonal prism
775:Augmented triangular prism
605:Elongated square bipyramid
535:Equilateral square pyramid
357:The last three operations—
194:
2414:
2355:
2246:
2233:tridiminished icosahedron
2223:triaugmented dodecahedron
2197:
2174:augmented hexagonal prism
2138:
2060:pentagonal orthobirotunda
1965:
1896:
1845:
1714:Paper Models of Polyhedra
1670:10.1007/978-3-540-38361-1
1664:. Springer. p. 114.
1355:10.1007/978-981-15-4470-5
1291:10.1007/978-3-030-76687-0
1256:10.1007/978-3-030-42653-8
1250:. Springer. p. 282.
1153:10.1007/978-3-319-64123-2
1026:tridiminished icosahedron
845:Tridiminished icosahedron
835:Triaugmented dodecahedron
800:Augmented hexagonal prism
700:Pentagonal orthobirotunda
69:Definition and background
2262:augmented truncated cube
2040:pentagonal orthobicupola
2025:triangular orthobicupola
1912:elongated square pyramid
1697:Gagnon, Sylvain (1982).
1349:. Springer. p. 62.
1285:. Springer. p. 23.
860:Augmented truncated cube
680:Pentagonal orthobicupola
665:Triangular orthobicupola
570:Elongated square pyramid
263:Triangular orthobicupola
2045:pentagonal gyrobicupola
1985:elongated square cupola
1615:Elemente der Mathematik
1343:Uehara, Ryuhei (2020).
1178:The Mathematics Teacher
685:Pentagonal gyrobicupola
625:Elongated square cupola
165:near-miss Johnson solid
2418:List of Johnson solids
2381:augmented sphenocorona
2208:augmented dodecahedron
1398:10.4153/CJM-1966-021-8
1141:Diudea, M. V. (2018).
965:Augmented sphenocorona
820:Augmented dodecahedron
519:
492:
472:
210:
197:List of Johnson solids
191:Naming and enumeration
2371:snub square antiprism
1771:attempts to discover
1418:. Consultants Bureau.
1042:snub square antiprism
955:Snub square antiprism
520:
518:{\displaystyle J_{1}}
493:
473:
471:{\displaystyle J_{n}}
226:), together with the
204:
127:A Johnson solid is a
2391:hebesphenomegacorona
2030:square orthobicupola
1937:pentagonal bipyramid
1932:triangular bipyramid
1412:Zalgaller, Victor A.
1215:. pp. 237–246.
1190:10.5951/MT.81.4.0261
1070:composite polyhedron
1054:hebesphenomegacorona
998:elementary polyhedra
975:Hebesphenomegacorona
670:Square orthobicupola
595:Pentagonal bipyramid
590:Triangular bipyramid
502:
482:
455:
259:pentagonal bipyramid
2420:, a sortable table)
2035:square gyrobicupola
1765:by Vladimir Bulatov
1707:Structural Topology
1608:"An enduring error"
1221:10.1145/73833.73860
675:Square gyrobicupola
253:) or unlike faces (
43:uniform polyhedrons
2249:Archimedean solids
1888:pentagonal rotunda
1868:pentagonal pyramid
1778:2020-10-31 at the
1740:Weisstein, Eric W.
1729:by George W. Hart.
1719:2013-02-26 at the
1314:Kaplan, Craig S.;
1022:pentagonal rotunda
1006:pentagonal pyramid
560:Pentagonal rotunda
540:Pentagonal pyramid
515:
488:
468:
211:
2425:
2424:
2358:elementary solids
1883:pentagonal cupola
1873:triangular cupola
1802:Conway operations
1679:978-3-540-38361-1
1589:978-0-521-55432-9
1470:Hartshorne, Robin
1364:978-981-15-4470-5
1300:978-3-030-76687-0
1265:978-3-030-42653-8
1162:978-3-319-64123-2
1121:Archimedean solid
1117:vertex-transitive
1018:pentagonal cupola
1010:triangular cupola
555:Pentagonal cupola
545:Triangular cupola
491:{\displaystyle n}
153:Archimedean solid
35:convex polyhedron
2445:
2396:disphenocingulum
2386:sphenomegacorona
1832:
1825:
1818:
1809:
1808:
1753:
1752:
1710:
1704:
1684:
1683:
1655:
1649:
1648:
1640:
1630:
1612:
1604:GrĂĽnbaum, Branko
1600:
1594:
1593:
1569:
1563:
1562:
1552:
1530:
1524:
1523:
1505:
1496:
1490:
1489:
1466:
1460:
1459:
1431:
1420:
1419:
1408:
1402:
1401:
1378:
1369:
1368:
1340:
1331:
1330:
1324:
1311:
1305:
1304:
1276:
1270:
1269:
1241:
1235:
1234:
1208:
1202:
1201:
1173:
1167:
1166:
1138:
1058:disphenocingulum
1050:sphenomegacorona
980:Disphenocingulum
970:Sphenomegacorona
524:
522:
521:
516:
514:
513:
497:
495:
494:
489:
477:
475:
474:
469:
467:
466:
344:can be found in
330:
321:
185:Zalgaller (1969)
176:Victor Zalgaller
133:regular polygons
111:stella octangula
99:
90:
81:
63:Victor Zalgaller
39:regular polygons
37:whose faces are
33:, is a strictly
2453:
2452:
2448:
2447:
2446:
2444:
2443:
2442:
2428:
2427:
2426:
2421:
2410:
2401:bilunabirotunda
2366:snub disphenoid
2351:
2242:
2200:Platonic solids
2193:
2134:
2020:gyrobifastigium
1961:
1892:
1841:
1836:
1780:Wayback Machine
1743:"Johnson Solid"
1721:Wayback Machine
1702:
1693:
1688:
1687:
1680:
1656:
1652:
1610:
1601:
1597:
1590:
1570:
1566:
1531:
1527:
1503:
1497:
1493:
1486:
1467:
1463:
1432:
1423:
1409:
1405:
1382:Johnson, Norman
1379:
1372:
1365:
1341:
1334:
1322:
1316:Hart, George W.
1312:
1308:
1301:
1277:
1273:
1266:
1242:
1238:
1231:
1209:
1205:
1174:
1170:
1163:
1139:
1135:
1130:
1086:Rupert property
1078:
1062:bilunabirotunda
1038:snub disphenoid
994:
985:Bilunabirotunda
950:Snub disphenoid
660:Gyrobifastigium
509:
505:
503:
500:
499:
483:
480:
479:
462:
458:
456:
453:
452:
395:parabiaugmented
355:
354:
353:
352:
333:
332:
331:
323:
322:
199:
193:
125:
124:
123:
122:
102:
101:
100:
92:
91:
83:
82:
71:
17:
12:
11:
5:
2451:
2441:
2440:
2438:Johnson solids
2423:
2422:
2415:
2412:
2411:
2409:
2408:
2403:
2398:
2393:
2388:
2383:
2378:
2373:
2368:
2362:
2360:
2353:
2352:
2350:
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2253:
2251:
2244:
2243:
2241:
2240:
2235:
2230:
2225:
2220:
2215:
2210:
2204:
2202:
2195:
2194:
2192:
2191:
2186:
2181:
2176:
2171:
2166:
2161:
2156:
2151:
2145:
2143:
2136:
2135:
2133:
2132:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2062:
2057:
2052:
2047:
2042:
2037:
2032:
2027:
2022:
2017:
2012:
2007:
2002:
1997:
1992:
1987:
1982:
1976:
1974:
1963:
1962:
1960:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1903:
1901:
1894:
1893:
1891:
1890:
1885:
1880:
1875:
1870:
1865:
1863:square pyramid
1859:
1857:
1843:
1842:
1839:Johnson solids
1835:
1834:
1827:
1820:
1812:
1806:
1805:
1795:
1766:
1760:
1759:by Jim McNeill
1754:
1735:
1730:
1727:Johnson Solids
1724:
1711:
1692:
1691:External links
1689:
1686:
1685:
1678:
1650:
1628:10.4171/EM/120
1595:
1588:
1582:. p. 91.
1564:
1543:(3): 255–261.
1525:
1514:(3): 483–512.
1491:
1484:
1461:
1442:(5): 329–352.
1421:
1403:
1370:
1363:
1332:
1306:
1299:
1271:
1264:
1236:
1229:
1203:
1184:(4): 261–265.
1168:
1161:
1132:
1131:
1129:
1126:
1125:
1124:
1109:
1077:
1074:
1002:square pyramid
993:
992:
987:
982:
977:
972:
967:
962:
957:
952:
947:
942:
937:
932:
927:
922:
917:
912:
907:
902:
897:
892:
887:
882:
877:
872:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
602:
597:
592:
587:
582:
577:
572:
567:
562:
557:
552:
547:
542:
537:
531:
527:Johnson (1966)
512:
508:
487:
465:
461:
449:
448:
441:
435:
429:
423:
413:
335:
334:
325:
324:
316:
315:
314:
313:
312:
311:
310:
302:
296:
266:
205:An example is
195:Main article:
192:
189:
180:Johnson (1966)
172:Norman Johnson
149:Platonic solid
104:
103:
94:
93:
85:
84:
76:
75:
74:
73:
72:
70:
67:
59:Norman Johnson
15:
9:
6:
4:
3:
2:
2450:
2439:
2436:
2435:
2433:
2419:
2413:
2407:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2363:
2361:
2359:
2354:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2254:
2252:
2250:
2245:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2205:
2203:
2201:
2196:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2146:
2144:
2142:
2137:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1977:
1975:
1973:
1969:
1964:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1904:
1902:
1900:
1895:
1889:
1886:
1884:
1881:
1879:
1878:square cupola
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1860:
1858:
1856:
1852:
1848:
1844:
1840:
1833:
1828:
1826:
1821:
1819:
1814:
1813:
1810:
1803:
1799:
1796:
1793:
1789:
1785:
1781:
1777:
1774:
1773:CRF polychora
1770:
1767:
1764:
1761:
1758:
1755:
1750:
1749:
1744:
1741:
1736:
1734:
1731:
1728:
1725:
1722:
1718:
1715:
1712:
1708:
1700:
1695:
1694:
1681:
1675:
1671:
1667:
1663:
1662:
1654:
1646:
1641:Reprinted in
1638:
1634:
1629:
1624:
1621:(3): 89–101.
1620:
1616:
1609:
1605:
1599:
1591:
1585:
1581:
1577:
1576:
1568:
1560:
1556:
1551:
1546:
1542:
1538:
1537:
1529:
1521:
1517:
1513:
1509:
1502:
1495:
1487:
1485:9780387986500
1481:
1477:
1476:
1471:
1465:
1457:
1453:
1449:
1445:
1441:
1437:
1430:
1428:
1426:
1417:
1413:
1407:
1399:
1395:
1391:
1387:
1383:
1377:
1375:
1366:
1360:
1356:
1352:
1348:
1347:
1339:
1337:
1328:
1321:
1317:
1310:
1302:
1296:
1292:
1288:
1284:
1283:
1275:
1267:
1261:
1257:
1253:
1249:
1248:
1240:
1232:
1230:0-89791-318-3
1226:
1222:
1218:
1214:
1207:
1199:
1195:
1191:
1187:
1183:
1179:
1172:
1164:
1158:
1154:
1150:
1146:
1145:
1137:
1133:
1122:
1118:
1114:
1110:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1082:
1081:
1073:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1027:
1023:
1019:
1015:
1014:square cupola
1011:
1007:
1003:
999:
991:
988:
986:
983:
981:
978:
976:
973:
971:
968:
966:
963:
961:
958:
956:
953:
951:
948:
946:
943:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
566:
563:
561:
558:
556:
553:
551:
550:Square cupola
548:
546:
543:
541:
538:
536:
533:
532:
530:
528:
510:
506:
485:
463:
459:
446:
442:
439:
436:
433:
430:
427:
424:
421:
417:
414:
411:
407:
406:
405:
402:
400:
396:
392:
388:
384:
383:tridiminished
380:
376:
372:
368:
364:
360:
351:
347:
343:
339:
329:
320:
308:
307:
303:
300:
297:
294:
290:
286:
282:
279:indicates an
278:
277:gyroelongated
274:
270:
267:
264:
260:
256:
252:
248:
245:
244:
243:
241:
237:
233:
229:
225:
221:
217:
208:
203:
198:
188:
186:
181:
177:
173:
168:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
120:
116:
112:
108:
98:
89:
80:
66:
64:
60:
56:
52:
48:
44:
40:
36:
32:
28:
27:Johnson solid
24:
19:
2376:sphenocorona
1838:
1791:
1787:
1783:
1746:
1706:
1660:
1653:
1644:
1618:
1614:
1598:
1574:
1567:
1540:
1534:
1528:
1511:
1507:
1494:
1474:
1464:
1439:
1435:
1415:
1406:
1389:
1385:
1345:
1326:
1309:
1281:
1274:
1246:
1239:
1212:
1206:
1181:
1177:
1171:
1143:
1136:
1079:
1046:sphenocorona
995:
960:Sphenocorona
450:
444:
443:The suffix -
437:
431:
425:
419:
415:
409:
403:
399:metabigyrate
398:
394:
390:
386:
382:
378:
374:
370:
366:
362:
359:augmentation
358:
356:
341:
337:
336:Examples of
304:
298:
284:
276:
271:indicates a
268:
254:
250:
246:
212:
169:
126:
30:
26:
20:
18:
1709:(6): 83–95.
1392:: 169–200.
232:Archimedean
2416:(See also
2139:Augmented
1723:Many links
1550:2210.00601
1128:References
1084:They have
1076:Properties
438:Megacorona
426:Hebespheno
363:diminution
299:Diminished
240:antiprisms
2247:Modified
2198:Modified
1966:Modified
1897:Modified
1748:MathWorld
1575:Polyhedra
420:Dispheno-
285:Augmented
281:antiprism
269:Elongated
161:antiprism
113:, is not
2432:Category
1972:rotundae
1899:pyramids
1855:rotundae
1847:Pyramids
1776:Archived
1717:Archived
1606:(2009).
1472:(2000).
1414:(1969).
1329:: 21–28.
1318:(2001).
1198:27965792
478:, where
445:cingulum
379:bigyrate
367:gyration
234:solids,
228:Platonic
222:, and a
216:pyramids
141:colinear
137:coplanar
119:coplanar
53:. and a
47:pyramids
23:geometry
1968:cupolae
1851:cupolae
1637:2520469
1456:0290245
289:pyramid
224:rotunda
220:cupolae
145:uniform
55:rotunda
51:cupolas
2356:Other
2141:prisms
1676:
1635:
1586:
1482:
1454:
1361:
1297:
1262:
1227:
1196:
1159:
1104:, and
1064:, and
1020:, and
432:Corona
416:Spheno
373:&
365:, and
306:Gyrate
293:cupola
251:ortho-
238:, and
236:prisms
129:convex
121:faces.
115:convex
1703:(PDF)
1611:(PDF)
1545:arXiv
1504:(PDF)
1323:(PDF)
1194:JSTOR
391:meta-
387:Para-
342:meta-
338:para-
273:prism
255:gyro-
159:, or
157:prism
1970:and
1853:and
1674:ISBN
1584:ISBN
1480:ISBN
1359:ISBN
1295:ISBN
1260:ISBN
1225:ISBN
1157:ISBN
410:lune
375:Tri-
348:and
340:and
230:and
174:and
61:and
25:, a
1666:doi
1623:doi
1555:doi
1541:131
1516:doi
1444:doi
1440:291
1394:doi
1351:doi
1287:doi
1252:doi
1217:doi
1186:doi
1149:doi
371:Bi-
291:or
247:Bi-
21:In
2434::
1849:,
1745:.
1705:.
1672:.
1633:MR
1631:.
1619:64
1617:.
1613:.
1578:.
1553:.
1539:.
1512:21
1510:.
1506:.
1452:MR
1450:.
1438:.
1424:^
1390:18
1388:.
1373:^
1357:.
1335:^
1325:.
1293:.
1258:.
1223:.
1192:.
1182:81
1180:.
1155:.
1100:,
1096:,
1092:,
1060:,
1056:,
1052:,
1048:,
1044:,
1040:,
1036:,
1032:,
1028:,
1016:,
1012:,
1008:,
1004:,
529::
408:A
361:,
283:.
218:,
178:.
167:.
155:,
151:,
65:.
49:,
1831:e
1824:t
1817:v
1792:F
1788:R
1784:C
1782:(
1751:.
1682:.
1668::
1639:.
1625::
1592:.
1561:.
1557::
1547::
1522:.
1518::
1488:.
1458:.
1446::
1400:.
1396::
1367:.
1353::
1303:.
1289::
1268:.
1254::
1233:.
1219::
1200:.
1188::
1165:.
1151::
1123:.
1108:.
1024:—
511:1
507:J
486:n
464:n
460:J
214:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
