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Semiregular polytope

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94: 83: 72: 54: 1122: 790: 43: 1739: 1668: 1597: 184:, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. 1223: 1126: 1373: 1981: 1287: 1526: 1061:
additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures:
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Makarov, P. V. (1988). "On the derivation of four-dimensional semi-regular polytopes". Voprosy Diskret. Geom.
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There are also hyperbolic uniform honeycombs composed of only regular cells (
135: 2226: 2203:; Whitrow, G. J. (1950). "World-structure and non-Euclidean honeycombs". 119: 107: 2152: 1069:– 1)-ic semi-check (analogous to a single rank or file of a chessboard) 169: 797:
in Euclidean 3-space has alternating tetrahedral and octahedral cells.
789: 42: 1120: 29: 203:. The only semiregular polytopes in higher dimensions are the 2275:(1900). "On the regular and semi-regular figures in space of 1284:, 3D honeycombs, which include uniform tilings as cells: 801:
Semiregular polytopes can be extended to semiregular
1065:
Hypercubic honeycomb prism, named by Gosset as the (
229:
Gosset's 4-polytopes (with his names in parentheses)
213:, where the rectified 5-cell is the special case of 1129:has tetrahedral and two types of octahedral cells. 805:. The semiregular Euclidean honeycombs are the 2185:(3rd ed.). New York: Dover Publications. 1134: 1740:Alternated order-6 hexagonal tiling honeycomb 1669:Alternated order-5 hexagonal tiling honeycomb 1598:Alternated order-4 hexagonal tiling honeycomb 168:have identical meanings, because all uniform 8: 2255:The Semiregular Polytopes of the Hyperspaces 144:The Semiregular Polytopes of the Hyperspaces 1127:hyperbolic tetrahedral-octahedral honeycomb 222: 1374:Rectified order-4 square tiling honeycomb 788: 1982:Tetrahedral-triangular tiling honeycomb 1288:Rectified order-6 tetrahedral honeycomb 977:(9-ic check) (8D Euclidean honeycomb), 218: 2257:. Groningen: University of Groningen. 1058: 2137:(1991). "The semiregular polytopes". 2011:9D hyperbolic paracompact honeycomb: 1527:Alternated hexagonal tiling honeycomb 27:Isogonal polytope with regular facets 7: 1073:Alternated hexagonal slab honeycomb 146:which included a wider definition. 1811:Alternated square tiling honeycomb 1455:Alternated order-6 cubic honeycomb 1148:Alternated order-5 cubic honeycomb 913:Gyrated alternated cubic honeycomb 811:gyrated alternated cubic honeycomb 25: 2140:Commentarii Mathematici Helvetici 1331:Rectified square tiling honeycomb 1252:Tetrahedron-icosahedron honeycomb 2206:Proceedings of the Royal Society 2103: 2098: 2093: 2088: 2083: 2078: 2073: 2068: 2063: 2058: 2053: 2048: 2043: 2038: 2033: 2028: 2023: 2001: 1996: 1991: 1986: 1973: 1968: 1963: 1958: 1953: 1948: 1943: 1935: 1930: 1925: 1920: 1915: 1902: 1897: 1892: 1887: 1873: 1868: 1863: 1858: 1853: 1845: 1840: 1835: 1830: 1825: 1820: 1815: 1802: 1797: 1792: 1787: 1782: 1774: 1769: 1764: 1759: 1754: 1749: 1744: 1731: 1726: 1721: 1716: 1711: 1703: 1698: 1693: 1688: 1683: 1678: 1673: 1660: 1655: 1650: 1645: 1640: 1632: 1627: 1622: 1617: 1612: 1607: 1602: 1589: 1584: 1579: 1574: 1569: 1561: 1556: 1551: 1546: 1541: 1536: 1531: 1517: 1512: 1507: 1502: 1497: 1489: 1484: 1479: 1474: 1469: 1464: 1459: 1446: 1441: 1436: 1431: 1426: 1421: 1416: 1408: 1403: 1398: 1393: 1388: 1383: 1378: 1365: 1360: 1355: 1350: 1345: 1340: 1335: 1322: 1317: 1312: 1307: 1302: 1297: 1292: 1271: 1266: 1261: 1256: 1243: 1238: 1233: 1228: 1224:Tetrahedral-octahedral honeycomb 1210: 1205: 1200: 1195: 1190: 1182: 1177: 1172: 1167: 1162: 1157: 1152: 1107: 1102: 1097: 1092: 1087: 1082: 1077: 1049: 1044: 1039: 1034: 1029: 1024: 1019: 1014: 1009: 1004: 999: 994: 989: 984: 979: 957: 952: 947: 942: 937: 932: 927: 922: 917: 915:(Complex tetroctahedric check), 899: 894: 889: 884: 879: 871: 866: 861: 856: 851: 846: 841: 833:Tetrahedral-octahedral honeycomb 807:tetrahedral-octahedral honeycomb 795:tetrahedral-octahedral honeycomb 775: 770: 765: 760: 755: 750: 745: 740: 735: 730: 725: 720: 715: 694: 689: 684: 679: 674: 669: 664: 659: 654: 649: 644: 623: 618: 613: 608: 603: 598: 593: 585: 580: 575: 570: 565: 560: 555: 550: 545: 524: 519: 514: 509: 504: 499: 494: 486: 481: 476: 471: 466: 461: 456: 451: 446: 421: 416: 411: 406: 401: 393: 388: 383: 378: 373: 368: 363: 355: 350: 345: 340: 335: 330: 325: 312: 307: 302: 297: 292: 287: 282: 269: 264: 259: 254: 249: 244: 239: 92: 81: 70: 52: 41: 1911:Order-4 square tiling honeycomb 839:(Simple tetroctahedric check), 1282:Paracompact uniform honeycombs 1: 2296:Mat. Issled. Akad. Nauk. Mold 1883:Cubic-square tiling honeycomb 1142:Hyperbolic uniform honeycombs 1075:(tetroctahedric semi-check), 187:The three convex semiregular 59:Complex tetroctahedric check 48:Simple tetroctahedric check 2341: 1135:Coxeter & Whitrow 1950 837:alternated cubic honeycomb 966:Semiregular E-honeycomb: 221:for four dimensions, and 176:. However, since not all 118:is usually taken to be a 64: 35: 2282:Messenger of Mathematics 709:(8-ic semi-regular), an 223:Blind & Blind (1991) 638:(7-ic semi-regular), a 539:(6-ic semi-regular), a 440:(5-ic semi-regular), a 430:Semiregular E-polytopes 225:for higher dimensions. 156:three-dimensional space 2227:10.1098/rspa.1950.0070 2121:Semiregular polyhedron 1130: 798: 1218:quasiregular polytope 1124: 1117:Hyperbolic honeycombs 907:quasiregular polytope 792: 158:and below, the terms 18:Semi-regular polytope 785:Euclidean honeycombs 432:in higher dimensions 160:semiregular polytope 149: 116:semiregular polytope 2219:1950RSPSA.201..417C 1879:(Also quasiregular) 1523:(Also quasiregular) 323:(Tetricosahedric), 140:longer list in 1912 32: 2153:10.1007/BF02566640 1131: 799: 280:(Octicosahedric), 278:Rectified 600-cell 237:(Tetroctahedric), 201:rectified 600-cell 30: 2325:Uniform polytopes 2213:(1066): 417–437. 2201:Coxeter, H. S. M. 2182:Regular Polytopes 2177:Coxeter, H. S. M. 1144:, 3D honeycombs: 178:uniform polyhedra 132:regular polytopes 124:vertex-transitive 104: 103: 31:Gosset's figures 16:(Redirected from 2332: 2311: 2302:: 139–150, 177. 2290: 2268: 2246: 2196: 2172: 2108: 2107: 2106: 2102: 2101: 2097: 2096: 2092: 2091: 2087: 2086: 2082: 2081: 2077: 2076: 2072: 2071: 2067: 2066: 2062: 2061: 2057: 2056: 2052: 2051: 2047: 2046: 2042: 2041: 2037: 2036: 2032: 2031: 2027: 2026: 2006: 2005: 2004: 2000: 1999: 1995: 1994: 1990: 1989: 1978: 1977: 1976: 1972: 1971: 1967: 1966: 1962: 1961: 1957: 1956: 1952: 1951: 1947: 1946: 1940: 1939: 1938: 1934: 1933: 1929: 1928: 1924: 1923: 1919: 1918: 1907: 1906: 1905: 1901: 1900: 1896: 1895: 1891: 1890: 1878: 1877: 1876: 1872: 1871: 1867: 1866: 1862: 1861: 1857: 1856: 1850: 1849: 1848: 1844: 1843: 1839: 1838: 1834: 1833: 1829: 1828: 1824: 1823: 1819: 1818: 1807: 1806: 1805: 1801: 1800: 1796: 1795: 1791: 1790: 1786: 1785: 1779: 1778: 1777: 1773: 1772: 1768: 1767: 1763: 1762: 1758: 1757: 1753: 1752: 1748: 1747: 1736: 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2291: 2269: 2263: 2247: 2197: 2191: 2173: 2147:(1): 150–154. 2128: 2125: 2124: 2123: 2116: 2113: 2112: 2111: 2110: 2109: 2016: 2009: 2008: 2007: 1979: 1908: 1880: 1808: 1737: 1666: 1595: 1524: 1452: 1371: 1328: 1279: 1278: 1277: 1249: 1221: 1137:), including: 1118: 1115: 1114: 1113: 1070: 1056: 1055: 972: 964: 963: 910: 816: 786: 783: 782: 781: 704: 700: 633: 629: 534: 530: 435: 433: 427: 318: 275: 232: 230: 219:Makarov (1988) 208: 151: 148: 112:Thorold Gosset 102: 101: 90: 88:Octicosahedric 79: 77:Tetroctahedric 67: 66: 62: 61: 50: 38: 37: 36:3D honeycombs 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 2337: 2326: 2323: 2322: 2320: 2309: 2305: 2301: 2297: 2292: 2288: 2284: 2283: 2279:dimensions". 2278: 2274: 2270: 2266: 2264:1-4181-7968-X 2260: 2256: 2252: 2248: 2244: 2240: 2236: 2232: 2228: 2224: 2220: 2216: 2212: 2208: 2207: 2202: 2198: 2194: 2192:0-486-61480-8 2188: 2184: 2183: 2178: 2174: 2170: 2166: 2162: 2158: 2154: 2150: 2146: 2142: 2141: 2136: 2131: 2130: 2126: 2122: 2119: 2118: 2114: 2020: 2013: 2012: 2010: 1983: 1980: 1912: 1909: 1884: 1881: 1812: 1809: 1741: 1738: 1670: 1667: 1599: 1596: 1528: 1525: 1456: 1453: 1375: 1372: 1332: 1329: 1289: 1286: 1285: 1283: 1280: 1253: 1250: 1225: 1222: 1219: 1149: 1146: 1145: 1143: 1140: 1139: 1138: 1136: 1128: 1123: 1116: 1074: 1071: 1068: 1064: 1063: 1062: 1060: 1059:Gosset (1900) 976: 969: 968: 967: 914: 911: 908: 838: 834: 831: 830: 829: 827: 822: 820: 813:(3D) and the 812: 808: 804: 796: 791: 784: 712: 708: 701: 641: 637: 630: 542: 538: 531: 443: 439: 436: 434: 431: 428: 322: 319: 279: 276: 236: 233: 231: 228: 227: 226: 224: 220: 216: 212: 207: 202: 198: 194: 190: 185: 183: 179: 175: 171: 167: 166: 161: 157: 150:Gosset's list 147: 145: 141: 137: 133: 129: 125: 121: 117: 113: 109: 100: 95: 91: 89: 84: 80: 78: 73: 69: 68: 65:4D polytopes 63: 60: 55: 51: 49: 44: 40: 39: 34: 19: 2299: 2295: 2286: 2280: 2276: 2254: 2210: 2204: 2180: 2144: 2138: 1132: 1066: 1057: 965: 823: 800: 321:Snub 24-cell 214: 205: 197:snub 24-cell 186: 163: 159: 153: 143: 115: 105: 2251:Elte, E. L. 2133:Blind, G.; 189:4-polytopes 138:compiled a 2127:References 826:honeycombs 803:honeycombs 711:8-polytope 640:7-polytope 541:6-polytope 442:5-polytope 438:5-demicube 2243:120322123 2169:119695696 2135:Blind, R. 2019:honeycomb 975:honeycomb 819:honeycomb 211:polytopes 136:E.L. Elte 2319:Category 2289:: 43–48. 2253:(1912). 2179:(1973). 2115:See also 707:polytope 636:polytope 537:polytope 191:are the 172:must be 170:polygons 122:that is 120:polytope 108:geometry 2308:0958024 2235:0041576 2215:Bibcode 2161:1090169 824:Gosset 182:regular 174:regular 2306:  2261:  2241:  2233:  2189:  2167:  2159:  1216:(Also 905:(Also 821:(8D). 809:(3D), 130:being 128:facets 2239:S2CID 2165:S2CID 110:, by 2259:ISBN 2187:ISBN 1125:The 793:The 199:and 180:are 162:and 2300:103 2223:doi 2211:201 2149:doi 835:or 591:or 399:or 154:In 142:as 106:In 2321:: 2304:MR 2298:. 2287:29 2285:. 2237:. 2231:MR 2229:. 2221:. 2209:. 2163:. 2157:MR 2155:. 2145:66 2143:. 2017:21 1984:, 1941:= 1913:, 1885:, 1851:↔ 1813:, 1780:↔ 1742:, 1709:↔ 1671:, 1638:↔ 1600:, 1567:↔ 1529:, 1495:↔ 1457:, 1414:↔ 1376:, 1333:, 1290:, 1254:, 1226:, 1188:↔ 1150:, 973:21 877:↔ 828:: 817:21 713:, 705:21 642:, 634:21 543:, 535:21 492:↔ 444:, 361:, 209:21 195:, 134:. 2310:. 2277:n 2267:. 2245:. 2225:: 2217:: 2195:. 2171:. 2151:: 2015:6 1220:) 1067:n 971:5 909:) 815:5 703:4 632:3 533:2 215:k 206:k 20:)

Index

Semi-regular polytope

Simple tetroctahedric check

Complex tetroctahedric check

Tetroctahedric

Octicosahedric

Tetricosahedric
geometry
Thorold Gosset
polytope
vertex-transitive
facets
regular polytopes
E.L. Elte
longer list in 1912
three-dimensional space
uniform polytope
polygons
regular
uniform polyhedra
regular
4-polytopes
rectified 5-cell
snub 24-cell
rectified 600-cell
k21 polytopes

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