1949:
1574:
1277:
303:
788:
781:
774:
767:
1960:
1585:
1288:
314:
802:
1828:
1453:
1164:
175:
760:
1014:
957:
918:
861:
217:
210:
1870:
1863:
1495:
1488:
1199:
1934:, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an
1559:, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an
1262:, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an
288:{3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an
2054:
2203:
677:
480:
470:
2199:
2041:
1812:
1068:
747:
2127:
2107:
2085:
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1058:
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347:
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2008:
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131:
2036:
2016:
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1807:
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63:
2003:
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159:
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337:
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126:
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106:
88:
78:
68:
2217:
2117:
1505:
2189:
227:
2059:
1684:
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1314:
1093:
38:
2076:
1483:
1263:
1194:
961:
922:
865:
806:
289:
205:
2045:, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is = .
1665:, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is = .
2111:
1953:
1887:
1578:
1512:
1281:
1216:
307:
234:
1948:
1573:
1276:
787:
780:
773:
766:
302:
1923:
1548:
1251:
269:
17:
2138:
1938:
1563:
1266:
292:
1971:
1927:
1691:
1596:
1552:
1321:
1255:
1100:
325:
273:
45:
2185:
1959:
1584:
1287:
313:
2123:
2103:
2095:
2081:
1931:
1823:
1556:
1448:
1259:
1159:
373:
369:
285:
170:
2071:
1911:
1536:
1239:
397:
357:
257:
1703:
1333:
1110:
376:
55:
2195:
2155:
2211:
1827:
1452:
1163:
801:
174:
2168:
1919:
1544:
1247:
265:
759:
2200:
Kleinian, a tool for visualizing
Kleinian groups, Geometry and the Imagination
1869:
1862:
1494:
1487:
1198:
1013:
956:
917:
860:
216:
209:
2177:
1848:
1907:
1838:
1532:
1463:
1235:
1174:
253:
185:
2171:
2088:. (Tables I and II: Regular polytopes and honeycombs, pp. 294â296)
1473:
1184:
195:
2099:
2151:
2169:
Spherical Video: {3,6,â} honeycomb with parabolic Möbius transform
2147:
1672:
1302:
1081:
356:, with alternating types or colors of triangular tiling cells. In
26:
2144:
Lorentzian
Coxeter groups and Boyd-Maxwell ball packings
2130:(Chapters 16â17: Geometries on Three-manifolds I, II)
1970:
It has a second construction as a uniform honeycomb,
1595:
It has a second construction as a uniform honeycomb,
324:
It has a second construction as a uniform honeycomb,
1943:
1568:
1271:
297:
2152:Visualizing Hyperbolic Honeycombs arXiv:1511.02851
2135:Sphere Packings and Hyperbolic Reflection Groups
2055:Convex uniform honeycombs in hyperbolic space
8:
1675:
1305:
1084:
29:
385:
2190:{7,3,3} Honeycomb Meets Plane at Infinity
2112:Regular Honeycombs in Hyperbolic Space
2080:, 3rd. ed., Dover Publications, 1973.
1676:Order-6-infinite triangular honeycomb
2137:, JOURNAL OF ALGEBRA 79,78-97 (1982)
2092:The Beauty of Geometry: Twelve Essays
1916:order-6-infinite triangular honeycomb
1669:Order-6-infinite triangular honeycomb
18:Order-6-infinite triangular honeycomb
7:
25:
2039:
2034:
2029:
2024:
2019:
2014:
2006:
2001:
1996:
1991:
1986:
1981:
1976:
1958:
1947:
1936:infinite-order triangular tiling
1930:{3,6,â}. It has infinitely many
1868:
1861:
1826:
1810:
1805:
1800:
1795:
1790:
1785:
1777:
1772:
1767:
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1644:
1639:
1631:
1626:
1621:
1616:
1611:
1606:
1601:
1583:
1572:
1555:{3,6,6}. It has infinitely many
1493:
1486:
1451:
1435:
1430:
1425:
1420:
1415:
1407:
1402:
1397:
1392:
1387:
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582:
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501:
496:
483:
478:
473:
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453:
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438:
433:
428:
423:
364:Related polytopes and honeycombs
350:
345:
340:
335:
330:
312:
301:
215:
208:
173:
157:
152:
147:
142:
137:
129:
124:
119:
114:
109:
104:
99:
91:
86:
81:
76:
71:
66:
61:
2142:Hao Chen, Jean-Philippe Labbé,
2121:The Shape of Space, 2nd edition
1896:
1886:
1876:
1854:
1844:
1834:
1819:
1702:
1690:
1680:
1521:
1511:
1501:
1479:
1469:
1459:
1444:
1332:
1320:
1310:
1306:Order-6-6 triangular honeycomb
1224:
1215:
1205:
1190:
1180:
1170:
1155:
1109:
1099:
1089:
1085:Order-6-5 triangular honeycomb
242:
233:
223:
201:
191:
181:
166:
54:
44:
34:
30:Order-6-4 triangular honeycomb
1974:{3,(6,â,6)}, Coxeter diagram,
1599:{3,(6,3,6)}, Coxeter diagram,
1541:order-6-6 triangular honeycomb
1299:Order-6-6 triangular honeycomb
1244:order-6-3 triangular honeycomb
1078:Order-6-5 triangular honeycomb
262:order-6-4 triangular honeycomb
1:
2094:(1999), Dover Publications,
1918:is a regular space-filling
1543:is a regular space-filling
1246:is a regular space-filling
368:It a part of a sequence of
264:is a regular space-filling
2234:
388:
360:the half symmetry is = .
2060:List of regular polytopes
1561:order-6 triangular tiling
409:
396:
1264:order-5 hexagonal tiling
328:{3,6}, Coxeter diagram,
290:order-4 hexagonal tiling
1258:{3,6,5}. It has five
2218:Regular 3-honeycombs
372:and honeycombs with
1954:Poincaré disk model
1579:Poincaré disk model
1282:Poincaré disk model
308:Poincaré disk model
1939:vertex arrangement
1912:hyperbolic 3-space
1564:vertex arrangement
1537:hyperbolic 3-space
1267:vertex arrangement
1240:hyperbolic 3-space
389:{3,6,p} polytopes
293:vertex arrangement
258:hyperbolic 3-space
2186:{7,3,3} Honeycomb
2077:Regular Polytopes
1968:
1967:
1932:triangular tiling
1904:
1903:
1685:Regular honeycomb
1593:
1592:
1557:triangular tiling
1529:
1528:
1315:Regular honeycomb
1296:
1295:
1260:triangular tiling
1232:
1231:
1094:Regular honeycomb
1075:
1074:
374:triangular tiling
370:regular polychora
322:
321:
286:triangular tiling
250:
249:
39:Regular honeycomb
16:(Redirected from
2225:
2133:George Maxwell,
2118:Jeffrey R. Weeks
2044:
2043:
2042:
2038:
2037:
2033:
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1944:
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1728:
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1723:
1722:
1718:
1717:
1713:
1712:
1704:Coxeter diagrams
1692:SchlÀfli symbols
1673:
1664:
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1334:Coxeter diagrams
1322:SchlÀfli symbols
1303:
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386:
358:Coxeter notation
355:
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219:
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95:
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69:
65:
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56:Coxeter diagrams
46:SchlÀfli symbols
27:
21:
2233:
2232:
2228:
2227:
2226:
2224:
2223:
2222:
2208:
2207:
2182:Visual insights
2165:
2068:
2051:
2040:
2035:
2030:
2025:
2020:
2015:
2013:
2007:
2002:
1997:
1992:
1987:
1982:
1977:
1975:
1972:SchlÀfli symbol
1963:
1952:
1928:SchlÀfli symbol
1892:
1866:
1811:
1806:
1801:
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1600:
1597:SchlÀfli symbol
1588:
1577:
1553:SchlÀfli symbol
1517:
1491:
1436:
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1291:
1280:
1256:SchlÀfli symbol
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1115:
1111:Coxeter diagram
1101:SchlÀfli symbol
1080:
1067:
1062:
1057:
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366:
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326:SchlÀfli symbol
317:
306:
282:
274:SchlÀfli symbol
213:
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2209:
2206:
2205:
2202:4 March 2014.
2196:Danny Calegari
2193:
2175:
2174:, Roice Nelson
2164:
2163:External links
2161:
2160:
2159:
2156:Henry Segerman
2154:Roice Nelson,
2149:
2140:
2131:
2115:
2089:
2067:
2064:
2063:
2062:
2057:
2050:
2047:
1966:
1965:
1964:Ideal surface
1956:
1902:
1901:
1898:
1894:
1893:
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1699:
1694:
1688:
1687:
1682:
1678:
1677:
1670:
1667:
1591:
1590:
1589:Ideal surface
1581:
1527:
1526:
1523:
1519:
1518:
1515:
1509:
1508:
1503:
1499:
1498:
1481:
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1476:
1471:
1467:
1466:
1461:
1457:
1456:
1446:
1442:
1441:
1336:
1330:
1329:
1324:
1318:
1317:
1312:
1308:
1307:
1300:
1297:
1294:
1293:
1292:Ideal surface
1284:
1230:
1229:
1226:
1222:
1221:
1219:
1213:
1212:
1207:
1203:
1202:
1192:
1188:
1187:
1182:
1178:
1177:
1172:
1168:
1167:
1157:
1153:
1152:
1113:
1107:
1106:
1103:
1097:
1096:
1091:
1087:
1086:
1079:
1076:
1073:
1072:
1010:
953:
914:
857:
798:
792:
791:
784:
777:
770:
763:
756:
752:
751:
674:
603:
560:
489:
416:
412:
411:
408:
405:
401:
400:
395:
391:
390:
365:
362:
320:
319:
318:Ideal surface
310:
281:
278:
248:
247:
244:
240:
239:
237:
231:
230:
225:
221:
220:
203:
199:
198:
193:
189:
188:
183:
179:
178:
168:
164:
163:
58:
52:
51:
48:
42:
41:
36:
32:
31:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2230:
2219:
2216:
2215:
2213:
2204:
2201:
2197:
2194:
2191:
2188:(2014/08/01)
2187:
2183:
2179:
2176:
2173:
2170:
2167:
2166:
2162:
2157:
2153:
2150:
2148:
2145:
2141:
2139:
2136:
2132:
2129:
2128:0-8247-0709-5
2125:
2122:
2119:
2116:
2113:
2110:(Chapter 10,
2109:
2108:0-486-40919-8
2105:
2101:
2097:
2093:
2090:
2087:
2086:0-486-61480-8
2083:
2079:
2078:
2073:
2070:
2069:
2065:
2061:
2058:
2056:
2053:
2052:
2048:
2046:
1973:
1961:
1957:
1955:
1950:
1946:
1945:
1942:
1940:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1899:
1895:
1891:
1889:
1888:Coxeter group
1885:
1882:
1879:
1875:
1871:
1864:
1860:
1857:
1855:Vertex figure
1853:
1850:
1847:
1843:
1840:
1837:
1833:
1829:
1825:
1822:
1818:
1707:
1705:
1701:
1695:
1693:
1689:
1686:
1683:
1679:
1674:
1668:
1666:
1598:
1586:
1582:
1580:
1575:
1571:
1570:
1567:
1565:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1524:
1520:
1516:
1514:
1513:Coxeter group
1510:
1507:
1504:
1500:
1496:
1489:
1485:
1482:
1480:Vertex figure
1478:
1475:
1472:
1468:
1465:
1462:
1458:
1454:
1450:
1447:
1443:
1337:
1335:
1331:
1325:
1323:
1319:
1316:
1313:
1309:
1304:
1298:
1289:
1285:
1283:
1278:
1274:
1273:
1270:
1268:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1227:
1223:
1220:
1218:
1217:Coxeter group
1214:
1211:
1208:
1204:
1200:
1196:
1193:
1191:Vertex figure
1189:
1186:
1183:
1179:
1176:
1173:
1169:
1165:
1161:
1158:
1154:
1114:
1112:
1108:
1104:
1102:
1098:
1095:
1092:
1088:
1083:
1077:
1020:
1015:
1011:
963:
958:
954:
924:
919:
915:
867:
862:
858:
808:
803:
799:
794:
793:
789:
785:
782:
778:
775:
771:
768:
764:
761:
757:
754:
753:
679:
678:{3,6,∞}
675:
607:
604:
564:
561:
493:
490:
420:
417:
414:
413:
406:
403:
402:
399:
393:
392:
387:
384:
382:
378:
375:
371:
363:
361:
359:
327:
315:
311:
309:
304:
300:
299:
296:
294:
291:
287:
279:
277:
275:
271:
267:
263:
259:
255:
245:
241:
238:
236:
235:Coxeter group
232:
229:
226:
222:
218:
211:
207:
204:
202:Vertex figure
200:
197:
194:
190:
187:
184:
180:
176:
172:
169:
165:
59:
57:
53:
49:
47:
43:
40:
37:
33:
28:
19:
2192:(2014/08/14)
2181:
2143:
2134:
2120:
2091:
2075:
1969:
1920:tessellation
1915:
1905:
1698:{3,(6,â,6)}
1594:
1545:tessellation
1540:
1530:
1328:{3,(6,3,6)}
1248:tessellation
1243:
1233:
491:
407:Paracompact
380:
367:
323:
284:It has four
283:
266:tessellation
261:
251:
2114:) Table III
1845:Edge figure
1470:Edge figure
1181:Edge figure
1019:{6,∞}
410:Noncompact
192:Edge figure
2066:References
1897:Properties
1867:{(6,â,6)}
1522:Properties
1492:{(6,3,6)}
1225:Properties
243:Properties
2178:John Baez
1924:honeycomb
1549:honeycomb
1252:honeycomb
276:{3,6,4}.
270:honeycomb
2212:Category
2146:, (2013)
2100:99-35678
2049:See also
1908:geometry
1900:Regular
1533:geometry
1525:Regular
1236:geometry
1228:Regular
1105:{3,6,5}
280:Geometry
254:geometry
246:Regular
50:{3,6,4}
2172:YouTube
2072:Coxeter
1926:) with
1906:In the
1881:{â,6,3}
1696:{3,6,â}
1551:) with
1531:In the
1506:{6,6,3}
1326:{3,6,6}
1254:) with
1234:In the
1210:{5,6,3}
797:figure
606:{3,6,6}
563:{3,6,5}
492:{3,6,4}
419:{3,6,3}
379:: {3,6,
272:) with
252:In the
228:{4,6,3}
214:r{6,6}
2158:(2015)
2126:
2106:
2098:
2084:
1914:, the
1539:, the
1242:, the
838:
795:Vertex
755:Image
460:
394:Space
260:, the
1859:{6,â}
1835:Faces
1824:{3,6}
1820:Cells
1484:{6,6}
1460:Faces
1449:{3,6}
1445:Cells
1195:{6,5}
1171:Faces
1160:{3,6}
1156:Cells
962:{6,6}
923:{6,5}
866:{6,4}
807:{6,3}
415:Name
404:Form
377:cells
206:{6,4}
182:Faces
171:{3,6}
167:Cells
2124:ISBN
2104:ISBN
2096:LCCN
2082:ISBN
1922:(or
1877:Dual
1681:Type
1547:(or
1502:Dual
1311:Type
1250:(or
1206:Dual
1090:Type
676:...
268:(or
224:Dual
35:Type
1910:of
1849:{â}
1839:{3}
1535:of
1474:{6}
1464:{3}
1238:of
1185:{5}
1175:{3}
256:of
196:{4}
186:{3}
2214::
2198:,
2184::
2180:,
2102:,
2074:,
2012:=
1941:.
1783:=
1637:=
1566:.
1413:=
1269:.
383:}
295:.
135:=
398:H
381:p
20:)
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