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109:
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1100:
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490:
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461:
428:
406:
417:
832:, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an
1163:{â,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal tilings existing around each vertex in an
1245:
340:
1390:
1255:
329:
334:
324:
204:
All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an
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1043:
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93:
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68:
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139:
38:
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154:
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511:
393:
386:
379:
218:
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17:
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591:
193:
45:
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351:
229:
1319:
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527:
454:
246:
104:
1267:
1140:
829:
809:
723:
465:
443:
279:
177:
896:, with alternating types or colors of cells. In Coxeter notation the half symmetry is = .
533:
934:
603:
314:
55:
1382:
1351:
1398:
208:
522:
500:
438:
1148:
817:
432:
185:
372:
516:
494:
449:
410:
108:
365:
1387:
Kleinian, a tool for visualizing
Kleinian groups, Geometry and the Imagination
1099:
1090:
769:
555:
505:
358:
1364:
1079:
1069:
762:
544:
427:
1136:
805:
271:
173:
129:
119:
1058:
727:
489:
471:
460:
405:
1284:. (Tables I and II: Regular polytopes and honeycombs, pp. 294â296)
748:
738:
416:
309:
1295:
1347:
1343:
903:
572:
26:
1236:, with alternating types or colors of apeirogonal tiling cells.
421:
1340:
Lorentzian
Coxeter groups and Boyd-Maxwell ball packings
1326:(Chapters 16â17: Geometries on Three-manifolds I, II)
1199:
It has a second construction as a uniform honeycomb,
859:
It has a second construction as a uniform honeycomb,
1348:Visualizing Hyperbolic Honeycombs arXiv:1511.02851
1331:Sphere Packings and Hyperbolic Reflection Groups
1246:Convex uniform honeycombs in hyperbolic space
8:
906:
575:
29:
259:
1377:{7,3,3} Honeycomb Meets Plane at Infinity
1172:
841:
213:
907:Order-3-infinite apeirogonal honeycomb
1308:Regular Honeycombs in Hyperbolic Space
1276:, 3rd. ed., Dover Publications, 1973.
1145:order-3-infinite apeirogonal honeycomb
900:Order-3-infinite apeirogonal honeycomb
1333:, JOURNAL OF ALGEBRA 79,78-97 (1982)
1288:The Beauty of Geometry: Twelve Essays
1256:Infinite-order dodecahedral honeycomb
7:
25:
18:Order-8 octagonal tiling honeycomb
1230:
1225:
1220:
1215:
1210:
1205:
1187:
1176:
1165:infinite-order triangular tiling
1159:{â,3,â}. It has infinitely many
1098:
1089:
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241:Related polytopes and honeycombs
228:
217:
107:
91:
86:
81:
76:
71:
66:
61:
1338:Hao Chen, Jean-Philippe Labbé,
1317:The Shape of Space, 2nd edition
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125:
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54:
44:
34:
30:Order-3-7 heptagonal honeycomb
1203:{â,(3,â,3)}, Coxeter diagram,
863:{8,(3,4,3)}, Coxeter diagram,
576:Order-3-8 octagonal honeycomb
182:order-3-7 heptagonal honeycomb
1:
814:order-3-8 octagonal honeycomb
569:Order-3-8 octagonal honeycomb
1290:(1999), Dover Publications,
1147:is a regular space-filling
816:is a regular space-filling
263:{p,3,p} regular honeycombs
245:It a part of a sequence of
1436:
1161:order-3 apeirogonal tiling
262:
1251:List of regular polytopes
834:order-8 triangular tiling
300:
278:
206:order-7 triangular tiling
184:a regular space-filling
1410:Infinite-order tilings
828:{8,3,8}. It has eight
1420:Regular 3-honeycombs
1183:Poincaré disk model
852:Poincaré disk model
341:{∞,3,∞}
224:Poincaré disk model
1405:Heptagonal tilings
1168:vertex arrangement
1141:hyperbolic 3-space
837:vertex arrangement
810:hyperbolic 3-space
178:hyperbolic 3-space
1373:{7,3,3} Honeycomb
1273:Regular Polytopes
1197:
1196:
1133:
1132:
916:Regular honeycomb
857:
856:
830:octagonal tilings
802:
801:
585:Regular honeycomb
566:
565:
247:regular polychora
238:
237:
170:
169:
39:Regular honeycomb
16:(Redirected from
1427:
1329:George Maxwell,
1314:Jeffrey R. Weeks
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923:SchlÀfli symbols
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592:SchlÀfli symbols
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1369:Visual insights
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1201:SchlÀfli symbol
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861:SchlÀfli symbol
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826:SchlÀfli symbol
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194:SchlÀfli symbol
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46:SchlÀfli symbol
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1392:
1389:4 March 2014.
1383:Danny Calegari
1380:
1360:
1359:External links
1357:
1356:
1355:
1352:Henry Segerman
1350:Roice Nelson,
1345:
1336:
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1263:
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1195:
1194:
1193:Ideal surface
1185:
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234:Ideal surface
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1324:0-8247-0709-5
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1309:
1306:(Chapter 10,
1305:
1304:0-486-40919-8
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1282:0-486-61480-8
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1117:Coxeter group
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1086:Vertex figure
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155:Coxeter group
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136:Vertex figure
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1415:3-honeycombs
1379:(2014/08/14)
1368:
1339:
1330:
1316:
1287:
1271:
1198:
1149:tessellation
1144:
1134:
929:{â,(3,â,3)}
858:
818:tessellation
813:
803:
598:{8,(3,4,3)}
298:Paracompact
276:Euclidean E
254:
250:
244:
203:
186:tessellation
181:
171:
1310:) Table III
1076:Edge figure
745:Edge figure
561:{3,∞}
477:{∞,3}
301:Noncompact
126:Edge figure
1399:Categories
1262:References
1126:Properties
1111:self-dual
1103:{(3,â,3)}
795:Properties
780:self-dual
767:{(3,8,3)}
163:Properties
149:self-dual
1365:John Baez
1153:honeycomb
822:honeycomb
196:{7,3,7}.
190:honeycomb
1342:, (2013)
1296:99-35678
1240:See also
1137:geometry
1129:Regular
806:geometry
798:Regular
295:Compact
200:Geometry
174:geometry
166:Regular
50:{7,3,7}
1268:Coxeter
1155:) with
1135:In the
927:{â,3,â}
824:) with
804:In the
596:{8,3,8}
485:figure
335:{8,3,8}
330:{7,3,7}
325:{6,3,6}
320:{5,3,5}
315:{4,3,4}
310:{3,3,3}
292:Affine
289:Finite
192:) with
172:In the
1354:(2015)
1322:
1302:
1294:
1280:
1143:, the
812:, the
483:Vertex
401:Cells
347:Image
268:Space
180:, the
1095:{3,â}
1066:Faces
1055:{â,3}
1051:Cells
759:{3,8}
735:Faces
724:{8,3}
720:Cells
550:{3,8}
539:{3,7}
528:{3,6}
517:{3,5}
506:{3,4}
495:{3,3}
466:{8,3}
455:{7,3}
444:{6,3}
433:{5,3}
422:{4,3}
411:{3,3}
306:Name
286:Form
140:{3,7}
116:Faces
105:{7,3}
101:Cells
1320:ISBN
1300:ISBN
1292:LCCN
1278:ISBN
1151:(or
1108:Dual
912:Type
820:(or
777:Dual
581:Type
188:(or
146:Dual
35:Type
1139:of
1080:{â}
1070:{â}
808:of
749:{8}
739:{8}
339:...
257:}:
253:,3,
176:of
130:{7}
120:{7}
1401::
1385:,
1371::
1367:,
1298:,
1270:,
1170:.
1014:â
839:.
683:=
211:.
280:H
272:S
255:p
251:p
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
