546:
521:
390:
757:
417:
364:
811:
377:
434:
172:
534:
558:
234:
225:
216:
207:
198:
344:
189:
163:
782:
149:
isogonal – demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.
1018:, Physics Department, Oregon State University, Corvallis, Presented at Mosaic2000, Millennial Open Symposium on the Arts and Interdisciplinary Computing, 21–24 August 2000, Seattle, WA
469:
notation sequencing the faces around each vertex. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration.
1015:
787:
910:
The
Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History
1024:
621:(edge-transitive). (Definition varies among authors; e.g. some exclude solids with dihedral symmetry, or nonconvex solids.)
867:
820:
1076:
968:
941:
1019:
762:
142:
91:
1199:
1179:
1174:
1131:
1106:
1234:
686:
1159:
900:
Coxeter, The
Densities of the Regular Polytopes II, p54-55, "hexagram" vertex figure of h{5/2,5}.
593:
551:
443:
24:
1184:
1069:
609:
1585:
1525:
1164:
1039:
485:
998:
1469:
1239:
1169:
1111:
888:
816:
799:
670:
545:
466:
259:
8:
1575:
1550:
1520:
1515:
1474:
1189:
539:
494:
310: = 2, 3, ...) with reflection lines across the mid-edge points.
77:
Technically, one says that for any two vertices there exists a symmetry of the polytope
1611:
1606:
1580:
1121:
682:
625:
573:
461:
357:
86:
948:
917:
862:
1560:
1154:
1062:
1036:
995:
964:
937:
770:
769:
because it contains two transitivity classes of vertices. This polyhedron is made of
629:
if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular.
353:
297:
78:
59:
520:
433:
1089:
876:
746:
701:
697:
674:
645:
369:
271:
251:
756:
1555:
1535:
1500:
1219:
1194:
1126:
884:
841:
835:
742:
586:
526:
389:
67:
66:
of the figure. This implies that each vertex is surrounded by the same kinds of
1565:
1545:
1510:
1505:
1136:
1116:
416:
242:
131:
97:
71:
957:
1600:
1540:
1391:
1284:
1204:
1146:
952:
693:
425:
1029:
810:
376:
363:
171:
1570:
1440:
1396:
1360:
1350:
1345:
1012:
395:
267:
135:
47:
20:
1479:
1386:
1365:
1355:
533:
1484:
1340:
1330:
1214:
1025:
Steven Dutch uses the term k-uniform for enumerating k-isogonal tilings
880:
557:
233:
224:
215:
206:
43:
197:
1459:
1449:
1426:
1416:
1406:
1335:
1244:
1209:
1044:
1003:
657:
650:
614:
602:
578:
349:
343:
282:
278:
255:
188:
180:
585:(edge-transitive); this implies that every face is the same kind of
162:
1464:
1454:
1411:
1370:
1299:
1289:
1279:
1098:
666:
618:
598:
582:
109:
82:
63:
35:
31:
1054:
1034:
1421:
1401:
1314:
1309:
1304:
1294:
1269:
1224:
1085:
803:
774:
382:
39:
993:
1229:
947:
781:
1274:
568:
Isogonal polyhedra and 2D tilings may be further classified:
745:. They can be represented visually with colors by different
95:
on its vertices, or that the vertices lie within a single
665:
These definitions can be extended to higher-dimensional
85:
onto the second. Other ways of saying this are that the
404:
707:
153:
956:
613:if every face is a regular polygon but it is not
1598:
661:dimensions: Isogonal polytopes and tessellations
281:which alternate two edge lengths, for example a
130:is a synonym borrowed from modern ideas such as
70:in the same or reverse order, and with the same
730:transitivity classes. A more restrictive term,
16:Polytope or tiling whose vertices are identical
455:and 2D tiling has a single kind of vertex. An
372:with 6 identical vertices and 2 edge lengths.
1070:
385:with blue and red radial lines of reflection
1077:
1063:
1051:(Also uses term k-uniform for k-isogonal)
398:with one vertex type, and two edge types
108:-dimensional isogonal figure exist on an
861:
1599:
1058:
1035:
994:
868:Discrete & Computational Geometry
641:if all the edges are the same length.
1084:
936:, Cambridge University Press 1997,
821:final stellation of the icosahedron
719:A polytope or tiling may be called
13:
1013:Isogonal Kaleidoscopical Polyhedra
635:if its elements are also isogonal.
126:has long been used for polyhedra.
14:
1623:
987:
459:with all regular faces is also a
405:Isogonal polyhedra and 2D tilings
865:(1997), "Isogonal prismatoids",
809:
780:
755:
556:
544:
532:
519:
432:
415:
388:
375:
362:
342:
232:
223:
214:
205:
196:
187:
170:
161:
154:Isogonal polygons and apeirogons
903:
894:
855:
763:truncated rhombic dodecahedron
696:of an isogonal polytope is an
408:
157:
1:
963:. W. H. Freeman and Company.
848:
700:, which is transitive on its
277:Some even-sided polygons and
270:of an isogonal polygon is an
74:between corresponding faces.
751:
471:
465:and can be represented by a
312:
7:
1040:"Demiregular tessellations"
829:
145: – which is
10:
1628:
944:, p. 369 Transitivity
798:). This tiling is made of
753:
601:(edge-transitive) but not
529:(ditrigonal trapezoprism)
517:
503:
18:
1493:
1439:
1379:
1323:
1262:
1253:
1145:
1097:
1030:List of n-uniform tilings
999:"Vertex-transitive graph"
914:Metamorphoses of polygons
687:convex uniform honeycombs
493:
143:pseudorhombicuboctahedron
104:All vertices of a finite
62:are equivalent under the
562:A hyper-truncated cube
552:truncated cuboctahedron
444:truncated square tiling
25:vertex-transitive graph
920:, Figure 1. Parameter
741:constructed only from
581:(face-transitive) and
87:group of automorphisms
726:if its vertices form
617:(face-transitive) or
292:All planar isogonal 2
260:regular star polygons
1310:Nonagon/Enneagon (9)
1240:Tangential trapezoid
959:Tilings and Patterns
800:equilateral triangle
467:vertex configuration
1422:Megagon (1,000,000)
1190:Isosceles trapezoid
1016:Vladimir L. Bulatov
979:tiling, p. 65
932:Peter R. Cromwell,
683:uniform 4-polytopes
681:, for example, the
540:rhombicuboctahedron
474:
473:Isogonal polyhedra
457:isogonal polyhedron
453:isogonal polyhedron
411:
1392:Icositetragon (24)
1037:Weisstein, Eric W.
996:Weisstein, Eric W.
881:10.1007/PL00009307
844:(Isohedral figure)
788:demiregular tiling
653:(face-transitive).
605:(face-transitive).
472:
462:uniform polyhedron
409:
358:vertex arrangement
354:crossed rectangles
1594:
1593:
1435:
1434:
1412:Myriagon (10,000)
1397:Triacontagon (30)
1361:Heptadecagon (17)
1351:Pentadecagon (15)
1346:Tetradecagon (14)
1285:Quadrilateral (4)
1155:Antiparallelogram
981:k-uniform tilings
838:(Isotoxal figure)
827:
826:
747:uniform colorings
739:k-isogonal figure
675:uniform polytopes
566:
565:
449:
448:
410:Isogonal tilings
402:
401:
356:sharing the same
298:dihedral symmetry
248:
247:
128:Vertex-transitive
92:acts transitively
56:vertex-transitive
1619:
1407:Chiliagon (1000)
1387:Icositrigon (23)
1366:Octadecagon (18)
1356:Hexadecagon (16)
1260:
1259:
1079:
1072:
1065:
1056:
1055:
1050:
1049:
1009:
1008:
974:
962:
949:Grünbaum, Branko
925:
907:
901:
898:
892:
891:
863:Grünbaum, Branko
859:
813:
784:
759:
752:
743:regular polygons
737:is defined as a
715:-uniform figures
698:isohedral figure
560:
548:
536:
523:
475:
436:
419:
412:
394:Isogonal "star"
392:
381:Isogonal convex
379:
366:
346:
313:
272:isotoxal polygon
252:regular polygons
236:
227:
218:
209:
200:
191:
174:
165:
158:
116:
107:
89:of the polytope
1627:
1626:
1622:
1621:
1620:
1618:
1617:
1616:
1597:
1596:
1595:
1590:
1489:
1443:
1431:
1375:
1341:Tridecagon (13)
1331:Hendecagon (11)
1319:
1255:
1249:
1220:Right trapezoid
1141:
1093:
1083:
990:
971:
953:Shephard, G. C.
929:
928:
918:Branko Grünbaum
908:
904:
899:
895:
860:
856:
851:
842:Face-transitive
836:Edge-transitive
832:
815:2-isogonal 9/4
814:
785:
760:
717:
663:
587:regular polygon
561:
549:
537:
527:hexagonal prism
524:
498:
489:
481:
442:
407:
393:
380:
367:
347:
337:
331:
325:
319:
305:
243:skew apeirogons
156:
132:symmetry groups
110:
105:
28:
17:
12:
11:
5:
1625:
1615:
1614:
1609:
1592:
1591:
1589:
1588:
1583:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1546:Pseudotriangle
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1497:
1495:
1491:
1490:
1488:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1452:
1446:
1444:
1437:
1436:
1433:
1432:
1430:
1429:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1389:
1383:
1381:
1377:
1376:
1374:
1373:
1368:
1363:
1358:
1353:
1348:
1343:
1338:
1336:Dodecagon (12)
1333:
1327:
1325:
1321:
1320:
1318:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1282:
1277:
1272:
1266:
1264:
1257:
1251:
1250:
1248:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1177:
1172:
1167:
1162:
1157:
1151:
1149:
1147:Quadrilaterals
1143:
1142:
1140:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1103:
1101:
1095:
1094:
1082:
1081:
1074:
1067:
1059:
1053:
1052:
1032:
1027:
1022:
1010:
989:
988:External links
986:
985:
984:
969:
945:
927:
926:
902:
893:
853:
852:
850:
847:
846:
845:
839:
831:
828:
825:
824:
807:
778:
773:and flattened
716:
711:-isogonal and
706:
662:
656:
655:
654:
649:if it is also
642:
636:
630:
622:
606:
597:if it is also
590:
577:if it is also
564:
563:
554:
542:
530:
516:
515:
512:
509:
506:
502:
501:
496:
492:
487:
483:
479:
447:
446:
438:
437:
429:
428:
421:
420:
406:
403:
400:
399:
386:
373:
360:
339:
338:
335:
332:
329:
326:
323:
320:
317:
301:
246:
245:
238:
237:
229:
228:
220:
219:
211:
210:
202:
201:
193:
192:
184:
183:
176:
175:
167:
166:
155:
152:
98:symmetry orbit
15:
9:
6:
4:
3:
2:
1624:
1613:
1610:
1608:
1605:
1604:
1602:
1587:
1586:Weakly simple
1584:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1526:Infinite skew
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1498:
1496:
1492:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1447:
1445:
1442:
1441:Star polygons
1438:
1428:
1427:Apeirogon (∞)
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1393:
1390:
1388:
1385:
1384:
1382:
1378:
1372:
1371:Icosagon (20)
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1328:
1326:
1322:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1267:
1265:
1261:
1258:
1252:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1205:Parallelogram
1203:
1201:
1200:Orthodiagonal
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1180:Ex-tangential
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1152:
1150:
1148:
1144:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1104:
1102:
1100:
1096:
1091:
1087:
1080:
1075:
1073:
1068:
1066:
1061:
1060:
1057:
1047:
1046:
1041:
1038:
1033:
1031:
1028:
1026:
1023:
1021:
1017:
1014:
1011:
1006:
1005:
1000:
997:
992:
991:
982:
978:
972:
970:0-7167-1193-1
966:
961:
960:
954:
950:
946:
943:
942:0-521-55432-2
939:
935:
931:
930:
923:
919:
915:
911:
906:
897:
890:
886:
882:
878:
874:
870:
869:
864:
858:
854:
843:
840:
837:
834:
833:
822:
819:(face of the
818:
812:
808:
805:
801:
797:
793:
789:
783:
779:
776:
772:
768:
764:
758:
754:
750:
748:
744:
740:
736:
734:
729:
725:
723:
714:
710:
705:
703:
699:
695:
690:
688:
684:
680:
676:
672:
671:tessellations
668:
660:
652:
648:
647:
643:
640:
637:
634:
631:
628:
627:
623:
620:
616:
612:
611:
607:
604:
600:
596:
595:
594:Quasi-regular
591:
588:
584:
580:
576:
575:
571:
570:
569:
559:
555:
553:
547:
543:
541:
535:
531:
528:
522:
518:
513:
510:
507:
504:
499:
490:
484:
477:
476:
470:
468:
464:
463:
458:
454:
445:
440:
439:
435:
431:
430:
427:
426:square tiling
423:
422:
418:
414:
413:
397:
391:
387:
384:
378:
374:
371:
365:
361:
359:
355:
351:
345:
341:
340:
333:
327:
321:
315:
314:
311:
309:
304:
299:
295:
290:
288:
284:
280:
275:
273:
269:
265:
261:
257:
253:
244:
240:
239:
235:
231:
230:
226:
222:
221:
217:
213:
212:
208:
204:
203:
199:
195:
194:
190:
186:
185:
182:
178:
177:
173:
169:
168:
164:
160:
159:
151:
148:
144:
139:
137:
133:
129:
125:
120:
118:
114:
102:
100:
99:
94:
93:
88:
84:
83:isometrically
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
26:
22:
1530:
1380:>20 sides
1315:Decagon (10)
1300:Heptagon (7)
1290:Pentagon (5)
1280:Triangle (3)
1175:Equidiagonal
1043:
1002:
980:
976:
975:(p. 33
958:
933:
921:
913:
909:
905:
896:
875:(1): 13–52,
872:
866:
857:
802:and regular
795:
791:
766:
738:
732:
731:
727:
721:
720:
718:
712:
708:
691:
678:
664:
658:
644:
638:
633:Semi-uniform
632:
624:
610:Semi-regular
608:
592:
572:
567:
538:A distorted
525:A distorted
460:
456:
452:
450:
396:tetradecagon
307:
302:
293:
291:
286:
276:
263:
249:
146:
140:
136:graph theory
127:
123:
121:
112:
103:
96:
90:
76:
55:
51:
29:
21:graph theory
1576:Star-shaped
1551:Rectilinear
1521:Equilateral
1516:Equiangular
1480:Hendecagram
1324:11–20 sides
1305:Octagon (8)
1295:Hexagon (6)
1270:Monogon (1)
1112:Equilateral
1020:VRML models
500:, order 48
491:, order 24
482:, order 12
441:A distorted
296:-gons have
58:if all its
1601:Categories
1581:Tangential
1485:Dodecagram
1263:1–10 sides
1254:By number
1235:Tangential
1215:Right kite
977:k-isogonal
912:, (1994),
849:References
792:2-isogonal
767:2-isogonal
550:A shallow
424:Distorted
350:rectangles
279:apeirogons
256:apeirogons
181:apeirogons
81:the first
64:symmetries
44:polyhedron
1612:Polytopes
1607:Polyhedra
1561:Reinhardt
1470:Enneagram
1460:Heptagram
1450:Pentagram
1417:65537-gon
1275:Digon (2)
1245:Trapezoid
1210:Rectangle
1160:Bicentric
1122:Isosceles
1099:Triangles
1045:MathWorld
1004:MathWorld
934:Polyhedra
817:enneagram
804:hexagonal
796:2-uniform
724:-isogonal
667:polytopes
651:isohedral
639:Scaliform
615:isohedral
603:isohedral
579:isohedral
368:Isogonal
348:Isogonal
283:rectangle
241:Isogonal
179:Isogonal
122:The term
1536:Isotoxal
1531:Isogonal
1475:Decagram
1465:Octagram
1455:Hexagram
1256:of sides
1185:Harmonic
1086:Polygons
955:(1987).
830:See also
790:is also
775:hexagons
735:-uniform
679:isogonal
619:isotoxal
599:isotoxal
583:isotoxal
508:3.4.4.4
370:hexagram
287:isogonal
264:isogonal
124:isogonal
60:vertices
52:isogonal
38:(e.g. a
36:polytope
32:geometry
1556:Regular
1501:Concave
1494:Classes
1402:257-gon
1225:Rhombus
1165:Crossed
889:1453440
806:faces.
771:squares
673:. All
626:Uniform
574:Regular
383:octagon
117:-sphere
79:mapping
46:) or a
40:polygon
1566:Simple
1511:Cyclic
1506:Convex
1230:Square
1170:Cyclic
1132:Obtuse
1127:Kepler
967:
940:
887:
702:facets
514:3.8.8
511:4.6.8
505:4.4.6
285:, are
266:. The
72:angles
48:tiling
23:, see
1541:Magic
1137:Right
1117:Ideal
1107:Acute
794:(and
786:This
761:This
646:Noble
1571:Skew
1195:Kite
1090:List
965:ISBN
938:ISBN
924:=2.0
694:dual
692:The
685:and
677:are
669:and
352:and
268:dual
262:are
258:and
250:All
141:The
134:and
68:face
34:, a
19:For
877:doi
765:is
451:An
147:not
115:−1)
54:or
50:is
42:or
30:In
1603::
1042:.
1001:.
951:;
916:,
885:MR
883:,
873:18
871:,
823:)
777:.
749:.
704:.
689:.
480:3d
306:,
300:(D
289:.
274:.
254:,
138:.
119:.
101:.
1092:)
1088:(
1078:e
1071:t
1064:v
1048:.
1007:.
983:)
973:.
922:t
879::
733:k
728:k
722:k
713:k
709:k
659:N
589:.
497:h
495:O
488:h
486:T
478:D
336:7
334:D
330:4
328:D
324:3
322:D
318:2
316:D
308:n
303:n
294:n
113:n
111:(
106:n
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
