549:
542:
535:
528:
521:
35:
594:
587:
366:
335:
24:
303:
289:
275:
261:
352:
608:
601:
615:
322:
1055:
409:
Different notations are used, sometimes with a comma (,) and sometimes a period (.) separator. The period operator is useful because it looks like a product and an exponent notation can be used. For example, 3.5.3.5 is sometimes written as (3.5).
657:
Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in the
1495:
Pp. 58–64, Tilings of regular polygons a.b.c.... (Tilings by regular polygons and star polygons) pp. 95–97, 176, 283, 614–620, Monohedral tiling symbol . pp. 632–642 hollow tilings.
682:
the number of turns around a circle. For example, "3/2" means a triangle that has vertices that go around twice, which is the same as backwards once. Similarly "5/3" is a backwards pentagram 5/2.
459:
This notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron.
714:
NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if its defect is zero. It can represent a tiling of the hyperbolic plane if its defect is negative.
717:
For uniform polyhedra, the angle defect can be used to compute the number of vertices. Descartes' theorem states that all the angle defects in a topological sphere must sum to 4
695:
836:>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6. In fact, all these configurations with three faces meeting at each vertex turn out to exist.
152:(the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as
98:
A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "
1288:
773:
Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible.
1136:
1408:
199:
470:
has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration.
1194:
1506:
573:
724:
Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices, which is 4
1258:
Resources for
Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins
1543:
644:
1488:
1457:
639:
578:
634:
699:
919:
568:
562:
556:
498:
497:
of {5/2,5} which expands to an explicit vertex configuration 5/2.5/2.5/2.5/2.5/2 or combined as (5/2). The
490:
1025:
898:
489:
For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The
1301:
Deza, Michel; Shtogrin, Mikhail (2000), "Uniform partitions of 3-space, their relatives and embedding",
1445:
915:
986:
691:
392:
91:
there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (
513:, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration (3.3.3.3.3)/2 or (3)/2.
1014:
923:
894:
463:
231:
92:
828:. Therefore, potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.
548:
969:
886:
704:
226:
has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for
1369:
1179:
1167:
1021:
902:
878:
1210:
696:
Tiling by regular polygons § Combinations of regular polygons that can meet at a vertex
541:
1332:
1110:
1094:
1058:
890:
8:
982:
965:
1152:
1133:
136:. The notation is cyclic and therefore is equivalent with different starting points, so
1538:
1509:(p. 289 Vertex figures, uses comma separator, for Archimedean solids and tilings).
1310:
1267:
1029:
622:
534:
527:
520:
502:
501:, {5/2,3} has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2). The
227:
88:
1468:
1350:
1101:
This notation represents a sequential count of the number of faces that exist at each
494:
414:
1522:
1502:
1484:
1453:
1346:
1102:
1043:
1010:
961:
839:
The number in parentheses is the number of vertices, determined by the angle defect.
708:
628:
510:
307:
194:
133:
130:
84:
28:
1246:
1032:(note that the two different orders of the same numbers give two different patterns)
34:
1437:
1320:
1279:
Structure and Form in Design: Critical Ideas for
Creative Practice By Michael Hann
1191:
909:
863:
370:
1328:
1198:
1140:
1106:
1081:
80:
1084:) and so they can be identified by a similar notation which is sometimes called
882:
740:
685:
418:
1477:
1532:
1472:
1441:, Cambridge University Press (1977) The Archimedean solids. Pp. 156–167.
1065:
1054:
976:
957:
586:
339:
219:
68:
39:
1416:. Pp. 101–115, pp. 118–119 Table I, Nets of Archimedean Duals, V.
593:
486:
has the symbol {5/2}, meaning it has 5 sides going around the centre twice.
1519:
1403:
1324:
1073:
857:
659:
479:
384:
365:
356:
188:
95:
polyhedra exist in mirror-image pairs with the same vertex configuration.)
76:
999:
849:
293:
265:
1450:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1222:
334:
936:
279:
72:
389:
A 0° angle defect will fill the
Euclidean plane with a regular tiling.
1355:
1315:
1117:, and alternating vertices of the rhombus contain 3 or 4 faces each.
1069:
1006:
942:
483:
467:
1412:(1952), (3rd edition, 1989, Stradbroke, England: Tarquin Pub.), 3.7
1370:
Divided
Spheres: Geodesics and the Orderly Subdivision of the Sphere
1234:
413:
The notation can also be considered an expansive form of the simple
23:
126:
122:
60:
1168:
Crystallography of
Quasicrystals: Concepts, Methods and Structures
302:
1114:
288:
274:
260:
216:
1467:
351:
230:
all the neighboring vertices are in the same plane and so this
1098:
uses square brackets around the symbol for isohedral tilings.
614:
607:
600:
1501:
2008, John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss,
1170:
by Walter
Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53
686:
All uniform vertex configurations of regular convex polygons
234:
can be used to visually represent the vertex configuration.
102:" describes a vertex that has 3 faces around it, faces with
853:
743:
3.8.8 has an angle defect of 30 degrees. Therefore, it has
478:
The notation also applies for nonconvex regular faces, the
326:
321:
1344:
456:. For example, an icosahedron is {3,5} = 3.3.3.3.3 or 3.
1289:
Symmetry-type graphs of
Platonic and Archimedean solids
1235:
Symmetry-type graphs of
Platonic and Archimedean solids
776:
Topological requirements limit existence. Specifically
700:
Uniform tiling § Expanded lists of uniform tilings
121:" indicates a vertex belonging to 4 faces, alternating
1247:
3. General Theorems: Regular and Semi-Regular Tilings
1088:. Cundy and Rollett prefixed these dual symbols by a
1476:
1109:. For example, V3.4.3.4 or V(3.4) represents the
505:, {5,5/2} has a pentagrammic vertex figure, with
1530:
1182:edited by David E. Laughlin, (2014) pp. 16–20
866:3.4.4 (6), 4.4.4 (8; also listed above), 4.4.
67:is a shorthand notation for representing the
1300:
1180:Physical Metallurgy: 3-Volume Set, Volume 1
1113:which is face-transitive: every face is a
222:showing the faces around the vertex. This
1314:
1163:
1161:
707:have vertex configurations with positive
20:
16:Notation for a polyhedron's vertex figure
1444:
1053:
383:A vertex needs at least 3 faces, and an
129:. This vertex configuration defines the
1173:
692:Archimedean_solid § Classification
1531:
1158:
1134:Uniform Solution for Uniform Polyhedra
945:3.3.3.3 (6; also listed above), 3.3.3.
237:
1345:
1049:
652:
721: radians or 720 degrees.
13:
1372:6.4.1 Cundy-Rollett symbol, p. 164
784:-gon is surrounded by alternating
14:
1555:
1513:
1303:European Journal of Combinatorics
1223:Uniform Polyhedra and their Duals
395:, the number of vertices is 720°/
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750:In particular it follows that {
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162:It has variously been called a
1520:Consistent Vertex Descriptions
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462:The notation is ambiguous for
436:-gons around each vertex. So {
241:
18:
1:
1483:. W. H. Freeman and Company.
1397:
509:is (5.5.5.5.5)/2 or (5)/2. A
318:
257:
243:Regular vertex figure nets, {
215:can also be represented as a
499:great stellated dodecahedron
491:small stellated dodecahedron
7:
1452:. Dover Publications, Inc.
678:is the number of sides and
144:The order is important, so
10:
1560:
1464:Uses Cundy-Rollett symbol.
689:
1544:Polytope notation systems
1414:The Archimedean Polyhedra
421:. The Schläfli notation {
382:
1499:The Symmetries of Things
1390:Cundy and Rollett (1952)
1120:
466:forms. For example, the
228:vertex-uniform polyhedra
1381:Laughlin (2014), p. 16
1325:10.1006/eujc.1999.0385
1155:Roman E. Maeder (1995)
1061:
193:for its usage for the
186:. It is also called a
1237:, Jurij KoviÄŤ, (2011)
1192:Archimedean Polyhedra
1153:The Uniform Polyhedra
1057:
705:Semiregular polyhedra
1479:Tilings and Patterns
1249:Kevin Mitchell, 1995
1111:rhombic dodecahedron
1095:Tilings and patterns
1064:The uniform dual or
1059:Rhombic dodecahedron
1020:semiregular tilings
981:semiregular tilings
914:semiregular tilings
507:vertex configuration
444:} can be written as
213:vertex configuration
65:vertex configuration
1409:Mathematical Models
1351:"Archimedean solid"
1005:Archimedean solids
956:Archimedean solids
877:Archimedean solids
255:
238:Variations and uses
200:Mathematical Models
197:in their 1952 book
79:as the sequence of
1430:vertically-regular
1347:Weisstein, Eric W.
1197:2017-07-05 at the
1139:2015-11-27 at the
1086:face configuration
1078:vertically-regular
1062:
1050:Face configuration
662:notation of sides
503:great dodecahedron
393:Descartes' theorem
242:
195:Archimedean solids
191:and Rollett symbol
176:vertex arrangement
164:vertex description
148:is different from
1523:Stella (software)
1507:978-1-56881-220-5
1406:and Rollett, A.,
1211:Uniform Polyhedra
792:-gons, so either
653:Inverted polygons
650:
649:
511:great icosahedron
482:. For example, a
419:regular polyhedra
407:
406:
399:(4Ď€ radians/
134:icosidodecahedron
131:vertex-transitive
89:uniform polyhedra
57:
56:
29:Icosidodecahedron
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1469:GrĂĽnbaum, Branko
1463:
1446:Williams, Robert
1435:Peter Cromwell,
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1092:. In contrast,
1082:face-transitive
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1042:regular tiling
998:Platonic solid
975:regular tiling
935:Platonic solid
908:regular tiling
780:implies that a
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117:For example, "
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42:represented as
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1291:Jurij KoviÄŤ
1225:Robert Webb
1213:Jim McNeill
1044:3.3.3.3.3.3
1013:(60) (both
820:is even or
808:is even or
796:is even or
745:720/30 = 24
739:Example: A
361:Defect 36°
330:Defect 90°
298:Defect 60°
184:face-vector
168:vertex type
1533:Categories
1398:References
1070:bipyramids
1024:(chiral),
993:Quintuples
943:antiprisms
930:Quadruples
788:-gons and
770:vertices.
747:vertices.
690:See also:
666:such that
645:V(3.5/2)/2
452:times) or
377:Defect 0°
346:Defect 0°
314:Defect 0°
73:polyhedron
1539:Polyhedra
1438:Polyhedra
1404:Cundy, H.
1356:MathWorld
1105:around a
1037:Sextuples
1030:3.3.4.3.4
1026:3.3.3.4.4
1022:3.3.3.3.6
1011:3.3.3.3.5
1007:3.3.3.3.4
1000:3.3.3.3.3
832:(for any
760:4 / (2 -
579:(3.5/2)/2
484:pentagram
468:snub cube
217:polygonal
127:pentagons
123:triangles
83:around a
1475:(1987).
1448:(1979).
1432:symbols.
1195:Archived
1137:Archived
674:, where
631:= (3)/2
625:= (5)/2
565:= (5/2)
559:= (5/2)
493:has the
446:p.p.p...
429:} means
142:5.3.5.3.
61:geometry
1428:... as
1333:1791208
1115:rhombus
987:3.4.6.4
983:3.6.3.6
977:4.4.4.4
970:3.4.5.4
966:3.4.4.4
962:3.5.3.5
958:3.4.3.4
937:3.3.3.3
916:3.12.12
901:(120),
887:3.10.10
843:Triples
824:equals
812:equals
800:equals
764:(1 - 2/
732:or 720/
635:V.3.5/2
629:{3,5/2}
623:{5,5/2}
563:{5/2,3}
557:{5/2,5}
150:3.5.3.5
146:3.3.5.5
138:3.5.3.5
119:3.5.3.5
114:sides.
45:3.5.3.5
1505:
1487:
1456:
1331:
1143:(1993)
1103:vertex
1076:, are
1015:chiral
1009:(24),
968:(24),
964:(30),
960:(12),
920:4.6.12
899:4.6.10
897:(48),
893:(24),
889:(60),
885:(24),
881:(12),
864:prisms
816:, and
758:} has
734:defect
730:defect
698:, and
640:V3.5/3
464:chiral
401:defect
397:defect
110:, and
93:Chiral
87:. For
85:vertex
77:tiling
1311:arXiv
1121:Notes
924:4.8.8
910:6.6.6
905:(60).
903:5.6.6
895:4.6.8
891:4.6.6
883:3.8.8
879:3.6.6
858:5.5.5
856:(8),
854:4.4.4
852:(4),
850:3.3.3
778:p.q.r
670:<2
574:3.5/3
569:3.5/2
371:{6,3}
357:{5,3}
340:{4,4}
327:{4,3}
308:{3,6}
294:{3,5}
280:{3,4}
266:{3,3}
189:Cundy
154:(3.5)
100:a.b.c
81:faces
71:of a
49:(3.5)
1503:ISBN
1485:ISBN
1454:ISBN
1107:face
1072:and
1002:(12)
972:(60)
860:(20)
417:for
251:} =
125:and
63:, a
1321:doi
939:(6)
664:p/q
403:).
391:By
359:= 5
296:= 3
282:= 3
268:= 3
75:or
59:In
47:or
1535::
1471:;
1353:,
1349:,
1329:MR
1327:,
1319:,
1307:21
1305:,
1160:^
1028:,
985:,
949:(2
922:,
918:,
870:(2
768:))
736:.
711:.
694:,
373:=
342:=
310:=
211:A
203:.
182:,
178:,
174:,
170:,
166:,
159:.
106:,
1493:.
1462:.
1426:c
1424:.
1422:b
1420:.
1418:a
1323::
1313::
1090:V
1080:(
1017:)
953:)
951:n
947:n
874:)
872:n
868:n
834:n
830:n
826:q
822:p
818:r
814:r
810:p
806:q
802:r
798:q
794:p
790:r
786:q
782:p
766:a
762:b
756:b
754:,
752:a
728:/
726:Ď€
719:Ď€
680:q
676:p
672:q
668:p
454:p
450:q
448:(
442:q
440:,
438:p
434:p
431:q
427:q
425:,
423:p
387:.
375:6
344:4
312:3
253:p
249:q
247:,
245:p
156:2
112:c
108:b
104:a
51:2
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