560:
553:
546:
539:
532:
46:
605:
598:
377:
346:
35:
314:
300:
286:
272:
363:
619:
612:
626:
333:
1066:
420:
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).
668:
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
1506:
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.
693:
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.
470:
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.
725:
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.
728:
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
706:
847:>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.
163:(the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as
109:
A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "
1299:
784:
Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible.
1147:
1419:
210:
481:
has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration.
1205:
1517:
584:
735:
Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices, which is 4
1269:
Resources for
Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins
1554:
655:
17:
1499:
1468:
650:
589:
645:
710:
930:
579:
573:
567:
509:
508:
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
501:
1036:
909:
500:
For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The
1312:
Deza, Michel; Shtogrin, Mikhail (2000), "Uniform partitions of 3-space, their relatives and embedding",
1456:
926:
997:
702:
403:
102:
there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (
524:, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration (3.3.3.3.3)/2 or (3)/2.
1025:
934:
905:
474:
242:
103:
839:. 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.
559:
980:
897:
715:
237:
has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for
1380:
1190:
1178:
1032:
913:
889:
1221:
707:
Tiling by regular polygons § Combinations of regular polygons that can meet at a vertex
552:
1343:
1121:
1105:
1069:
901:
8:
993:
976:
1163:
1144:
147:. The notation is cyclic and therefore is equivalent with different starting points, so
1549:
1520:(p. 289 Vertex figures, uses comma separator, for Archimedean solids and tilings).
1321:
1278:
1040:
633:
545:
538:
531:
513:
512:, {5/2,3} has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2). The
238:
99:
1479:
1361:
1112:
This notation represents a sequential count of the number of faces that exist at each
505:
425:
1533:
1513:
1495:
1464:
1357:
1113:
1054:
1021:
972:
850:
The number in parentheses is the number of vertices, determined by the angle defect.
719:
639:
521:
318:
205:
144:
141:
95:
39:
1257:
1043:(note that the two different orders of the same numbers give two different patterns)
45:
1448:
1331:
1290:
Structure and Form in Design: Critical Ideas for
Creative Practice By Michael Hann
1202:
920:
874:
381:
1339:
1209:
1151:
1117:
1092:
91:
1095:) and so they can be identified by a similar notation which is sometimes called
893:
751:
696:
429:
1488:
1543:
1483:
1452:, Cambridge University Press (1977) The Archimedean solids. Pp. 156–167.
1076:
1065:
987:
968:
597:
350:
230:
79:
50:
1427:. Pp. 101–115, pp. 118–119 Table I, Nets of Archimedean Duals, V.
604:
497:
has the symbol {5/2}, meaning it has 5 sides going around the centre twice.
1530:
1414:
1335:
1084:
868:
670:
490:
395:
376:
367:
199:
106:
polyhedra exist in mirror-image pairs with the same vertex configuration.)
87:
1010:
860:
304:
276:
1461:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1233:
345:
947:
290:
83:
400:
A 0° angle defect will fill the
Euclidean plane with a regular tiling.
1366:
1326:
1128:, and alternating vertices of the rhombus contain 3 or 4 faces each.
1080:
1017:
953:
494:
478:
1423:(1952), (3rd edition, 1989, Stradbroke, England: Tarquin Pub.), 3.7
1381:
Divided
Spheres: Geodesics and the Orderly Subdivision of the Sphere
1245:
424:
The notation can also be considered an expansive form of the simple
34:
137:
133:
71:
1179:
Crystallography of
Quasicrystals: Concepts, Methods and Structures
313:
1125:
299:
285:
271:
227:
1478:
362:
241:
all the neighboring vertices are in the same plane and so this
1109:
uses square brackets around the symbol for isohedral tilings.
625:
618:
611:
1512:
2008, John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss,
1181:
by Walter
Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53
697:
All uniform vertex configurations of regular convex polygons
245:
can be used to visually represent the vertex configuration.
113:" describes a vertex that has 3 faces around it, faces with
864:
754:
3.8.8 has an angle defect of 30 degrees. Therefore, it has
489:
The notation also applies for nonconvex regular faces, the
337:
332:
1355:
467:. For example, an icosahedron is {3,5} = 3.3.3.3.3 or 3.
1300:
Symmetry-type graphs of
Platonic and Archimedean solids
1246:
Symmetry-type graphs of
Platonic and Archimedean solids
787:
Topological requirements limit existence. Specifically
711:
Uniform tiling § Expanded lists of uniform tilings
132:" indicates a vertex belonging to 4 faces, alternating
1258:
3. General Theorems: Regular and Semi-Regular Tilings
1099:. Cundy and Rollett prefixed these dual symbols by a
1487:
1120:. For example, V3.4.3.4 or V(3.4) represents the
516:, {5,5/2} has a pentagrammic vertex figure, with
1541:
1193:edited by David E. Laughlin, (2014) pp. 16–20
877:3.4.4 (6), 4.4.4 (8; also listed above), 4.4.
78:is a shorthand notation for representing the
1311:
1191:Physical Metallurgy: 3-Volume Set, Volume 1
1124:which is face-transitive: every face is a
233:showing the faces around the vertex. This
1325:
1174:
1172:
718:have vertex configurations with positive
31:
27:Notation for a polyhedron's vertex figure
1455:
1064:
394:A vertex needs at least 3 faces, and an
140:. This vertex configuration defines the
1184:
703:Archimedean_solid § Classification
14:
1542:
1169:
1145:Uniform Solution for Uniform Polyhedra
956:3.3.3.3 (6; also listed above), 3.3.3.
248:
1356:
1060:
663:
732: radians or 720 degrees.
24:
1383:6.4.1 Cundy-Rollett symbol, p. 164
795:-gon is surrounded by alternating
25:
1566:
1524:
1314:European Journal of Combinatorics
1234:Uniform Polyhedra and their Duals
406:, the number of vertices is 720°/
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761:In particular it follows that {
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173:It has variously been called a
1531:Consistent Vertex Descriptions
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473:The notation is ambiguous for
447:-gons around each vertex. So {
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1:
1494:. W. H. Freeman and Company.
1408:
520:is (5.5.5.5.5)/2 or (5)/2. A
329:
268:
254:Regular vertex figure nets, {
226:can also be represented as a
510:great stellated dodecahedron
502:small stellated dodecahedron
7:
1463:. Dover Publications, Inc.
689:is the number of sides and
155:The order is important, so
10:
1571:
1475:Uses Cundy-Rollett symbol.
700:
1555:Polytope notation systems
1425:The Archimedean Polyhedra
432:. The Schläfli notation {
393:
1510:The Symmetries of Things
1401:Cundy and Rollett (1952)
1131:
477:forms. For example, the
239:vertex-uniform polyhedra
1392:Laughlin (2014), p. 16
1336:10.1006/eujc.1999.0385
1166:Roman E. Maeder (1995)
1072:
204:for its usage for the
197:. It is also called a
1248:, Jurij KoviÄŤ, (2011)
1203:Archimedean Polyhedra
1164:The Uniform Polyhedra
1068:
716:Semiregular polyhedra
1490:Tilings and Patterns
1260:Kevin Mitchell, 1995
1122:rhombic dodecahedron
1106:Tilings and patterns
1075:The uniform dual or
1070:Rhombic dodecahedron
1031:semiregular tilings
992:semiregular tilings
925:semiregular tilings
518:vertex configuration
455:} can be written as
224:vertex configuration
76:vertex configuration
1420:Mathematical Models
1362:"Archimedean solid"
1016:Archimedean solids
967:Archimedean solids
888:Archimedean solids
266:
249:Variations and uses
211:Mathematical Models
208:in their 1952 book
90:as the sequence of
1441:vertically-regular
1358:Weisstein, Eric W.
1208:2017-07-05 at the
1150:2015-11-27 at the
1097:face configuration
1089:vertically-regular
1073:
1061:Face configuration
673:notation of sides
514:great dodecahedron
404:Descartes' theorem
253:
206:Archimedean solids
202:and Rollett symbol
187:vertex arrangement
175:vertex description
159:is different from
18:Face configuration
1534:Stella (software)
1518:978-1-56881-220-5
1417:and Rollett, A.,
1222:Uniform Polyhedra
803:-gons, so either
664:Inverted polygons
661:
660:
522:great icosahedron
493:. For example, a
430:regular polyhedra
418:
417:
410:(4Ď€ radians/
145:icosidodecahedron
142:vertex-transitive
100:uniform polyhedra
68:
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40:Icosidodecahedron
16:(Redirected from
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1480:GrĂĽnbaum, Branko
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1446:Peter Cromwell,
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1103:. In contrast,
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1009:Platonic solid
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946:Platonic solid
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791:implies that a
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1236:Robert Webb
1224:Jim McNeill
1055:3.3.3.3.3.3
1024:(60) (both
831:is even or
819:is even or
807:is even or
756:720/30 = 24
750:Example: A
372:Defect 36°
341:Defect 90°
309:Defect 60°
195:face-vector
179:vertex type
1544:Categories
1409:References
1081:bipyramids
1035:(chiral),
1004:Quintuples
954:antiprisms
941:Quadruples
799:-gons and
781:vertices.
758:vertices.
701:See also:
677:such that
656:V(3.5/2)/2
463:times) or
388:Defect 0°
357:Defect 0°
325:Defect 0°
84:polyhedron
1550:Polyhedra
1449:Polyhedra
1415:Cundy, H.
1367:MathWorld
1116:around a
1048:Sextuples
1041:3.3.4.3.4
1037:3.3.3.4.4
1033:3.3.3.3.6
1022:3.3.3.3.5
1018:3.3.3.3.4
1011:3.3.3.3.3
843:(for any
771:4 / (2 -
590:(3.5/2)/2
495:pentagram
479:snub cube
228:polygonal
138:pentagons
134:triangles
94:around a
1486:(1987).
1459:(1979).
1443:symbols.
1206:Archived
1148:Archived
685:, where
642:= (3)/2
636:= (5)/2
576:= (5/2)
570:= (5/2)
504:has the
457:p.p.p...
440:} means
153:5.3.5.3.
72:geometry
1439:... as
1344:1791208
1126:rhombus
998:3.4.6.4
994:3.6.3.6
988:4.4.4.4
981:3.4.5.4
977:3.4.4.4
973:3.5.3.5
969:3.4.3.4
948:3.3.3.3
927:3.12.12
912:(120),
898:3.10.10
854:Triples
835:equals
823:equals
811:equals
775:(1 - 2/
743:or 720/
646:V.3.5/2
640:{3,5/2}
634:{5,5/2}
574:{5/2,3}
568:{5/2,5}
161:3.5.3.5
157:3.3.5.5
149:3.5.3.5
130:3.5.3.5
125:sides.
56:3.5.3.5
1516:
1498:
1467:
1342:
1154:(1993)
1114:vertex
1087:, are
1026:chiral
1020:(24),
979:(24),
975:(30),
971:(12),
931:4.6.12
910:4.6.10
908:(48),
904:(24),
900:(60),
896:(24),
892:(12),
875:prisms
827:, and
769:} has
745:defect
741:defect
709:, and
651:V3.5/3
475:chiral
412:defect
408:defect
121:, and
104:Chiral
98:. For
96:vertex
88:tiling
1322:arXiv
1132:Notes
935:4.8.8
921:6.6.6
916:(60).
914:5.6.6
906:4.6.8
902:4.6.6
894:3.8.8
890:3.6.6
869:5.5.5
867:(8),
865:4.4.4
863:(4),
861:3.3.3
789:p.q.r
681:<2
585:3.5/3
580:3.5/2
382:{6,3}
368:{5,3}
351:{4,4}
338:{4,3}
319:{3,6}
305:{3,5}
291:{3,4}
277:{3,3}
200:Cundy
165:(3.5)
111:a.b.c
92:faces
82:of a
60:(3.5)
1514:ISBN
1496:ISBN
1465:ISBN
1118:face
1083:and
1013:(12)
983:(60)
871:(20)
428:for
262:} =
136:and
74:, a
1332:doi
950:(6)
675:p/q
414:).
402:By
370:= 5
307:= 3
293:= 3
279:= 3
86:or
70:In
58:or
1546::
1482:;
1364:,
1360:,
1340:MR
1338:,
1330:,
1318:21
1316:,
1171:^
1039:,
996:,
960:(2
933:,
929:,
881:(2
779:))
747:.
722:.
705:,
384:=
353:=
321:=
222:A
214:.
193:,
189:,
185:,
181:,
177:,
170:.
117:,
1504:.
1473:.
1437:c
1435:.
1433:b
1431:.
1429:a
1334::
1324::
1101:V
1091:(
1028:)
964:)
962:n
958:n
885:)
883:n
879:n
845:n
841:n
837:q
833:p
829:r
825:r
821:p
817:q
813:r
809:q
805:p
801:r
797:q
793:p
777:a
773:b
767:b
765:,
763:a
739:/
737:Ď€
730:Ď€
691:q
687:p
683:q
679:p
465:p
461:q
459:(
453:q
451:,
449:p
445:p
442:q
438:q
436:,
434:p
398:.
386:6
355:4
323:3
264:p
260:q
258:,
256:p
167:2
123:c
119:b
115:a
62:2
20:)
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