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The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2) possible forms, but the one where the generator point is on all the mirrors is impossible. The symbol that would normally refer to that is reused for the snub tilings.
70:
A Wythoff symbol consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different
Wythoff symbols from different symmetry generators. For example, the regular
307:, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.
23:
Example
Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.
102:
With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in
Euclidean or hyperbolic space.
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114:
on a fundamental triangle. This point must be chosen at equal distance from all edges that it does not lie on, and a perpendicular line is then dropped from it to each such edge.
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which is designated for the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.
202:). The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following:
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was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.
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The fundamental triangles are drawn in alternating colors as mirror images. The sequence of triangles (
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In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The
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63:, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the
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1100:
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129:, represent the corners of the Schwarz triangle used in the construction, which are
1719:
Greg Egan's applet to display uniform polyhedra using
Wythoff's construction method
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84:
56:
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respectively. The triangle is also represented with the same numbers, written (
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The eight forms for the
Wythoff constructions from a general triangle (
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Example spherical, euclidean and hyperbolic tilings on right triangles
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1414:
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1234:
1603:
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
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40:
1630:(Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
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indicates that the generator lies in the interior of the triangle.
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The
Wythoff symbol is functionally similar to the more general
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1110:
894:
72:
1724:
A Shadertoy renderization of
Wythoff's construction method
1647:(Chapter 3: Wythoff's Construction for Uniform Polytopes)
1694:
237:
indicates that the generator lies on the edge between
1749:
218:indicates that the generator lies on the corner
280:the bar if the corresponding mirror is active.
1735:that generated many of the images on the page.
110:The Wythoff construction begins by choosing a
1598:List of uniform polyhedra by Schwarz triangle
1731:Free educational software for Windows by
1664:
1450:
1335:
1220:
1105:
990:
875:
760:
1622:, Third edition, (1973), Dover edition,
331:≥ 7). Hyperbolic tilings are shown as a
95:, as well as 2 2 2 | with 3 colors and D
26:
18:
117:The three numbers in Wythoff's symbol,
1750:
1738:
1695:
1637:The Beauty of Geometry: Twelve Essays
1588:Uniform tilings in hyperbolic plane
75:can be represented by 3 | 2 4 with
13:
14:
1774:
1741:"Hyperbolic Planar Tessellations"
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1672:. Cambridge University Press.
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1:
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47:is a notation representing a
1639:, Dover Publications, 1999,
319:3 2) change from spherical (
283:A special use is the symbol
7:
1653:, Longuet-Higgins, Miller,
1566:
10:
1779:
323:= 3, 4, 5), to Euclidean (
83:, and 2 4 | 2 as a square
1763:Polytope notation systems
1593:List of uniform polyhedra
756:
55:or plane tiling within a
16:Notation for tesselations
1583:List of uniform tilings
59:. It was first used by
305:Coxeter-Dynkin diagram
36:
24:
327:= 6), to hyperbolic (
30:
22:
1661:1954, 246 A, 401–50.
49:Wythoff construction
1714:The Wythoff symbol
1697:Weisstein, Eric W.
1578:Regular polyhedron
276:values are listed
87:with 2 colors and
53:uniform polyhedron
37:
25:
1670:Polyhedron Models
1666:Wenninger, Magnus
1655:Uniform polyhedra
1619:Regular Polytopes
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757:7 forms and snub
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1700:"Wythoff symbol"
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1573:Regular polytope
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65:Coxeter diagram
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1689:External links
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1684:pp. 9–10.
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45:Wythoff symbol
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1729:KaleidoTile 3
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1739:Hatch, Don.
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1659:Phil. Trans.
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1550:| ∞ 3 2
1537:∞ 3 2 |
1524:∞ 3 | 2
1511:∞ | 3 2
1498:2 ∞ | 3
1487:2 | ∞ 3
1474:2 3 | ∞
1461:3 | ∞ 2
1437:| 8 3 2
1424:8 3 2 |
1411:8 3 | 2
1398:8 | 3 2
1385:2 8 | 3
1372:2 | 8 3
1359:2 3 | 8
1346:3 | 8 2
1322:| 7 3 2
1309:7 3 2 |
1296:7 3 | 2
1283:7 | 3 2
1270:2 7 | 3
1257:2 | 7 3
1244:2 3 | 7
1231:3 | 7 2
1207:| 6 3 2
1194:6 3 2 |
1181:6 3 | 2
1168:6 | 3 2
1155:2 6 | 3
1142:2 | 6 3
1129:2 3 | 6
1116:3 | 6 2
1092:| 5 3 2
1079:5 3 2 |
1066:5 3 | 2
1053:5 | 3 2
1040:2 5 | 3
1027:2 | 5 3
1014:2 3 | 5
1001:3 | 5 2
977:| 4 3 2
964:4 3 2 |
951:4 3 | 2
938:4 | 3 2
925:2 4 | 3
912:2 | 4 3
899:2 3 | 4
886:3 | 4 2
862:| 3 3 2
849:3 3 2 |
836:3 3 | 2
823:3 | 3 2
810:2 3 | 3
797:2 | 3 3
784:2 3 | 3
771:3 | 3 2
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38:
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106:Description
1752:Categories
1609:References
99:symmetry.
1758:Polyhedra
1705:MathWorld
1559:3.3.3.3.∞
1446:3.3.3.3.8
1331:3.3.3.3.7
1216:3.3.3.3.6
1101:3.3.3.3.5
986:3.3.3.3.4
871:3.3.3.3.3
410:2 |
401:| 2
366:2 |
1668:(1974).
1567:See also
1495:3.∞.3.∞
93:symmetry
81:symmetry
41:geometry
1651:Coxeter
1634:Coxeter
1615:Coxeter
1533:3.4.∞.4
1452:(∞ 3 2)
1420:3.4.8.4
1381:3.8.3.8
1368:3.16.16
1337:(8 3 2)
1305:3.4.7.4
1266:3.7.3.7
1253:3.14.14
1222:(7 3 2)
1190:3.4.6.4
1151:3.6.3.6
1138:3.12.12
1107:(6 3 2)
1075:3.4.5.4
1036:3.5.3.5
1023:3.10.10
992:(5 3 2)
960:3.4.4.4
921:3.4.3.4
877:(4 3 2)
845:3.4.3.4
806:3.3.3.3
762:(3 3 2)
413:|
388:|
379:|
360:|
349:|
190:radians
187:
171:
167:
151:
147:
131:
61:Coxeter
1676:
1643:
1626:
1433:4.6.16
1318:4.6.14
1203:4.6.12
1088:4.6.10
278:before
169:, and
125:, and
43:, the
1546:4.6.∞
1507:∞.6.6
1483:3.∞.∞
1394:8.6.6
1279:7.6.6
1164:6.6.6
1049:5.6.6
973:4.6.8
934:4.6.6
908:3.8.8
858:4.6.6
819:3.6.6
793:3.6.6
85:prism
51:of a
33:p q r
1674:ISBN
1641:ISBN
1624:ISBN
742:3.3.
241:and
73:cube
746:.3.
732:4.2
729:.4
725:.4.
39:In
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1657:,
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268:,
232:|
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1708:.
1682:.
1520:3
1470:∞
1407:3
1355:8
1292:3
1240:7
1177:3
1125:6
1062:3
1010:5
947:3
895:4
832:3
780:3
748:q
744:p
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179:/
175:π
163:q
159:/
155:π
143:p
139:/
135:π
127:r
123:q
119:p
89:D
79:h
77:O
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