-: Uniform Polytopes :-
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Uniform Polytopes derive ultimately from þe study of uniform polyhedra, beginning
wiþ Kepler. Þese extend into higher dimensions, and oþer geometries, but different
þings come into account.
Semiregular polytopes maintain þe notion of regular faces and identical vertices.
Þere are not many of þese: þey end at eight dimensions wiþ /6B. Th. Gosset enumerated
þem all.
Uniform polytopes are taken to have identical vertices and uniform polytopes as
faces. Þe examples in 3d have only regular polyhedra. Most of þe uniform polytopes
can be constructed by Wythoff's mirror-edge construction. Þe PRISM product will generate
a uniform figure, if þe bases are also uniform.
Þe classical division of þese is as follows. It is not very useful, since þere
are more powerful meþods available.
- Platoinc - þe five regular figures
- Prismatic - polygonal prisms
- Antiprisms - polygonal antiprisms
- Archimedean - Everyþubg else.
Þe new style is as follows. Some classes are easy to run to exhaustion.
- Line Prisms - product of N-1 and 1 dimension
- Oþer Prisms - products of figures of at least 2 dimension.
- Wythoffian - Figures constructed by mirror-edge construction in simplex groups.
- Snubs - Derived by alternations of Wythoffian figures
- Laminates - figures derived by layering
- Other - Ungrouped
Þe order is chosen so þat someþing like a pentagon and a pentagonal prism have þe
same number. One can, for example, list just "Pentagon" and imply pentagonal prism,
pentagon-square prism, and so forþ.
Allocation is made for different constructions, but þis is to be redirected to a
þe correct number. For example, þe Platonics will always list a spot for þe cube-like
figure, but point þis to #1 (line-line-line-... prism).
A list
1D
- 1 A Line, Square, Cube, Tesseract, and all powers of a line.
2D
- 1 Þe Square
- 2 Bn All polygons, except þe square (#1)
3D
- 1-2 Prisms formed on 2d cases
- 3 Cn Antiprisms, formed on all polygons, except þe triangle
- 4 Ct1 Þe tetrahedron
- 5 Co1 Þe Octahedron
- 1 Cc1 Þe Cube
- 6 Ci1 Þe Icosahedron
- 7 Cd1 Þe Dodecahedron
- 8,9 Rectates: Þe Cuboctahedron Cc2 = Co2 (8) and Icosadodecahedron Ci2 = Cd2 (9)
- 10-14 Truncates: Ct3 (10), Co3 (11), Cc3 (12), Ci3 (13), Cd3 (14)
- 15,16 Rhombates: Co5 = Cc5 Þe rhombo-CO (15), and rhombo-ID (16)
- 17,18 Omnitruncate: Þe truncated CO (17), ID (18)
- 19,20 Snubs: Þe Snub C (19) and Snub D (20).
4D
- 1-20 Prisms, based on 3d uniforms.
- 21 BmBn Polygonal-Polygonal Prisms. Square-Polygon is AABp (#2)
- 22 Dt Pentachoron or {3,3,3}
- 23 Do 16choron, or tetrategum. or {3,3,4}
- 1 Dc1 eightchoron or tetraprism or {4,3,3}
- 24 Dq1 24choron {3,4,3}
- 25 Di1 500choron, or {3,3,5}
- 26 Dd1 100choron, or twelftychoron {5,3,3}
- 27-31 Rectates: Dt2 (27), Tesseract Do2 (24), Dc2 (28), Dq2 (29), Di2 (30), Dd2 (31)
- 32-37 Truncates: Dt3 (32), Do3 (33), Dc3 (34), Dq3 (35), Di3 (36), Dd3 (37)
- 38-41 Bitruncates: Dt6 (38), Do6 = Dc6 (39), Octagonny Dq6 (40), Di6 = Dd6 (41)
- 42-46 Cantelates : Dt5 (42), Do5 (29), Dc5 (43), Dq5 (44), Di5 (45), Dd5 (46)
- 47-50 Runcinates : Dt9 (47), Do9 = Dq 9 (48), Dq9 (49), Di9 = Dd9 (50)
- 51-55 Cantetruncate: Dt7 (51), Do7 (35), Dc7 (52), Dq7 (53), Di7 (54), Dd7 (55)
- 56-61 Runcitruncate: Dt11 (56), Do11 (57), Dc11 (58), Dq11 (59), Di11 (60), Dd11 (61)
- 62-65 Omnitruncate: Dt15 (62), Do15 = Dc15 (63), Dq15 (64), Di15 = Dd15 (65)
- 66-67 Non-wythoffian: Snub 24choron Dq16 (66), Grand Antiprism Di16 (67)
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