Semiregular polytope: Difference between revisions
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'''Semiregular figures Gosset enumerated:''' (his names in parentheses) |
'''Semiregular figures Gosset enumerated:''' (his names in parentheses) |
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* [[Convex uniform honeycomb]]s, two 3D honeycombs: |
* [[Convex uniform honeycomb]]s, two 3D honeycombs: |
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*#[[Tetrahedral-octahedral honeycomb]] (Simple tetroctahedric check), {{CDD|node_h1|4|node|3|node|4|node}} ↔ {{CDD|nodes_10ru|split2|node|4|node}} |
*#[[Tetrahedral-octahedral honeycomb]] (Simple tetroctahedric check), {{CDD|node_h1|4|node|3|node|4|node}} ↔ {{CDD|nodes_10ru|split2|node|4|node}} (Also [[quasiregular polytope]]) |
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*#[[Gyrated alternated cubic honeycomb]] (Complex tetroctahedric check), {{CDD|node|3|node|6|node_h|2x|node_h|infin|node}} |
*#[[Gyrated alternated cubic honeycomb]] (Complex tetroctahedric check), {{CDD|node|3|node|6|node_h|2x|node_h|infin|node}} |
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* [[Uniform polychoron|Uniform polychora]], three [[4-polytope]]s: |
* [[Uniform polychoron|Uniform polychora]], three [[4-polytope]]s: |
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*[[Hyperbolic uniform honeycomb]]s, 3D honeycombs: |
*[[Hyperbolic uniform honeycomb]]s, 3D honeycombs: |
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*# [[Alternated order-5 cubic honeycomb]], {{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}} |
*# [[Alternated order-5 cubic honeycomb]], {{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}} (Also [[quasiregular polytope]]) |
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*# [[Hyperbolic tetrahedral-octahedral honeycomb|Tetrahedral-octahedral honeycomb]], {{CDD|label4|branch|3ab|branch_10l}} |
*# [[Hyperbolic tetrahedral-octahedral honeycomb|Tetrahedral-octahedral honeycomb]], {{CDD|label4|branch|3ab|branch_10l}} |
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*# [[Tetrahedron-icosahedron honeycomb]], {{CDD|label5|branch|3ab|branch_10l}} |
*# [[Tetrahedron-icosahedron honeycomb]], {{CDD|label5|branch|3ab|branch_10l}} |
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*# [[Rectified order-6 tetrahedral honeycomb]], {{CDD|node|3|node_1|3|node|6|node}} |
*# [[Rectified order-6 tetrahedral honeycomb]], {{CDD|node|3|node_1|3|node|6|node}} |
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*# [[Rectified square tiling honeycomb]], {{CDD|node|4|node_1|4|node|3|node}} |
*# [[Rectified square tiling honeycomb]], {{CDD|node|4|node_1|4|node|3|node}} |
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*# [[Alternated |
*# [[Alternated order-6 cubic honeycomb]], {{CDD|node_h1|4|node|3|node|6|node}} ↔ {{CDD|nodes_10ru|split2|node|6|node}} (Also quasiregular) |
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*# [[Alternated hexagonal tiling honeycomb]], {{CDD||node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}} |
*# [[Alternated hexagonal tiling honeycomb]], {{CDD||node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}} |
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*# [[Alternated square tiling honeycomb]], {{CDD|node_h1|4|node|4|node|3|node}} ↔ {{CDD|nodes_10ru|split2-44|node|3|node}} |
*# [[Alternated square tiling honeycomb]], {{CDD|node_h1|4|node|4|node|3|node}} ↔ {{CDD|nodes_10ru|split2-44|node|3|node}} (Also quasiregular) |
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*# [[Cubic-square tiling honeycomb]], {{CDD|label4|branch_10r|4a4b|branch}} |
*# [[Cubic-square tiling honeycomb]], {{CDD|label4|branch_10r|4a4b|branch}} |
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*# [[Tetrahedral-triangular tiling honeycomb]], {{CDD|label6|branch|3ab|branch_10l}} |
*# [[Tetrahedral-triangular tiling honeycomb]], {{CDD|label6|branch|3ab|branch_10l}} |
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Revision as of 02:50, 23 May 2014
| 3D honeycombs | ||
|---|---|---|
 Simple tetroctahedric check |
 Complex tetroctahedric check | |
| 4D polytopes | ||
 Tetroctahedric |
 Octicosahedric |
 Tetricosahedric |
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.
Gosset's list
In three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions.
The three convex semiregular polychora (4-polytopes) are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the k21 polytopes, where the rectified 5-cell is the special case of k = 0.
Semiregular polytopes can be extended to semiregular honeycombs. The semiregular Euclidean honeycombs are the tetrahedral-octahedral honeycomb (3D), gyrated alternated cubic honeycomb (3D) and the 521 honeycomb (8D).
Semiregular figures Gosset enumerated: (his names in parentheses)
- Convex uniform honeycombs, two 3D honeycombs:
- Tetrahedral-octahedral honeycomb (Simple tetroctahedric check),
↔ 
(Also quasiregular polytope) - Gyrated alternated cubic honeycomb (Complex tetroctahedric check),
- Tetrahedral-octahedral honeycomb (Simple tetroctahedric check),
- Uniform polychora, three 4-polytopes:
- Rectified 5-cell (Tetroctahedric),
- Rectified 600-cell (Octicosahedric),
- Snub 24-cell (Tetricosahedric),
- Rectified 5-cell (Tetroctahedric),
- Semiregular E-polytopes, four polytopes, and one honeycomb:
- 5-demicube (5-ic semi-regular), a 5-polytope, ↔
- 221 polytope (6-ic semi-regular), a 6-polytope,
- 321 polytope (7-ic semi-regular), a 7-polytope,
- 421 polytope (8-ic semi-regular), an 8-polytope,
- 521 honeycomb (9-ic check) (8D Euclidean honeycomb),
- 5-demicube (5-ic semi-regular), a 5-polytope,
Beyond Gosset's list
There are also hyperbolic uniform honeycombs composed of only regular cells, including:
- Hyperbolic uniform honeycombs, 3D honeycombs:
- Paracompact uniform honeycombs, 3D honeycombs, which include uniform tilings as cells:
- Rectified order-6 tetrahedral honeycomb,
- Rectified square tiling honeycomb,
- Alternated order-6 cubic honeycomb, ↔ (Also quasiregular)
- Alternated hexagonal tiling honeycomb, ↔
- Alternated square tiling honeycomb, ↔
(Also quasiregular) - Cubic-square tiling honeycomb,
- Tetrahedral-triangular tiling honeycomb,
- Rectified order-6 tetrahedral honeycomb,
- 9D hyperbolic paracompact honeycomb:
- 621 honeycomb (10-ic check),
- 621 honeycomb (10-ic check),
See also
References
- Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. ISBN 0-486-61480-8.
- Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
- Elte, E. L. (1912). The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen. ISBN 1-4181-7968-X.