Ditrigonary polyhedra
In geometry, a ditrigonary polyhedron is a uniform star polyhedron with Wythoff symbol: 3 | p q. There are three of them, each including two types of faces, being of triangles, pentagons, or pentagrams. Their vertex figures have the same vertex arrangement, but different edges.
They have 20 vertices, shared with the regular dodecahedron. They are also related to the compound of five cubes which shares the same vertex arrangement and the same edge arrangement.
| Type | Regular | Compound | Ditrigonary | ||
|---|---|---|---|---|---|
| Name | Dodecahedron | Five cubes | Small ditrigonal icosidodecahedron | Ditrigonal dodecadodecahedron | Great ditrigonal icosidodecahedron |
| Vertices | 20 | ||||
| Edges | 30 | 60 | |||
| Faces | 12 {5} | 30 {4} | 32 20 {3}, 12 {5/2} | 24 12 {5}, 12 {5/2} | 32 20 {3}, 12 {5} |
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| Wythoff symbol | 3 | 2 5 | 3 | 5/2 3 | 3 | 5/3 5 | 3 | 3/2 5 | |
| Coxeter diagram |     |
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Related polytopes
Norman Johnson discovered three related antiprism-like star polytopes, published in 1966 in his Ph.D. Dissertation, now named the Johnson antiprisms. These have these ditrigonary star polyhedra as their bases.[1] They all have 40 vertices, 40 total cells, and 180 total faces. They have 184 (small ditrigonary icosidodecahedral antiprism), 168 (ditrigonary dodecadodecahedral antiprism), and 184 (great ditrigonary icosidodecahedral antiprism) edges respectively. Stella4D software can render these as models 966, 967, and 968. Their Coxeter-Dynkin diagrams are 
, , and respectively.
References
- ↑ Johnson, 1966
- Coxeter, H.S.M., M.S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401–450.
- The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Skilling, J. (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 278: 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333
- Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57–110, 1993. Zvi Har’El, Kaleido software, Images, dual images
- Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48–57, 1993.
External links
- Category 20: Miscellaneous star polychora 966:Sidtidap, 967:ditdidap, and 968:gidtidap are commonly referred to as the Johnson Antiprisms, for they were discovered by Norman Johnson, they also form a regiment.
- Richard Klitzing, Uniform star polychoron 966, small-ditrigonal-icosidodecahedron antiprism (sidtidap)
- Richard Klitzing, Uniform star polychoron 967, ditrigonal-dodecadodecahedron antiprism (ditdidap)
- Richard Klitzing, Uniform star polychoron 968, great-ditrigonal-icosidodecahedron antiprism (gidtidap)
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