Ditrigonal polyhedron

In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal.[1]

Ditrigonal vertex figures

There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.[1]

The three uniform star polyhedron with Wythoff symbol of the form 3 | p q or 3/2 | p q are ditrigonal, at least if p and q are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form p.q.p.q.p.q or (p.q)³ with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ditrigonal means "having two sets of 3 angles").[2]

Type	Small ditrigonal icosidodecahedron	Ditrigonal dodecadodecahedron	Great ditrigonal icosidodecahedron
Image
Vertex figure
Vertex configuration	3.​^5⁄2.3.​^5⁄2.3.​^5⁄2	5.​^5⁄3.5.​^5⁄3.5.​^5⁄3	(3.5.3.5.3.5)/2
Faces	32 20 {3}, 12 { ​^5⁄2 }	24 12 {5}, 12 { ​^5⁄2 }	32 20 {3}, 12 {5}
Wythoff symbol	3 | 5/2 3	3 | 5/3 5	3 | 3/2 5
Coxeter diagram

Other uniform ditrigonal polyhedra

The small ditrigonal dodecicosidodecahedron and the great ditrigonal dodecicosidodecahedron are also uniform.

Their duals are respectively the small ditrigonal dodecacronic hexecontahedron and great ditrigonal dodecacronic hexecontahedron.[1]

See also

• Small complex icosidodecahedron
• Great complex icosidodecahedron

References

Notes

1. Har'El, 1993
2. Uniform Polyhedron, Mathworld (retrieved 10 June 2016)

Bibliography

• Coxeter, H.S.M., M.S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401–450.
• Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57–110, 1993. Zvi Har’El, Kaleido software, Images, dual images

Further reading

• Johnson, N.; 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