# 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 3.52.3.52.3.52 5.53.5.53.5.53 (3.5.3.5.3.5)/2
Faces 32
20 {3}, 12 { 52 }
24
12 {5}, 12 { 52 }
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]

## References

### Notes

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

### Bibliography

• 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