# Ditrigonal polyhedron

## 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]

## References

### Notes

- Har'El, 1993
- 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