Ditrigonal polyhedron

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

Ditrigonal vertex figures[edit]

There are five uniform ditrigonal polyhedra, all with icosahedral symmetry and 20 vertices.[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 ababab 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 Small ditrigonal icosidodecahedron.png Ditrigonal dodecadodecahedron.png Great ditrigonal icosidodecahedron.png
Vertex figure Small ditrigonal icosidodecahedron vertfig.png Ditrigonal dodecadodecahedron vertfig.png Great ditrigonal icosidodecahedron vertfig.png
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 Small ditrigonal icosidodecahedron cd.png Ditrigonal dodecadodecahedron cd.png Great ditrigonal icosidodecahedron cd.png

Other uniform ditrigonal polyhedra[edit]

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[edit]

References[edit]

Notes[edit]

  1. ^ a b c Har'El, 1993
  2. ^ Uniform Polyhedron, mathworld (retrieved 10 June 2016)

Bibliography[edit]

Further reading[edit]

  • Johnson, N.; The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 [1]
  • 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