Ditrigonal icosahedron
The ditrigonal icosahedron, or ditti, also called the excavated dodecahedron, is a noble selfdual semiuniform polyhedron. It consists of 20 propeller tripods, with six joining at a vertex. It has two lengths of edges, but it has no degrees of variance; the longer edges are times the length of the shorter edges. The longer edges are also the edges of a great stellated dodecahedron, while the shorter edges are the edges of a dodecahedron. It is a stellation of the icosahedron.
Ditrigonal icosahedron  

Rank  3 
Type  Semiuniform 
Notation  
Bowers style acronym  Ditti 
Elements  
Faces  20 propeller tripods 
Edges  30+30 
Vertices  20 
Vertex figure  Concave triambus 
Measures (edge lengths 1, )  
Edge length ratio  
Circumradius  
Central density  0 
Number of external pieces  60 
Level of complexity  3 
Related polytopes  
Army  Doe 
Regiment  Ditti 
Dual  Ditrigonal icosahedron 
Conjugate  Ditrigonal icosahedron 
Convex core  Icosahedron 
Abstract & topological properties  
Flag count  240 
Euler characteristic  –20 
Schläfli type  {6,6} 
Orientable  Yes 
Genus  11 
Properties  
Symmetry  H_{3}, order 120 
Flag orbits  2 
Convex  No 
Nature  Tame 
History  
Discovered by  Edmond Hess 
First discovered  1877 
The ditrigonal icosahedron is a twoorbit polyhedron, and also the only twoorbit polyhedron which is also noble. If one counts conjugacies as symmetries, the ditrigonal icosahedron is a regular polyhedron. Abstractly this polyhedron is a quotient of the order6 hexagonal tiling.
Extending the shorter edges while keeping the face planes results in a great icosahedron, while contracting the longer edges while keeping the face planes results in an icosahedron. Relatedly, it can be constructed by antitruncating the icosahedron or hypertruncating the great icosahedron until the resulting pentagons or pentagrams meet, causing the polyhedron to be exotic. Removing the pentagons or pentagrams (which form the faces of a dodecahedron or great stellated dodecahedron) results in the ditrigonal icosahedron.
This polyhedron appears as the vertex figure of the uniform invertidodecahedronary hecatonicosachoron.
Gallery edit

A single face highlighted of the ditrigonal icosahedron.
External links edit
 Klitzing, Richard. "ditti".
 Wikipedia contributors. "Excavated dodecahedron".
 Bowers, Jonathan. "Uniform Polyhedra and Other Three Dimensional Shapes".
 McCooey, David. "Hugel's Polyhedron"
 Hartley, Michael. "{6,6}*240c".
 Wedd, N. R11.5