Dodecadodecahedron
Dodecadodecahedron  

Rank  3 
Type  Uniform 
Notation  
Bowers style acronym  Did 
Coxeter diagram  o5/2x5o () 
Schläfli symbol  
Elements  
Faces  12 pentagons, 12 pentagrams 
Edges  60 
Vertices  30 
Vertex figure  Rectangle, edge lengths (1+√5)/2 and (√5–1)/2 
Petrie polygons  20 skew triambi 
Measures (edge length 1)  
Circumradius  1 
Volume  5 
Dihedral angle  
Central density  3 
Number of external pieces  72 
Level of complexity  6 
Related polytopes  
Army  Id, edge length 
Regiment  Did 
Dual  Medial rhombic triacontahedron 
Petrie dual  Petrial dodecadodecahedron 
Conjugate  Dodecadodecahedron 
Convex core  Dodecahedron 
Abstract & topological properties  
Flag count  240 
Euler characteristic  –6 
Schläfli type  {5,4} 
Surface  Bring's surface 
Orientable  Yes 
Genus  4 
Properties  
Symmetry  H_{3}, order 120 
Flag orbits  2 
Convex  No 
Nature  Tame 
The dodecadodecahedron, or did, is a quasiregular uniform polyhedron. It consists of 12 pentagons and 12 pentagrams, with two of each joining at a vertex. It can be derived as a rectified small stellated dodecahedron or great dodecahedron.
This polyhedron is abstractly regular, being a quotient of the order4 pentagonal tiling. Among the nonregular uniform polytopes, it shares this property with the ditrigonary dodecadodecahedron. Its realization may also be considered regular if one also counts conjugacies as symmetries.
Vertex coordinates[edit  edit source]
A dodecadodecahedron of side length 1 has vertex coordinates given by all permutations of
 ,
and even permutations of
 .
The first set of vertices corresponds to a scaled octahedron which can be inscribed into the dodecadodecahedron.
Related polyhedra[edit  edit source]
The dodecadodecahedron is the colonel of a threemember regiment that also includes the small dodecahemicosahedron and the great dodecahemicosahedron.
Gallery[edit  edit source]

A fundamental domain of the dodecadodecahedron in {5,4}.
External links[edit  edit source]
 Hartley, Michael. "{5,4}*240".
 Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#27).
 Bowers, Jonathan. "Batch 3: Id, Did, and Gid Facetings" (#1 under did).
 Klitzing, Richard. "did".
 Wikipedia contributors. "Dodecadodecahedron".
 McCooey, David. "Dodecadodecahedron"