Rank3
TypeUniform
Notation
Bowers style acronymDid
Coxeter diagramo5/2x5o ()
Schläfli symbol${\displaystyle \{5,4\}_{6}}$
${\displaystyle {\begin{Bmatrix}5\\5/2\end{Bmatrix}}}$
Elements
Faces12 pentagons, 12 pentagrams
Edges60
Vertices30
Vertex figureRectangle, edge lengths (1+5)/2 and (5–1)/2
Petrie polygons20 skew triambi
Measures (edge length 1)
Volume5
Dihedral angle${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density3
Number of external pieces72
Level of complexity6
Related polytopes
ArmyId, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentDid
DualMedial rhombic triacontahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count240
Euler characteristic–6
Schläfli type{5,4}
SurfaceBring's surface
OrientableYes
Genus4
Properties
SymmetryH3, order 120
Flag orbits2
ConvexNo
NatureTame

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 order-4 pentagonal tiling. Among the non-regular 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

A dodecadodecahedron of side length 1 has vertex coordinates given by all permutations of

• ${\displaystyle \left(\pm 1,\,0,\,0\right)}$,

and even permutations of

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}}\right)}$.

The first set of vertices corresponds to a scaled octahedron which can be inscribed into the dodecadodecahedron.

## Related polyhedra

The dodecadodecahedron is the colonel of a three-member regiment that also includes the small dodecahemicosahedron and the great dodecahemicosahedron.