(Redirected from Did) Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymDid
Coxeter diagramo5/2x5o (       )
Schläfli symbol$\{5,4\}_6$ $\left\{5\atop{5/2}\right\}$ 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$\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$ Central density3
Number of external pieces72
Level of complexity6
Related polytopes
ArmyId, edge length $\frac{\sqrt5-1}{2}$ 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
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

• $\left(±1,\,0,\,0\right),$ and even permutations of

• $\left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12\right).$ 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.

o5o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great dodecahedron gad {5,5/2} x5o5/2o (     )
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o (     )
Dodecadodecahedron did r{5,5/2} o5x5/2o (     )
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x (     )
Small stellated dodecahedron sissid {5/2,5} o5o5/2x (     )
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x (     )
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x (     )
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s (     )