Dodecadodecahedron

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Dodecadodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymDid
Coxeter diagramo5/2x5o ()
Schläfli symbol
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)
Circumradius1
Volume5
Dihedral angle
Central density3
Number of external pieces72
Level of complexity6
Related polytopes
ArmyId, edge length
RegimentDid
DualMedial rhombic triacontahedron
Petrie dualPetrial dodecadodecahedron
ConjugateDodecadodecahedron
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[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 three-member regiment that also includes the small dodecahemicosahedron and the great dodecahemicosahedron.

Gallery[edit | edit source]

External links[edit | edit source]