Dodecadodecahedron

Dodecadodecahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
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	H3, order 120
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 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.

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
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 ()