Ditrigonary dodecadodecahedron

The ditrigonary dodecadodecahedron, or ditdid, is a quasiregular uniform polyhedron. It consists of 12 pentagons and 12 pentagrams, with three of each joining at a vertex.

Ditrigonary dodecadodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Ditdid
Coxeter diagram	x5/3o3o5*a ()
Schläfli symbol	$\{5,6\}_4$
Elements
Faces	12 pentagons, 12 pentagrams
Edges	60
Vertices	20
Vertex figure	Tripod, edge lengths (√5–1)/2 and (1+√5)/2
Petrie polygons	30 skew rhombi
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} \approx 0.86603$
Volume	0
Dihedral angle	$\arccos\left(\frac{\sqrt5}{5}\right) \approx 63.43495^\circ$
Central density	4
Number of external pieces	192
Level of complexity	11
Related polytopes
Army	Doe, edge length $\frac{\sqrt5-1}{2}$
Regiment	Sidtid
Dual	Medial triambic icosahedron
Conjugate	Ditrigonary dodecadodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	240
Euler characteristic	–16
Schläfli type	{5,6}
Orientable	Yes
Genus	9
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

It is a faceting of the small ditrigonary icosidodecahedron, using its 12 pentagrams along with 12 additional pentagons.

This polyhedron is abstractly regular, being a quotient of the order-6 pentagonal tiling. Among the non-regular uniform polytopes, it shares this property with the dodecadodecahedron. Its realization may also be considered regular if one also counts conjugacies as symmetries.

This polyhedron is the vertex figure of the ditrigonary dishecatonicosachoron.

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the small ditrigonary icosidodecahedron.

External links

• Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#34).

• Klitzing, Richard. "ditdid".

• Wikipedia Contributors. "Ditrigonal dodecadodecahedron".
• McCooey, David. "Ditrigonal Dodecadodecahedron"

• Hartley, Michael. "{5,6}*240a".