Disrhombidodecahedron

Disrhombidodecahedron
(OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Dird
Elements
Components	12 pentagonal prisms
Faces	60 squares, 24 pentagons
Edges	60+120
Vertices	60
Vertex figure	Compound of two isosceles triangles, edge lengths (1+√5)/2, √2, √2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{15+2\sqrt5}{20}} ≈ 0.98672$
Volume	$3\sqrt{25+10\sqrt5} ≈ 20.64573$
Dihedral angles	4–4: 108°
	4–5: 90°
Central density	12
Related polytopes
Army	Semi-uniform Ti
Regiment	Dird
Dual	Compound of twelve pentagonal tegums
Conjugate	Great disrhombidodecahedron
Topological properties
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The disrhombidodecahedron, dird, or compound of twelve pentagonal prisms is a uniform polyhedron compound. It consists of 60 squares and 24 pentagons, with two pentagons and four squares joining at a vertex.

It can be formed by combining the two chiral forms of the chirorhombidodecahedron, which results in vertices pairing up and two components joining per vertex.

Its quotient prismatic equivalent is the pentagonal prismatic dodecadakoorthowedge, which is fourteen-dimensional.

Vertex coordinates

The vertices of a disrhombidodecahedron of edge length 1 are given by all permutations of:

• $\left(±\sqrt{\frac{5-2\sqrt5}{20}},\,±\sqrt{\frac{5+2\sqrt5}{20}},\,±\sqrt{\frac{5+2\sqrt5}{20}}\right),$

Plus all even permutations of:

• $\left(0,\,±\sqrt{\frac{5-\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),$
• $\left(±\sqrt{\frac{5-\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category C7: Chiral and Doubled Prismatics" (#41).

• Klitzing, Richard. "dird".

• Wikipedia Contributors. "Compound of twelve pentagonal prisms".