Chirorhombidodecahedron

Not to be confused with Rhombic dodecahedron.

Chirorhombidodecahedron
(OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Kred
Elements
Components	6 pentagonal prisms
Faces	30 squares, 12 pentagons
Edges	30+60
Vertices	60
Vertex figure	Isosceles triangle, edge lengths (1+√5)/2, √2, √2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{15+2\sqrt5}{20}} \approx 0.98672$
Volume	$\frac{3\sqrt{25+10\sqrt5}}{2} \approx 10.32286$
Dihedral angles	4–4: 108°
	4–5: 90°
Central density	6
Number of external pieces	180
Level of complexity	34
Related polytopes
Army	Semi-uniform Ti
Regiment	Kred
Dual	Compound of six pentagonal tegums
Conjugate	Great chirorhombidodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	360
Orientable	Yes
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

The chirorhombidodecahedron, rhombidodecahedron, kred, or compound of six pentagonal prisms is a uniform polyhedron compound. It consists of 30 squares and 12 pentagons, with one pentagon and two squares joining at a vertex.

Its quotient prismatic equivalent is the pentagonal prismatic hexateroorthowedge, which is eight-dimensional.

Vertex coordinates

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

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

Plus all even permutations of:

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

Related polyhedra

This compound is chiral. The compound of the two enantiomorphs is the disrhombidodecahedron.

External links

• Bowers, Jonathan. "Polyhedron Category C7: Chiral and Doubled Prismatics" (#40).

• Klitzing, Richard. "kred".

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