Compound of six pentagonal prisms

(Redirected from Rhombidodecahedron)
Compound of six pentagonal prisms
Rank3
TypeUniform
Notation
Bowers style acronymKred
Elements
Components6 pentagonal prisms
Faces30 squares, 12 pentagons
Edges30+60
Vertices60
Vertex figureIsosceles triangle, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+2{\sqrt {5}}}{20}}}\approx 0.98672}$
Volume${\displaystyle {\frac {3{\sqrt {25+10{\sqrt {5}}}}}{2}}\approx 10.32286}$
Dihedral angles4–4: 108°
4–5: 90°
Central density6
Number of external pieces180
Level of complexity34
Related polytopes
ArmySemi-uniform Ti, edge lengths ${\displaystyle {\sqrt {\frac {5-2{\sqrt {5}}}{5}}}}$ (pentagons), ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}}$ (between ditrigons)
RegimentKred
DualCompound of six pentagonal tegums
ConjugateCompound of six pentagrammic prisms
Convex coreDodecahedron
Abstract & topological properties
Flag count360
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

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:

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

Plus all even permutations of:

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

Related polyhedra

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