# Great chirorhombidodecahedron

The great chirorhombidodecahedron, gikrid, or compound of six pentagrammic prisms is a uniform polyhedron compound. It consists of 30 squares and 12 pentagrams, with one pentagram and two squares joining at a vertex.

Great chirorhombidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGikrid
Elements
Components6 pentagrammic prisms
Faces30 squares, 12 pentagrams
Edges30+60
Vertices60
Vertex figureIsosceles triangle, edge lengths (5–1)/2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{15-2\sqrt5}{20}} \approx 0.72553}$
Volume${\displaystyle \frac{3\sqrt{25-10\sqrt5}}{2} \approx 2.43690}$
Dihedral angles4–5/2: 90°
4–4: 36°
Central density12
Number of external pieces192
Level of complexity38
Related polytopes
ArmySrid
RegimentGikrid
DualCompound of six pentagrammic tegums
ConjugateChirorhombidodecahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count360
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

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

## Vertex coordinates

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

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

plus all even permutations of:

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

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