# Compound of six pentagrammic retroprisms

Compound of six pentagrammic retroprisms
Rank3
TypeUniform
Notation
Bowers style acronymGissed
Elements
Components6 pentagrammic retroprisms
Faces60 triangles, 12 pentagrams
Edges60+60
Vertices60
Vertex figureCrossed isosceles trapezoid, edge length 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Volume${\displaystyle 5-2{\sqrt {5}}\approx 0.52786}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
5/2–3: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Central density18
Number of external pieces1272
Level of complexity83
Related polytopes
ArmySemi-uniform Tid, edge lengths ${\displaystyle {\frac {5-{\sqrt {5}}}{10}}}$ (triangles), ${\displaystyle {\frac {5-2{\sqrt {5}}}{5}}}$ (between dipentagons)
RegimentGissed
DualCompound of six pentagrammic concave antitegums
ConjugateCompound of six pentagonal antiprisms
Convex corePentakis dodecahedron
Abstract & topological properties
Flag count480
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great inverted snub dodecahedron, gissed, or compound of six pentagrammic retroprisms is a uniform polyhedron compound. It consists of 60 triangles and 12 pentagrams, with one pentagram and three triangles joining at a vertex.

This compound can be formed by inscribing six pentagrammic retroprisms within a great icosahedron (each by removing one pair of opposite vertices) and then rotating each retroprism by 36° around its axis.

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

A double cover of this compound occurs as a special case of the great inverted disnub dodecahedron.

## Vertex coordinates

The vertices of a great inverted snub dodecahedron of edge length 1 are given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {5-2{\sqrt {5}}}{10}},\,0,\,\pm {\frac {5+3{\sqrt {5}}}{20}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5}}{5}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {\sqrt {5}}{10}}\right),}$
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{20}},\,\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{10}}\right).}$