# Compound of six pentagonal antiprisms

(Redirected from Great snub dodecahedron)
Compound of six pentagonal antiprisms
Rank3
TypeUniform
Notation
Bowers style acronymGassid
Elements
Components6 pentagonal antiprisms
Faces60 triangles, 12 pentagons
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge length 1, 1, 1, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Volume${\displaystyle 5+2{\sqrt {5}}\approx 9.47214}$
Dihedral angles3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
5–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Central density6
Number of external pieces600
Level of complexity33
Related polytopes
ArmySemi-uniform Tid, edge lengths ${\displaystyle {\frac {\sqrt {5}}{5}}}$ (triangles) and ${\displaystyle {\frac {3{\sqrt {5}}-5}{10}}}$ (between dipentagons)
RegimentGassid
DualCompound of six pentagonal antitegums
ConjugateCompound of six pentagrammic retroprisms
Convex hullSemi-uniform Tid
Convex coreDodecahedron
Abstract & topological properties
Flag count480
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great snub dodecahedron, gassid, or compound of six pentagonal antiprisms is a uniform polyhedron compound. It consists of 60 triangles and 12 pentagons, with one pentagon and three triangles joining at a vertex.

This compound can be formed by inscribing six pentagonal antiprisms within an icosahedron (each by removing one pair of opposite vertices) and then rotating each antiprism by 36º around its axis.

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

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

## Vertex coordinates

Coordinates for the vertices of a great snub dodecahedron of edge length 1 are given by all even permutations of:

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