# Compound of six pentagrammic antiprisms

(Redirected from Small snub dodecahedron)
Compound of six pentagrammic antiprisms
Rank3
TypeUniform
Notation
Bowers style acronymSassid
Elements
Components6 pentagrammic antiprisms
Faces60 triangles, 12 pentagrams
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge length 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+{\sqrt {5}}}{40}}}\approx 0.65643}$
Volume${\displaystyle {\sqrt {5{\sqrt {5}}}}\approx 3.34370}$
Dihedral angles5/2–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{3}}}\right)\approx 114.80110^{\circ }}$
3–3: ${\displaystyle \arccos \left({\frac {2-{\sqrt {5}}}{3}}\right)\approx 94.51323^{\circ }}$
Central density12
Number of external pieces360
Level of complexity66
Related polytopes
ArmyNon-uniform Snid
RegimentSassid
DualCompound of six pentagrammic antitegums
ConjugateCompound of six pentagrammic antiprisms
Convex coreNon-Catalan pentagonal hexecontahedron
Abstract & topological properties
Flag count480
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

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

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

## Vertex coordinates

The vertices of a small snub dodecahedron of edge length 1 are given by all even permutations and even sign changes of:

• ${\displaystyle \left({\sqrt {\frac {{\sqrt {5}}+{\sqrt {5({\sqrt {5}}-2)}}}{20}}},\,\pm {\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}-2{\sqrt {10({\sqrt {5}}-1)}}}{40}}}\right),}$
• ${\displaystyle \left({\sqrt {\frac {3{\sqrt {5}}-5}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,{\frac {\sqrt {5{\sqrt {5}}}}{10}}\right),}$
• ${\displaystyle \left(-{\sqrt {\frac {2{\sqrt {5}}+{\sqrt {10({\sqrt {5}}-1)}}}{20}}},\,\pm {\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}+2{\sqrt {10({\sqrt {5}}-1)}}}{40}}}\right),}$
• ${\displaystyle \left(-{\sqrt {\frac {{\sqrt {5}}-{\sqrt {5({\sqrt {5}}-2)}}}{20}}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}+2{\sqrt {15({\sqrt {5}}-2)}}}{20}}}\right),}$
• ${\displaystyle \left({\sqrt {\frac {{\sqrt {5}}+2{\sqrt {5({\sqrt {5}}-2)}}}{20}}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}-2{\sqrt {5({\sqrt {5}}-2)}}}{20}}}\right).}$

## Related polyhedra

This compound is chiral. The compound of the two enantiomorphs is the small disnub dodecahedron.