# Small snub dodecahedron

Small snub dodecahedron Rank3
TypeUniform
SpaceSpherical
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$\sqrt{\frac{15+\sqrt5}{40}} \approx 0.65643$ Volume$\sqrt{5\sqrt5} \approx 3.34370$ Dihedral angles5/2–3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}3}\right) \approx 114.80110^\circ$ 3–3: $\arccos\left(\frac{2-\sqrt5}3\right) \approx 94.51323^\circ$ Central density12
Number of external pieces360
Level of complexity66
Related polytopes
ArmyNon-uniform Snid
RegimentSassid
DualCompound of six pentagrammic antitegums
ConjugateSmall snub dodecahedron
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:

• $\left(\sqrt{\frac{\sqrt5+\sqrt{5(\sqrt5-2)}}{20}},\,\pm\sqrt{\frac{5-2\sqrt5}{20}},\,\pm\sqrt{\frac{5+\sqrt5-2\sqrt{10(\sqrt5-1)}}{40}}\right),$ • $\left(\sqrt{\frac{3\sqrt5-5}{40}},\,\sqrt{\frac{5-\sqrt5}{10}},\,\frac{\sqrt{5\sqrt5}}{10}\right),$ • $\left(-\sqrt{\frac{2\sqrt5+\sqrt{10(\sqrt5-1)}}{20}},\,\pm\sqrt{\frac{5-2\sqrt5}{20}},\,\pm\sqrt{\frac{5+\sqrt5+2\sqrt{10(\sqrt5-1)}}{40}}\right),$ • $\left(-\sqrt{\frac{\sqrt5-\sqrt{5(\sqrt5-2)}}{20}},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5+2\sqrt{15(\sqrt5-2)}}{20}}\right),$ • $\left(\sqrt{\frac{\sqrt5+2\sqrt{5(\sqrt5-2)}}{20}},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5-2\sqrt{5(\sqrt5-2)}}{20}}\right).$ ## Related polyhedra

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