Small snub dodecahedron

Small snub dodecahedron
	(OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sassid
Elements
Components	6 pentagrammic antiprisms
Faces	60 triangles, 12 pentagrams
Edges	60+60
Vertices	60
Vertex figure	Isosceles trapezoid, edge length 1, 1, 1, (√5–1)/2
Measures (edge length 1)
Circumradius
Volume
Dihedral angles	5/2–3:
	3–3:
Central density	12
Number of external pieces	360
Level of complexity	66
Related polytopes
Army	Non-uniform Snid
Regiment	Sassid
Dual	Compound of six pentagrammic antitegums
Conjugate	Small snub dodecahedron
Convex core	Non-Catalan pentagonal hexecontahedron
Abstract & topological properties
Flag count	480
Orientable	Yes
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

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.

Vertex coordinates[edit | edit source]

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

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