Small snub icosicosidodecahedron

Small snub icosicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Seside
Coxeter diagram	s5/2s3s3*a ()
Elements
Faces	60 triangles, 40 triangles as 20 hexagrams, 12 pentagrams
Edges	60+60+60
Vertices	60
Vertex figure	Mirror-symmetric hexagon, edge lengths 1, 1, 1, 1, 1, (√5–1)/2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {13+3{\sqrt {5}}+{\sqrt {102+46{\sqrt {5}}}}}}{4}}\approx 1.45819$
Volume	${\frac {45+39{\sqrt {5}}+5{\sqrt {302+150{\sqrt {5}}}}}{12}}\approx 21.53680$
Dihedral angles	5/2–3: $\arccos \left(-{\sqrt {\frac {15-2{\sqrt {5}}+2{\sqrt {30{\sqrt {5}}-65}}}{15}}}\right)\approx 161.02258^{\circ }$
	3–3: $\arccos \left(-{\frac {\sqrt {3+2{\sqrt {5}}}}{3}}\right)\approx 155.66804^{\circ }$
Central density	2
Number of external pieces	212
Level of complexity	15
Related polytopes
Army	Semi-uniform ti, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (pentagons), ${\frac {1-{\sqrt {5}}+{\sqrt {6{\sqrt {5}}-2}}}{4}}$ (between ditrigons)
Regiment	Seside
Dual	Small hexagonal hexecontahedron
Conjugate	Small retrosnub icosicosidodecahedron
Convex core	Truncated pentakis dodecahedron
Abstract & topological properties
Flag count	720
Euler characteristic	-8
Orientable	Yes
Genus	5
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small snub icosicosidodecahedron, or seside, is a uniform polyhedron. It consists of 60 snub triangles, 40 more triangles that create 20 hexagrams due to pairs falling in the same plane, and 12 pentagrams. Five triangles and one pentagram meet at each vertex.

It can be obtained as a holosnub truncated icosahedron, after adjusting all edge lengths to be equal.

Vertex coordinates

A small snub icosicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• $\left(0,\,\pm {\frac {3+{\sqrt {3+2{\sqrt {5}}}}}{4}},\,\pm {\frac {1-{\sqrt {5}}+{\sqrt {6{\sqrt {5}}-2}}}{8}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}+{\sqrt {3+2{\sqrt {5}}}}}{4}},\,\pm {\frac {{\sqrt {5}}-1+{\sqrt {6{\sqrt {5}}-2}}}{8}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {3+2{\sqrt {5}}}}}{4}},\,\pm {\frac {3+{\sqrt {5}}+{\sqrt {6{\sqrt {5}}-2}}}{8}}\right).$

Representations

A small snub icosicosidodecahedron has the following Coxeter diagrams:

• s5/2s3s3*a
• o5ß3ß (as holosnub)

Related polyhedra

o5/2o3o3*a truncations

Name	OBSA	CD diagram	Picture
Small ditrigonary icosidodecahedron	sidtid	x5/2o3o3*a ()
(degenerate, double cover of id)		x5/2x3o3*a ()
(degenerate, double cover of ike)		o5/2o3x3*a ()
Small icosicosidodecahedron	siid	x5/2o3x3*a ()
(degenerate, double cover of ti)		x5/2x3x3*a ()
Small snub icosicosidodecahedron	seside	s5/2s3s3*a ()

External links

• Bowers, Jonathan. "Polyhedron Category 6: Snubs" (#68).

• Klitzing, Richard. "seside".
• Wikipedia contributors. "Small snub icosicosidodecahedron".
• McCooey, David. "Small Snub Icosicosidodecahedron"