Small icosicosidodecahedron

The small icosicosidodecahedron, or siid, is a uniform polyhedron. It consists of 20 triangles, 12 pentagrams, and 20 hexagons. One triangle, one pentagram, and two hexagons join at each vertex.

Small icosicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Siid
Coxeter diagram	x5/2o3x3*a ()
Elements
Faces	20 triangles, 12 pentagrams, 20 hexagons
Edges	60+60
Vertices	60
Vertex figure	Isosceles trapezoid, edge lengths 1, √3, (√5–1)/2, √3
Measures (edge length 1)
Circumradius	$\sqrt{\frac{17+3\sqrt5}{8}} \approx 1.72149$
Volume	$\frac{30+29\sqrt5}{3} \approx 31.61533$
Dihedral angles	5/2–6: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) \approx 142.62263^\circ$
	3–6: $\arccos\left(-\frac{\sqrt5}{3}\right) \approx 138.18969^\circ$
Central density	2
Number of external pieces	92
Level of complexity	7
Related polytopes
Army	Semi-uniform Srid, edge lengths $\frac{\sqrt5-1}{2}$ (pentagons), 1 (triangles)
Regiment	Siid
Dual	Small icosacronic hexecontahedron
Conjugate	Great icosicosidodecahedron
Convex core	Truncated icosahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–8
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

A semi-uniform variant of this polyhedron can be constructed as a rectified small ditrigonary icosidodecahedron.

Vertex coordinates

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

• $\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{\sqrt5-1}{4}\right),$
• $\left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{\sqrt5}{2}\right),$
• $\left(±\frac12,\,±1,\,±\frac{3+\sqrt5}{4}\right).$

Representations

A small icosicosidodecahedron has the following Coxeter diagrams:

• x5/2o3x3*a
• ß5o3x (as holosnub)

Related polyhedra

The small icosicosidodecahedron is the colonel of a three-member regiment that also includes the small ditrigonal dodecicosidodecahedron and the small dodecicosahedron.

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 4: Trapeziverts" (#51).

• Klitzing, Richard. "siid".

• Wikipedia Contributors. "Small icosicosidodecahedron".
• McCooey, David. "Small Icosicosidodecahedron"