Small retrosnub icosicosidodecahedron

Small retrosnub icosicosidodecahedron
Type	Uniform star polyhedron
Elements	F = 112, E = 180 V = 60 (χ = −8)
Faces by sides	(40+60){3}+12{5/2}
Wythoff symbol	| 3/2 3/2 5/2
Symmetry group	Ih, [5,3], *532
Index references	U72, C91, W118
Dual polyhedron	Small hexagrammic hexecontahedron
Vertex figure	(35.5/3)/2
	Sirsid

3D model of a small retrosnub icosicosidodecahedron

In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U₇₂. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.^[1] It is given a Schläfli symbol ß{3⁄2,5}.

The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.

Convex hull[]

Its convex hull is a nonuniform truncated dodecahedron.

Truncated dodecahedron

Convex hull

Small retrosnub icosicosidodecahedron

Cartesian coordinates[]

Cartesian coordinates for the vertices of a small retrosnub icosicosidodecahedron are all the even permutations of

(±(1-ϕ−α), 0, ±(3−ϕα))

(±(ϕ-1−α), ±2, ±(2ϕ-1−ϕα))

(±(ϕ+1−α), ±2(ϕ-1), ±(1−ϕα))

where ϕ = (1+√5)/2 is the golden ratio and α = √3ϕ−2.

See also[]

• List of uniform polyhedra
• Small snub icosicosidodecahedron

References[]

External links[]

• Weisstein, Eric W. "Small retrosnub icosicosidodecahedron". MathWorld.
• Klitzing, Richard. "3D star small retrosnub icosicosidodecahedron".

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