Small retrosnub icosicosidodecahedron

Small retrosnub icosicosidodecahedron
Small retrosnub icosicosidodecahedron.png
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 Small retrosnub icosicosidodecahedron vertfig.png
Bowers acronym Sirsid

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{​32,5}.

3D model of a small retrosnub icosicosidodecahedron

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 hullEdit

Its convex hull is a nonuniform truncated dodecahedron.

Truncated dodecahedron
Convex hull
Small retrosnub icosicosidodecahedron

Cartesian coordinatesEdit

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.


  1. ^ Maeder, Roman. "72: small retrosnub icosicosidodecahedron". MathConsult.

See alsoEdit

External linksEdit