Small snub icosicosidodecahedron explained

In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U₃₂. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron,

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 icosahedron.

Cartesian coordinates

Let

\xi =- (32+ (12 \sqrt{ 1+4 \phi} ≈)/(-0 . 1332396008261379))/() be largest (least negative) zero of the polynomial P = x ^{2+3x+\phi} ^{-2} , where \phi is the golden ratio. Let the point p be given byp = \begin{pmatrix} \phi ^{-1} \xi + \phi ^{-3} \\ \xi \\ \phi ^{-2} \xi + \phi ^{-2} \end{pmatrix} .Let the matrix M be given byM = \begin{pmatrix} 1 / 2 & - \phi/ 2 & 1 /( 2 \phi) \\ \phi/ 2 & 1 /( 2 \phi) & -1 / 2 \\ 1 /( 2 \phi) & 1 / 2 & \phi/ 2 \end{pmatrix} .M is the rotation around the axis ( 1 , 0 , \phi) by an angle of 2 \pi/ 5 , counterclockwise. Let the linear transformations T _{0,} \ldots, T _{11} be the transformations which send a point (x, y, z) to the even permutations of (\pm x, \pm y, \pm z) with an even number of minus signs. The transformations T _{i} constitute the group of rotational symmetries of a regular tetrahedron.The transformations T _{i} M ^{j} (i = 0 ,\ldots, 11 , j = 0 ,\ldots, 4 ) constitute the group of rotational symmetries of a regular icosahedron.Then the 60 points T _{i} M ^{j} p are the vertices of a small snub icosicosidodecahedron. The edge length equals -2 \xi , the circumradius equals \sqrt{ -4 \xi - \phi ^{-2} }, and the midradius equals \sqrt{ - \xi} .

For a small snub icosicosidodecahedron whose edge length is 1,the circumradius is

R = (12 \sqrt{ (\xi -1)/(\xi))/() } \approx 1.4581903307387025Its midradius isr = (12 \sqrt{ (-1)/(\xi))/() } \approx 1.369787954633799

The other zero of

P plays a similar role in the description of the small retrosnub icosicosidodecahedron.

See also

• List of uniform polyhedra
• Small retrosnub icosicosidodecahedron