Small Snub Icosicosidodecahedron

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 $\xi=-\frac32+\frac12\sqrt\approx -0.1332396008261379$ be largest (least negative) zero of the polynomial $P=x^2+3x+\phi^$, where $\phi$ is the golden ratio. Let the point $p$ be given by
:$p= \begin \phi^\xi+\phi^ \\ \xi \\ \phi^\xi+\phi^ \end$.
Let the matrix $M$ be given by
:$M= \begin 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end$.
$M$ is the rotation around the axis $(1, 0, \phi)$ by an angle of $2\pi/5$, counterclockwise. Let the linear transformations $T_0, \ldots, T_$
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 $\sqrt$, and the midradius equals $\sqrt$.

For a small snub icosicosidodecahedron whose edge length is 1,
the circumradius is
:$R = \frac12\sqrt \approx 1.4581903307387025$
Its midradius is
:$r = \frac12\sqrt \approx 1.369787954633799$

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

External links

*
*

Uniform polyhedra

{{Polyhedron-stub