Login
Login

Small snub icosicosidodecahedron

{{Short description|Geometric figure}}

{{Uniform polyhedra db|Uniform polyhedron stat table|Seside}}

File:Small snub icosicosidododecahedron.stl

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, {{math|ß{3,5}.}}

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 icosahedron.

class=wikitable width=200
valign=top

|100px
Truncated icosahedron
(regular faces)

|100px
Convex hull
(isogonal hexagons)

|100px
Small snub icosicosidodecahedron

Cartesian coordinates

Let \xi=-\frac32+\frac12\sqrt{1+4\phi}\approx -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 by

:p=

\begin{pmatrix}

\phi^{-1}\xi+\phi^{-3} \\

\xi \\

\phi^{-2}\xi+\phi^{-2}

\end{pmatrix}

.

Let the matrix M be given by

:M=

\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 = \frac12\sqrt{\frac{\xi-1}{\xi}} \approx 1.4581903307387025

Its midradius is

:r = \frac12\sqrt{\frac{-1}{\xi}} \approx 1.369787954633799

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

See also

External links

  • {{mathworld | urlname = SmallSnubIcosicosidodecahedron| title = Small snub icosicosidodecahedron}}
  • {{KlitzingPolytopes|../incmats/seside.htm|3D star|small snub icosicosidodecahedron}}

Category:Uniform polyhedra

{{Polyhedron-stub}}