Small retrosnub icosicosidodecahedron

Last updated June 28, 2024

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}
Coxeter diagram
Wythoff symbol	\| 3/2 3/2 5/2
Symmetry group	I_h, [5,3], *532
Index references	U ₇₂, C ₉₁, W ₁₁₈
Dual polyhedron	Small hexagrammic hexecontahedron
Vertex figure	(3⁵.5/3)/2
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 $U 72$ . It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.^[1] It is given a Schläfli symbol sr{⁵/₃,³/₂}.

Convex hull

Its convex hull is a nonuniform truncated dodecahedron.

Truncated dodecahedron	Convex hull	Small retrosnub icosicosidodecahedron

Cartesian coordinates

Let $\xi =-{\frac {3}{2}}-{\frac {1}{2}}{\sqrt {1+4\phi }}\approx -2.866760399173862$ be the smallest (most 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={\frac {1}{2}}{\sqrt {\frac {\xi -1}{\xi }}}\approx 0.5806948001339209

Its midradius is

r={\frac {1}{2}}{\sqrt {\frac {-1}{\xi }}}\approx 0.2953073837589815

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

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References

↑ Maeder, Roman. "72: small retrosnub icosicosidodecahedron". MathConsult.
↑ Birrell, Robert J. (May 1992). The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron (M.S.). California State University.
↑ Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.

External links

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

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

[2] Birrell, Robert J. (May 1992). The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron (M.S.). California State University.

[3] Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.

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