Snub icosidodecadodecahedron

Snub icosidodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 104, E = 180 V = 60 (χ = −16)
Faces by sides	(20+60){3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	| 5/3 3 5
Symmetry group	I, [5,3]+, 532
Index references	U46, C58, W112
Dual polyhedron	Medial hexagonal hexecontahedron
Vertex figure	3.3.3.5.3.5/3
Bowers acronym	Sided

3D model of a snub icosidodecadodecahedron

In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₆. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.

Cartesian coordinates

Let ⁠$\rho =$⁠ 1.324717957244746... be the real zero of the polynomial ⁠$x^{3}-x-1$⁠. The constant ⁠$\rho$⁠ is known as the plastic ratio. Denote by ⁠$\phi =$⁠ 1.618033988749894... the golden ratio.

Let the point ⁠$p$⁠ be given by $$p={\begin{pmatrix}\rho \\(\phi -\rho )/(\rho +1)\\(\rho -\phi )/\rho \end{pmatrix}}$$ 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 snub icosidodecadodecahedron.

The edge length $\lambda =2{\sqrt {(\rho ^{5}-\phi ^{2})\rho ^{-5}}}$, the circumradius equals ${\sqrt {(\rho ^{5}-\phi ^{2})(1+\rho ^{-5})}},$ and the midradius ${\sqrt {\rho ^{5}-\phi ^{2}}}.$ To get a snub icosidodecadodecahedron with unit edge length, divide all above coordinates by ⁠$\lambda$⁠.

Properties

The snub icosidodecadodecahedron was among the ten snub polyhedra discovered by Coxeter and Miller in the early 1930's, but published only in 1954. Its equation is in Table 4 of Uniform polyhedra.

The measures listed on David McCooey's Visual Polyhedra can be written in terms of ⁠${\sqrt {5}}$⁠ and plastic constant ⁠$\rho$⁠.

The snub icosidodecadodecahedron of unit edge length

element	size
circumscribed radius	$R={\tfrac {1}{2}}{\sqrt {\rho ^{5}+1}}$
midscribed radius	$r={\tfrac {1}{2}}{\sqrt {\rho ^{5}}}$
pentagon center radius	$r_{5}={\tfrac {1}{2}}{\sqrt {\rho ^{4}-{\tfrac {2}{\sqrt {5}}}}}$
triangle center radius	$r_{3}={\tfrac {1}{2}}{\sqrt {\rho ^{5}-{\tfrac {1}{3}}}}$
pentagram center radius	$r_{\tfrac {5}{3}}=-{\tfrac {1}{2}}{\sqrt {\rho ^{4}+{\tfrac {2}{\sqrt {5}}}}}$
triangle-triangle angle	$\alpha _{33}=\arccos \left({\tfrac {1}{3}}(1-2\rho ^{2})\right)$
pentagram-triangle angle	$\alpha _{{\tfrac {5}{3}}3}=\arccos \left({\sqrt {{\tfrac {1}{3}}(u-v)}}\right)$
pentagon-triangle angle	$\alpha _{53}=\arccos \left(-{\sqrt {{\tfrac {1}{3}}(u+v)}}\right)$

With $u=2\rho ^{2}-2\rho +1,$ $v=(\rho ^{2}-3\rho +1){\tfrac {2}{\sqrt {5}}}.$

McCooey's metrics panel gives volume 4.620684..., but no formula. However, this solution (see Answer) is correct.

Graph

The skeleton of a snub icosidodecadodecahedron can be represented as a graph with 60 vertices and 180 edges, an Eulerian graph.

Related polyhedra

Medial hexagonal hexecontahedron

Medial hexagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 180 V = 104 (χ = −16)
Symmetry group	I, [5,3]+, 532
Index references	DU46
dual polyhedron	Snub icosidodecadodecahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

See also

• Snubs Jonathan Bowers
• Uniform snubs Robert Ferréol
• Sided Richard Klitzing
• List of uniform polyhedra

Further reading

• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 0730208..