Snub icosidodecadodecahedron

Template:Short description Template:Uniform polyhedra db File:Snub icosidodecadodecahedron.stl 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 Template:Tmath Template:Math be the real zero of the polynomial Template:Tmath. The constant Template:Tmath is known as the plastic ratio. Denote by Template:Tmath Template:Math the golden ratio.

Let the point Template:Tmath be given by $$p=\left(\begin{array}{c}\rho \\ (\varphi -\rho )/(\rho +1)\\ (\rho -\varphi )/\rho \end{array}\right)$$ Let the matrix Template:Tmath be given by $$M=\left(\begin{array}{ccc}1/2& -\varphi /2& 1/(2\varphi )\\ \varphi /2& 1/(2\varphi )& -1/2\\ 1/(2\varphi )& 1/2& \varphi /2\end{array}\right)$$

$M$ is the rotation around the axis $(1,0,\varphi )$ by an angle of $2\pi /5$, counterclockwise. Let the linear transformations $T_0,\dots ,T_{11}$ 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 $T_i{M}^j$ $(i=0,\dots ,11$, $j=0,\dots ,4)$ 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 $\lambda =2\sqrt{({\rho }^5-{\varphi }^2){\rho }^{-5}}$, the circumradius equals $\sqrt{({\rho }^5-{\varphi }^2)(1+{\rho }^{-5})},$ and the midradius $\sqrt{{\rho }^5-{\varphi }^2}.$ To get a snub icosidodecadodecahedron with unit edge length, divide all above coordinates by Template:Tmath.

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 Template:Plainlink can be written in terms of Template:Tmath and plastic constant Template:Tmath.

The snub icosidodecadodecahedron of unit edge length

element	size
circumscribed radius	$R=\frac{1}{2}\sqrt{{\rho }^5+1}$
midscribed radius	$r=\frac{1}{2}\sqrt{{\rho }^5}$
pentagon center radius	$r_5=\frac{1}{2}\sqrt{{\rho }^4-\frac{2}{\sqrt{5}}}$
triangle center radius	$r_3=\frac{1}{2}\sqrt{{\rho }^5-\frac{1}{3}}$
pentagram center radius	$r_{\frac{5}{3}}=-\frac{1}{2}\sqrt{{\rho }^4+\frac{2}{\sqrt{5}}}$
triangle-triangle angle	${\alpha }_{33}=\arccos (\frac{1}{3}(1-2{\rho }^2))$
pentagram-triangle angle	${\alpha }_{\frac{5}{3}3}=\arccos \left(\sqrt{\frac{1}{3}(u-v)}\right)$
pentagon-triangle angle	${\alpha }_{53}=\arccos (-\sqrt{\frac{1}{3}(u+v)})$

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

McCooey's metrics panel gives volume Template:Math, but no formula. However, Template:Plainlink (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

Template:Uniform polyhedra db 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

• Template:Cite book.