Snub dodecadodecahedron

Snub dodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 84, E = 150 V = 60 (χ = −6)
Faces by sides	60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	| 2 5/2 5
Symmetry group	I, [5,3]+, 532
Index references	U40, C49, W111
Dual polyhedron	Medial pentagonal hexecontahedron
Vertex figure	3.3.5/2.3.5
Bowers acronym	Siddid

3D model of a snub dodecadodecahedron

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₀. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.

Cartesian coordinates

Let $\xi \approx 1.2223809502469911$ be the smallest real zero of the polynomial $P=2x^{4}-5x^{3}+3x+1$. Denote by $\phi$ the golden ratio. Let the point $p$ be given by

$p={\begin{pmatrix}\phi ^{-2}\xi ^{2}-\phi ^{-2}\xi +\phi ^{-1}\\-\phi ^{2}\xi ^{2}+\phi ^{2}\xi +\phi \\\xi ^{2}+\xi \end{pmatrix}}$.

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 dodecadodecahedron. The edge length equals $2(\xi +1){\sqrt {\xi ^{2}-\xi }}$, the circumradius equals $(\xi +1){\sqrt {2\xi ^{2}-\xi }}$, and the midradius equals $\xi ^{2}+\xi$.

For a great snub icosidodecahedron whose edge length is 1, the circumradius is

$R={\frac {1}{2}}{\sqrt {\frac {2\xi -1}{\xi -1}}}\approx 1.2744398820380232$

Its midradius is

$r={\frac {1}{2}}{\sqrt {\frac {\xi }{\xi -1}}}\approx 1.1722614951149297$

The other real root of P plays a similar role in the description of the Inverted snub dodecadodecahedron

Related polyhedra

Medial pentagonal hexecontahedron

Medial pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]+, 532
Index references	DU40
dual polyhedron	Snub dodecadodecahedron

3D model of a medial pentagonal hexecontahedron

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

See also

• List of uniform polyhedra
• Inverted snub dodecadodecahedron

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208

External links

• Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld.