Snub dodecadodecahedron explained

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.^[1] It is given a Schläfli symbol as a snub great dodecahedron.

Cartesian coordinates

Let

\xi ≈ 1 . 2223809502469911 be the smallest real zero of the polynomial P =2 x ^{4-5x} ^{3+3x+1} . Denote by \phi the golden ratio. Let the point p be given byp = \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 byM = \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 = (12 \sqrt{ (2 \xi -1)/(\xi -1))/() } \approx 1.2744398820380232 Its midradius isr = (1)/(\sqrt{ 2) (\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

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

Notes and References

1. Web site: 40: snub dodecadodecahedron. Maeder. Roman. MathConsult.