Great snub icosidodecahedron

Let ${\displaystyle \xi \approx 0.3990206456527105}$  be the positive zero of the polynomial ${\displaystyle x^{3}+2x^{2}-\phi ^{-2}}$ , where ${\displaystyle \phi }$  is the golden ratio. Let the point ${\displaystyle p}$  be given by

${\displaystyle p={\begin{pmatrix}\xi \\\phi ^{-2}-\phi ^{-2}\xi \\-\phi ^{-3}+\phi ^{-1}\xi +2\phi ^{-1}\xi ^{2}\end{pmatrix}}}$ .

Let the matrix ${\displaystyle M}$  be given by

${\displaystyle M={\begin{pmatrix}1/2&-\phi /2&1/(2\phi )\\\phi /2&1/(2\phi )&-1/2\\1/(2\phi )&1/2&\phi /2\end{pmatrix}}}$ .

${\displaystyle M}$  is the rotation around the axis ${\displaystyle (1,0,\phi )}$  by an angle of ${\displaystyle 2\pi /5}$ , counterclockwise. Let the linear transformations ${\displaystyle T_{0},\ldots ,T_{11}}$  be the transformations which send a point ${\displaystyle (x,y,z)}$  to the even permutations of ${\displaystyle (\pm x,\pm y,\pm z)}$  with an even number of minus signs. The transformations ${\displaystyle T_{i}}$  constitute the group of rotational symmetries of a regular tetrahedron. The transformations ${\displaystyle T_{i}M^{j}}$  ${\displaystyle (i=0,\ldots ,11}$ , ${\displaystyle j=0,\ldots ,4)}$  constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points ${\displaystyle T_{i}M^{j}p}$  are the vertices of a great snub icosahedron. The edge length equals ${\displaystyle 2\xi {\sqrt {1-\xi }}}$ , the circumradius equals ${\displaystyle \xi {\sqrt {2-\xi }}}$ , and the midradius equals ${\displaystyle \xi }$ .

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

${\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 0.8160806747999234}$ 

Its midradius is

${\displaystyle r={\frac {1}{2}}{\sqrt {\frac {1}{1-\xi }}}\approx 0.6449710596467862}$ 

The four positive real roots of the sextic in R², $${\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0}$$  are, in order, the circumradii of the great retrosnub icosidodecahedron (U₇₄), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉) and snub dodecahedron (U₂₉).

The great pentagonal hexecontahedron (or great petaloid ditriacontahedron) is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.

Denote the golden ratio by ${\displaystyle \phi }$ . Let ${\displaystyle \xi \approx -0.199\,510\,322\,83}$  be the negative zero of the polynomial ${\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}$ . Then each pentagonal face has four equal angles of ${\displaystyle \arccos(\xi )\approx 101.508\,325\,512\,64^{\circ }}$  and one angle of ${\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 133.966\,697\,949\,42^{\circ }}$ . Each face has three long and two short edges. The ratio ${\displaystyle l}$  between the lengths of the long and the short edges is given by

${\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.315\,765\,089\,00}$ .

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 104.432\,268\,611\,86^{\circ }}$ . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial ${\displaystyle P}$  play a similar role in the description of the great inverted pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

• List of uniform polyhedra
• Great inverted snub icosidodecahedron
• Great retrosnub icosidodecahedron

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

• Weisstein, Eric W. "Great pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Great snub icosidodecahedron". MathWorld.