Great snub icosidodecahedron

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

:$p=$

\begin{pmatrix}

\xi \\

\phi^{-2}-\phi^{-2}\xi \\

-\phi^{-3}+\phi^{-1}\xi+2\phi^{-1}\xi^2

\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 $(1,\; 0,\; \backslash phi)$ by an angle of $2\backslash pi/5$, counterclockwise. Let the linear transformations $T\_0,\; \backslash ldots,\; T\_\{11\}$

be the transformations which send a point $(x,\; y,\; z)$ to the even permutations of $(\backslash pm\; x,\; \backslash pm\; y,\; \backslash pm\; z)$ with an even number of minus signs.

The transformations $T\_i$ constitute the group of rotational symmetries of a regular tetrahedron.

The transformations $T\_i\; M^j$ $(i\; =\; 0,\backslash ldots,\; 11$, $j\; =\; 0,\backslash ldots,\; 4)$ constitute the group of rotational symmetries of a regular icosahedron.

Then the 60 points $T\_i\; M^j\; p$ are the vertices of a great snub icosahedron. The edge length equals $2\backslash xi\backslash sqrt\{1-\backslash xi\}$, the circumradius equals $\backslash xi\backslash sqrt\{2-\backslash xi\}$, and the midradius equals $\backslash xi$.

For a great snub icosidodecahedron whose edge length is 1,

the circumradius is

:$R\; =\; \backslash frac12\backslash sqrt\{\backslash frac\{2-\backslash xi\}\{1-\backslash xi\}\}\; \backslash approx\; 0.8160806747999234$

Its midradius is

:$r=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{1\}\{1-\backslash xi\}\}\; \backslash approx\; 0.6449710596467862$

The four positive real roots of the sextic in {{math|R{{sup|2}}}},

$$4096R^\{12\}\; -\; 27648R^\{10\}\; +\; 47104R^8\; -\; 35776R^6\; +\; 13872R^4\; -\; 2696R^2\; +\; 209\; =\; 0$$

are, in order, the circumradii of the great retrosnub icosidodecahedron (U₇₄), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉) and snub dodecahedron (U₂₉).

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{{Uniform polyhedra db|Uniform dual polyhedron stat table|Gosid}}

File:Great pentagonal hexecontahedron.stl

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

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

The dihedral angle equals $\backslash arccos(\backslash xi/(\backslash xi+1))\backslash approx\; 104.432\backslash ,268\backslash ,611\backslash ,86^\{\backslash circ\}$. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial $P$ 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

• {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983}}

{{Reflist}}

• {{mathworld | urlname = GreatPentagonalHexecontahedron| title =Great pentagonal hexecontahedron}}
• {{mathworld | urlname = GreatSnubIcosidodecahedron| title = Great snub icosidodecahedron}}

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Category:Uniform polyhedra