Great inverted snub icosidodecahedron

Template:Short description Template:No footnotes Template:Uniform polyhedra db File:Great inverted snub icosidodecahedron.stl In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U₆₉. It is given a Schläfli symbol Template:Math and Coxeter-Dynkin diagram Template:CDD. In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.

Cartesian coordinates

Let $\xi \approx -0.5055605785332548$ be the largest (least negative) negative zero of the polynomial $x^3+2x^2-{\varphi }^{-2}$, where $\varphi$ is the golden ratio. Let the point $p$ be given by

$p=\left(\begin{array}{c}\xi \\ {\varphi }^{-2}-{\varphi }^{-2}\xi \\ -{\varphi }^{-3}+{\varphi }^{-1}\xi +2{\varphi }^{-1}{\xi }^2\end{array}\right)$.

Let the matrix $M$ 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 great snub icosahedron. The edge length equals $-2\xi \sqrt{1-\xi }$, the circumradius equals $-\xi \sqrt{2-\xi }$, and the midradius equals $-\xi$.

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

$R=\frac{1}{2}\sqrt{\frac{2-\xi }{1-\xi }}\approx 0.6450202372957795$

Its midradius is

$r=\frac{1}{2}\sqrt{\frac{1}{1-\xi }}\approx 0.4074936889340787$

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

Related polyhedra

Great inverted pentagonal hexecontahedron

Template:Uniform polyhedra db File:Great inverted pentagonal hexecontahedron.stl The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron. It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices.

It is the dual of the uniform great inverted snub icosidodecahedron. Template:-

Proportions

Denote the golden ratio by $\varphi$. Let $\xi \approx 0.25278028927$ be the smallest positive zero of the polynomial $P=8x^3-8x^2+{\varphi }^{-2}$. Then each pentagonal face has four equal angles of $\arccos (\xi )\approx 75.35790341742^{\circ }$ and one angle of $360^{\circ }-\arccos (-{\varphi }^{-1}+{\varphi }^{-2}\xi )\approx 238.56838633033^{\circ }$. 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

$l=\frac{2-4{\xi }^2}{1-2\xi }\approx 3.52805303481$.

The dihedral angle equals $\arccos (\xi /(\xi +1))\approx 78.35919906062^{\circ }$. 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 pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

See also

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

References

• Template:Citation p. 126

External links

• Template:Mathworld
• Template:Mathworld

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