Inverted snub dodecadodecahedron

Short description: Polyhedron with 84 faces

Inverted 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}
Wythoff symbol	| 5/3 2 5
Symmetry group	I, [5,3]+, 532
Index references	U60, C76, W114
Dual polyhedron	Medial inverted pentagonal hexecontahedron
Vertex figure	3.3.5.3.5/3
Bowers acronym	Isdid

File:Inverted snub dodecadodecahedron.stl In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U₆₀. It is given a Schläfli symbol sr{5/3,5}.

Cartesian coordinates

Let $\xi \approx 2.109759446579943$ be the largest real zero of the polynomial $P=2x^4-5x^3+3x+1$. Denote by $\varphi$ the golden ratio. Let the point $p$ be given by

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

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 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

$R=\frac{1}{2}\sqrt{\frac{2\xi -1}{\xi -1}}\approx 0.8516302281174128$

Its midradius is

$r=\frac{1}{2}\sqrt{\frac{\xi }{\xi -1}}\approx 0.6894012223976083$

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

Related polyhedra

Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]+, 532
Index references	DU60
dual polyhedron	Inverted snub dodecadodecahedron

File:Medial inverted pentagonal hexecontahedron.stl The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.

Proportions

Denote the golden ratio by $\varphi$, and let $\xi \approx -0.23699384345$ be the largest (least negative) real zero of the polynomial $P=8x^4-12x^3+5x+1$. Then each face has three equal angles of $\arccos (\xi )\approx 103.70918221953^{\circ }$, one of $\arccos ({\varphi }^2\xi +\varphi )\approx 3.99013042341^{\circ }$ and one of $360^{\circ }-\arccos ({\varphi }^{-2}\xi -{\varphi }^{-1})\approx 224.88232291799^{\circ }$. Each face has one medium length edge, two short and two long ones. If the medium length is $2$, then the short edges have length $$1-\sqrt{\frac{1-\xi }{{\varphi }^3-\xi }}\approx 0.47412646054,$$ and the long edges have length $$1+\sqrt{\frac{1-\xi }{{\varphi }^{-3}-\xi }}\approx 37.55187944854.$$ The dihedral angle equals $\arccos (\xi /(\xi +1))\approx 108.09571935234^{\circ }$. The other real zero of the polynomial $P$ plays a similar role for the medial pentagonal hexecontahedron.

See also

• List of uniform polyhedra
• Snub dodecadodecahedron

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5  p. 124

External links

• Weisstein, Eric W.. "Medial inverted pentagonal hexecontahedron". http://mathworld.wolfram.com/MedialInvertedPentagonalHexecontahedron.html. 
• Weisstein, Eric W.. "Inverted snub dodecadodecahedron". http://mathworld.wolfram.com/InvertedSnubDodecadodecahedron.html.