Inverted snub dodecadodecahedron

In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U₆₀[1]. It is given a Schläfli symbol sr{5/3,5}.

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

3D model of an inverted snub dodecadodecahedron

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of

(±2α, ±2, ±2β),

(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),

(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),

(±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and

(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),

with an even number of plus signs, where

β = (α²/τ+τ)/(ατ−1/τ),

where τ = (1+√5)/2 is the golden mean and α is the negative real root of τα⁴−α³+2α²−α−1/τ, or approximately −0.3352090. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

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

3D model of a medial inverted pentagonal hexecontahedron

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 ${\displaystyle \phi }$, and let ${\displaystyle \xi \approx -0.236\,993\,843\,45}$ be the largest (least negative) real zero of the polynomial ${\displaystyle P=8x^{4}-12x^{3}+5x+1}$. Then each face has three equal angles of ${\displaystyle \arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }}$, one of ${\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }}$ and one of ${\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 224.882\,322\,917\,99^{\circ }}$. Each face has one medium length edge, two short and two long ones. If the medium length is ${\displaystyle 2}$, then the short edges have length

${\displaystyle 1-{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.474\,126\,460\,54}$,

and the long edges have length

${\displaystyle 1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 37.551\,879\,448\,54}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }}$. The other real zero of the polynomial ${\displaystyle P}$ 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, MR 0730208 p. 124

1. Roman, Maeder. "60: inverted snub dodecadodecahedron". MathConsult.

External links

• Weisstein, Eric W. "Medial inverted pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Inverted snub dodecadodecahedron". MathWorld.