Inverted snub dodecadodecahedron

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}
Coxeter diagram
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

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

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of ${\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha \ ,&\pm \,2\ ,&\pm \,2\beta \ &{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}+\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi +\beta +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi -1{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi +1{\bigr ]},&\pm {\bigl [}-\alpha +{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta -{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi -1{\bigr ]},&\pm {\bigl [}\alpha -{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta +{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi -\beta +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi +1{\bigr ]}&{\Bigr )},\end{array}}$

with an even number of plus signs, where $\beta ={\frac {\ \ {\frac {\alpha ^{2}}{\varphi }}+\varphi \ \ }{\ \alpha \varphi -{\frac {1}{\varphi }}}}\ ,$ $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden ratio, and $α$ is the negative real root of $\varphi \alpha ^{4}-\alpha ^{3}+2\alpha ^{2}-\alpha -{\frac {1}{\varphi }}\quad \implies \quad \alpha \approx -0.3352090.$ Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Taking $α$ to be the positive root gives 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

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 $\phi$ , and let $\xi \approx -0.236\,993\,843\,45$ 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 $\arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }$ , one of $\arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }$ and one of $360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 224.882\,322\,917\,99^{\circ }$ . Each face has one medium length edge, two short and two long ones. If the medium length is $2$ , then the short edges have length $1-{\sqrt {\frac {1-\xi }{\phi ^{3}-\xi }}}\approx 0.474\,126\,460\,54,$ and the long edges have length $1+{\sqrt {\frac {1-\xi }{\phi ^{-3}-\xi }}}\approx 37.551\,879\,448\,54.$ The dihedral angle equals $\arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }$ . The other real zero of the polynomial $P$ plays a similar role for the medial pentagonal hexecontahedron.