Snub icosidodecadodecahedron

Polyhedron with 104 faces

Snub icosidodecadodecahedron

Type	Uniform star polyhedron
Elements	F = 104, E = 180 V = 60 (χ = −16)
Faces by sides	(20+60){3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	\| 5/3 3 5
Symmetry group	I, [5,3]⁺, 532
Index references	U₄₆, C₅₈, W₁₁₂
Dual polyhedron	Medial hexagonal hexecontahedron
Vertex figure	3.3.3.5.3.5/3
Bowers acronym	Sided

In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₆. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices.^[1] As the name indicates, it belongs to the family of snub polyhedra.

The circumradius of the snub icosidodecadodecahedron with unit edge length is

{\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}},

where ρ is the plastic constant, or the unique real root of ρ³ = ρ + 1.^[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of

{\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha ,&\pm \,2\gamma ,&\pm \,2\beta &{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}+\gamma \varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi +\beta +{\frac {\gamma }{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi -\gamma {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi +\gamma {\bigr ]},&\pm {\bigl [}-\alpha +{\frac {\beta }{\varphi }}-\gamma \varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta -{\frac {\gamma }{\varphi }}{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi -\gamma {\bigr ]},&\pm {\bigl [}\alpha -{\frac {\beta }{\varphi }}-\gamma \varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta +{\frac {\gamma }{\varphi }}{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}-\gamma \varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi -\beta +{\frac {\gamma }{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi +\gamma {\bigr ]}&{\Bigr )},\end{array}}

with an even number of plus signs, where $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden ratio; ρ is the plastic ratio, or the unique real solution to ρ³ = ρ + 1;

{\begin{aligned}\alpha &=\rho +1=\rho ^{3},\\[4pt]\beta &=\varphi ^{2}\rho ^{4}+\varphi ,\\[4pt]\gamma &=\rho ^{2}+\varphi \rho .\end{aligned}}

Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.^[3]

Related polyhedra

Medial hexagonal hexecontahedron

Medial hexagonal hexecontahedron

Type	Star polyhedron
Face
Elements	F = 60, E = 180 V = 104 (χ = −16)
Symmetry group	I, [5,3]⁺, 532
Index references	DU₄₆
dual polyhedron	Snub icosidodecadodecahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

References

^ Maeder, Roman. "46: snub icosidodecadodecahedron". MathConsult.
^ Weisstein, Eric W. "Snub icosidodecadodecahedron". MathWorld.
^ Skilling, John (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society A, 278 (1278): 111–135, doi:10.1098/rsta.1975.0022.