Small retrosnub icosicosidodecahedron
polyhedron with 112 faces
In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. It is given a Schläfli symbol sr{⁵/₃,³/₂}.
The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.
George Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity).
Convex hull
Its convex hull is a nonuniform truncated dodecahedron.
Cartesian coordinates
Let
ξ
=
−
3
2
−
1
2
1
+
4
ϕ
≈
−
2.86676039917
{\displaystyle \xi =-{\frac {3}{2}}-{\frac {1}{2}}{\sqrt {1+4\phi }}\approx -2.86676039917}
be the smallest (most negative) zero of the polynomial
P
=
x
2
+
3
x
+
ϕ
−
2
{\displaystyle P=x^{2}+3x+\phi ^{-2}}
, where
ϕ
{\displaystyle \phi }
is the golden ratio. Equivalently,
ξ
=
−
1
−
ϕ
+
ϕ
+
ϕ
+
⋯
=
−
1
−
β
{\displaystyle \xi =-1-{\sqrt {\phi +{\sqrt {\phi +{\sqrt {\phi +\cdots }}}}}}\,=-1-\beta }
where
β
≈
1.86676039917
{\displaystyle \beta \approx 1.86676039917}
(OEIS: A275828) is a root of
β
2
−
β
−
ϕ
=
0.
{\displaystyle \beta ^{2}-\beta -\phi =0.}
Let the point
p
{\displaystyle p}
be given by
p
=
(
ϕ
−
1
ξ
+
ϕ
−
3
ξ
ϕ
−
2
ξ
+
ϕ
−
2
)
{\displaystyle p={\begin{pmatrix}\phi ^{-1}\xi +\phi ^{-3}\\\xi \\\phi ^{-2}\xi +\phi ^{-2}\end{pmatrix}}}
.
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