Polyhedron with 84 faces
3D model of an 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}.
Cartesian coordinates
Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of
(
±
2
α
,
±
2
,
±
2
β
)
,
(
±
[
α
+
β
φ
+
φ
]
,
±
[
−
α
φ
+
β
+
1
φ
]
,
±
[
α
φ
+
β
φ
−
1
]
)
,
(
±
[
−
α
φ
+
β
φ
+
1
]
,
±
[
−
α
+
β
φ
−
φ
]
,
±
[
α
φ
+
β
−
1
φ
]
)
,
(
±
[
−
α
φ
+
β
φ
−
1
]
,
±
[
α
−
β
φ
−
φ
]
,
±
[
α
φ
+
β
+
1
φ
]
)
,
(
±
[
α
+
β
φ
−
φ
]
,
±
[
α
φ
−
β
+
1
φ
]
,
±
[
α
φ
+
β
φ
+
1
]
)
,
{\displaystyle {\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
β
=
α
2
φ
+
φ
α
φ
−
1
φ
,
{\displaystyle \beta ={\frac {\ \ {\frac {\alpha ^{2}{\varphi }+\varphi \ \ }{\ \alpha \varphi -{\frac {1}{\varphi }\ ,}
φ
=
1
+
5
2
{\displaystyle \varphi ={\tfrac {1+{\sqrt {5}{2}
is the
golden ratio , and
α is the negative real
root of
φ
α
4
−
α
3
+
2
α
2
−
α
−
1
φ
⟹
α
≈
−
0.3352090.
{\displaystyle \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
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
ϕ
{\displaystyle \phi }
, and let
ξ
≈
−
0.236
993
843
45
{\displaystyle \xi \approx -0.236\,993\,843\,45}
be the largest (least negative) real zero of the polynomial
P
=
8
x
4
−
12
x
3
+
5
x
+
1
{\displaystyle P=8x^{4}-12x^{3}+5x+1}
. Then each face has three equal angles of
arccos
(
ξ
)
≈
103.709
182
219
53
∘
{\displaystyle \arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }
, one of
arccos
(
ϕ
2
ξ
+
ϕ
)
≈
3.990
130
423
41
∘
{\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }
and one of
360
∘
−
arccos
(
ϕ
−
2
ξ
−
ϕ
−
1
)
≈
224.882
322
917
99
∘
{\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
2
{\displaystyle 2}
, then the short edges have length
1
−
1
−
ξ
ϕ
3
−
ξ
≈
0.474
126
460
54
,
{\displaystyle 1-{\sqrt {\frac {1-\xi }{\phi ^{3}-\xi }\approx 0.474\,126\,460\,54,}
and the long edges have length
1
+
1
−
ξ
ϕ
−
3
−
ξ
≈
37.551
879
448
54.
{\displaystyle 1+{\sqrt {\frac {1-\xi }{\phi ^{-3}-\xi }\approx 37.551\,879\,448\,54.}
The
dihedral angle equals
arccos
(
ξ
/
(
ξ
+
1
)
)
≈
108.095
719
352
34
∘
{\displaystyle \arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }
. The other real zero of the polynomial
P
{\displaystyle P}
plays a similar role for the
medial pentagonal hexecontahedron .
See also
References
External links