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Polyhedron with 60 faces
3D model of a medial hexagonal hexecontahedron
In geometry , the medial hexagonal hexecontahedron (or midly dentoid ditriacontahedron ) is a nonconvex isohedral polyhedron . It is the dual of the uniform snub icosidodecadodecahedron .
Proportions
The faces of the medial hexagonal hexecontahedron are irregular nonconvex hexagons. Denote the golden ratio by
ϕ
{\displaystyle \phi }
, and let
ξ
≈
−
0.377
438
833
12
{\displaystyle \xi \approx -0.377\,438\,833\,12}
be the real zero of the polynomial
8
x
3
−
4
x
2
+
1
{\displaystyle 8x^{3}-4x^{2}+1}
. The number
ξ
{\displaystyle \xi }
can be written as
ξ
=
−
1
/
(
2
ρ
)
{\displaystyle \xi =-1/(2\rho )}
, where
ρ
{\displaystyle \rho }
is the plastic ratio . Then each face has four equal angles of
arccos
(
ξ
)
≈
112.175
128
045
27
∘
{\displaystyle \arccos(\xi )\approx 112.175\,128\,045\,27^{\circ }}
, one of
arccos
(
ϕ
2
ξ
+
ϕ
)
≈
50.958
265
917
31
∘
{\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 50.958\,265\,917\,31^{\circ }}
and one of
360
∘
−
arccos
(
ϕ
−
2
ξ
−
ϕ
−
1
)
≈
220.341
221
901
59
∘
{\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 220.341\,221\,901\,59^{\circ }}
. Each face has two long edges, two of medium length and two short ones. If the medium edges have length
2
{\displaystyle 2}
, the long ones have length
1
+
(
1
−
ξ
)
/
(
−
ϕ
−
3
−
ξ
)
≈
4.121
448
816
41
{\displaystyle 1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 4.121\,448\,816\,41}
and the short ones
1
−
(
1
−
ξ
)
/
(
ϕ
3
−
ξ
)
≈
0.453
587
559
98
{\displaystyle 1-{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.453\,587\,559\,98}
. The dihedral angle equals
arccos
(
ξ
/
(
ξ
+
1
)
)
≈
127.320
132
197
62
∘
{\displaystyle \arccos(\xi /(\xi +1))\approx 127.320\,132\,197\,62^{\circ }}
.
References
External links