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 ϕ -->
{\displaystyle \phi }
ξ ξ -->
≈ ≈ -->
− − -->
0.377
438
833
12
{\displaystyle \xi \approx -0.377\,438\,833\,12}
8
x
3
− − -->
4
x
2
+
1
{\displaystyle 8x^{3}-4x^{2}+1}
ξ ξ -->
{\displaystyle \xi }
ξ ξ -->
=
− − -->
1
/
(
2
ρ ρ -->
)
{\displaystyle \xi =-1/(2\rho )}
ρ ρ -->
{\displaystyle \rho }
plastic ratio . Then each face has four equal angles of
arccos
-->
(
ξ ξ -->
)
≈ ≈ -->
112.175
128
045
27
∘ ∘ -->
{\displaystyle \arccos(\xi )\approx 112.175\,128\,045\,27^{\circ }}
arccos
-->
(
ϕ ϕ -->
2
ξ ξ -->
+
ϕ ϕ -->
)
≈ ≈ -->
50.958
265
917
31
∘ ∘ -->
{\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 50.958\,265\,917\,31^{\circ }}
360
∘ ∘ -->
− − -->
arccos
-->
(
ϕ ϕ -->
− − -->
2
ξ ξ -->
− − -->
ϕ ϕ -->
− − -->
1
)
≈ ≈ -->
220.341
221
901
59
∘ ∘ -->
{\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 220.341\,221\,901\,59^{\circ }}
2
{\displaystyle 2}
1
+
(
1
− − -->
ξ ξ -->
)
/
(
− − -->
ϕ ϕ -->
− − -->
3
− − -->
ξ ξ -->
)
≈ ≈ -->
4.121
448
816
41
{\displaystyle 1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 4.121\,448\,816\,41}
1
− − -->
(
1
− − -->
ξ ξ -->
)
/
(
ϕ ϕ -->
3
− − -->
ξ ξ -->
)
≈ ≈ -->
0.453
587
559
98
{\displaystyle 1-{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.453\,587\,559\,98}
dihedral angle equals
arccos
-->
(
ξ ξ -->
/
(
ξ ξ -->
+
1
)
)
≈ ≈ -->
127.320
132
197
62
∘ ∘ -->
{\displaystyle \arccos(\xi /(\xi +1))\approx 127.320\,132\,197\,62^{\circ }}
References
External links
