- Source: Medial pentagonal hexecontahedron
In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
Proportions
Denote the golden ratio by φ, and let
ξ
≈
−
0.409
037
788
014
42
{\displaystyle \xi \approx -0.409\,037\,788\,014\,42}
be the smallest (most 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
)
≈
114.144
404
470
43
∘
,
{\displaystyle \arccos(\xi )\approx 114.144\,404\,470\,43^{\circ },}
one of
arccos
(
φ
2
ξ
+
φ
)
≈
56.827
663
280
94
∘
{\displaystyle \arccos(\varphi ^{2}\xi +\varphi )\approx 56.827\,663\,280\,94^{\circ }}
and one of
arccos
(
φ
−
2
ξ
−
φ
−
1
)
≈
140.739
123
307
76
∘
.
{\displaystyle \arccos(\varphi ^{-2}\xi -\varphi ^{-1})\approx 140.739\,123\,307\,76^{\circ }.}
Each face has one medium length edge, two short and two long ones. If the medium length is 2, then the short edges have length
1
+
1
−
ξ
φ
3
−
ξ
≈
1.550
761
427
20
,
{\displaystyle 1+{\sqrt {\frac {1-\xi }{\varphi ^{3}-\xi }}}\approx 1.550\,761\,427\,20,}
and the long edges have length
1
+
1
−
ξ
−
φ
−
3
−
ξ
≈
3.854
145
870
08.
{\displaystyle 1+{\sqrt {\frac {1-\xi }{-\varphi ^{-3}-\xi }}}\approx 3.854\,145\,870\,08.}
The dihedral angle equals
arccos
(
ξ
ξ
+
1
)
≈
133.800
984
233
53
∘
.
{\displaystyle \arccos \left({\tfrac {\xi }{\xi +1}}\right)\approx 133.800\,984\,233\,53^{\circ }.}
The other real zero of the polynomial P plays a similar role for the medial inverted pentagonal hexecontahedron.
References
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
Uniform polyhedra and duals