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 Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links Weisstein, Eric W. "Medial hexagonal hexecontahedron". MathWorld.
