Medial hexagonal hexecontahedron

Template:Short description Template:Uniform polyhedra db File:Medial hexagonal hexecontahedron.stl 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 $\varphi$, and let $\xi \approx -0.37743883312$ be the real zero of the polynomial $8x^3-4x^2+1$. The number $\xi$ can be written as $\xi =-1/(2\rho )$, where $\rho$ is the plastic ratio. Then each face has four equal angles of $\arccos (\xi )\approx 112.17512804527^{\circ }$, one of $\arccos ({\varphi }^2\xi +\varphi )\approx 50.95826591731^{\circ }$ and one of $360^{\circ }-\arccos ({\varphi }^{-2}\xi -{\varphi }^{-1})\approx 220.34122190159^{\circ }$. Each face has two long edges, two of medium length and two short ones. If the medium edges have length $2$, the long ones have length $1+\sqrt{(1-\xi )/(-{\varphi }^{-3}-\xi )}\approx 4.12144881641$ and the short ones $1-\sqrt{(1-\xi )/({\varphi }^3-\xi )}\approx 0.45358755998$. The dihedral angle equals $\arccos (\xi /(\xi +1))\approx 127.32013219762^{\circ }$.

References

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External links

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