medial hexagonal hexecontahedron

The faces of the medial hexagonal hexecontahedron are irregular nonconvex hexagons. Denote the golden ratio by $\backslash phi$, and let $\backslash xi\backslash approx\; -0.377\backslash ,438\backslash ,833\backslash ,12$ be the real zero of the polynomial $8x^3-4x^2+1$. The number $\backslash xi$ can be written as $\backslash xi=-1/(2\backslash rho)$, where $\backslash rho$ is the plastic ratio. Then each face has four equal angles of $\backslash arccos(\backslash xi)\backslash approx\; 112.175\backslash ,128\backslash ,045\backslash ,27^\{\backslash circ\}$, one of $\backslash arccos(\backslash phi^2\backslash xi+\backslash phi)\backslash approx\; 50.958\backslash ,265\backslash ,917\backslash ,31^\{\backslash circ\}$ and one of $360^\{\backslash circ\}-\backslash arccos(\backslash phi^\{-2\}\backslash xi-\backslash phi^\{-1\})\backslash approx\; 220.341\backslash ,221\backslash ,901\backslash ,59^\{\backslash circ\}$. 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 $1+\backslash sqrt\{(1-\backslash xi)/(-\backslash phi^\{-3\}-\backslash xi)\}\backslash approx\; 4.121\backslash ,448\backslash ,816\backslash ,41$ and the short ones $1-\backslash sqrt\{(1-\backslash xi)/(\backslash phi^\{3\}-\backslash xi)\}\backslash approx\; 0.453\backslash ,587\backslash ,559\backslash ,98$. The dihedral angle equals $\backslash arccos(\backslash xi/(\backslash xi+1))\backslash approx\; 127.320\backslash ,132\backslash ,197\backslash ,62^\{\backslash circ\}$.

• {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983}}

• {{mathworld | urlname = MedialHexagonalHexecontahedron| title =Medial hexagonal hexecontahedron}}

Category:Dual uniform polyhedra

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