Medial pentagonal hexecontahedron

Denote the golden ratio by {{mvar|φ}}, and let $\backslash xi\backslash approx\; -0.409\backslash ,037\backslash ,788\backslash ,014\backslash ,42$ be the smallest (most negative) real zero of the polynomial $P=8x^4-12x^3+5x+1.$ Then each face has three equal angles of $\backslash arccos(\backslash xi)\backslash approx\; 114.144\backslash ,404\backslash ,470\backslash ,43^\{\backslash circ\},$ one of $\backslash arccos(\backslash varphi^2\backslash xi+\backslash varphi)\backslash approx\; 56.827\backslash ,663\backslash ,280\backslash ,94^\{\backslash circ\}$ and one of $\backslash arccos(\backslash varphi^\{-2\}\backslash xi-\backslash varphi^\{-1\})\backslash approx\; 140.739\backslash ,123\backslash ,307\backslash ,76^\{\backslash circ\}.$ 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

$$1\; +\; \backslash sqrt\{\backslash frac\{1-\backslash xi\}\{\backslash varphi^3-\backslash xi\}\}\; \backslash approx\; 1.550\backslash ,761\backslash ,427\backslash ,20,$$

and the long edges have length

$$1\; +\; \backslash sqrt\{\; \backslash frac\{1-\backslash xi\}\{-\backslash varphi^\{-3\}-\backslash xi\}\}\backslash approx\; 3.854\backslash ,145\backslash ,870\backslash ,08.$$

The dihedral angle equals $\backslash arccos\backslash left(\backslash tfrac\{\backslash xi\}\{\backslash xi+1\}\backslash right)\; \backslash approx\; 133.800\backslash ,984\backslash ,233\backslash ,53^\{\backslash circ\}.$ The other real zero of the polynomial {{mvar|P}} plays a similar role for the medial inverted pentagonal hexecontahedron.

• {{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 = MedialPentagonalHexecontahedron| title =Medial pentagonal hexecontahedron}}
• [https://web.archive.org/web/20171110075259/http://gratrix.net/polyhedra/uniform/summary/ Uniform polyhedra and duals]

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Category:Dual uniform polyhedra

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