Great pentagrammic hexecontahedron

Denote the golden ratio by $\backslash phi$. Let $\backslash xi\backslash approx\; 0.946\backslash ,730\backslash ,033\backslash ,56$ be the largest positive zero of the polynomial $P\; =\; 8x^3-8x^2+\backslash phi^\{-2\}$. Then each pentagrammic face has four equal angles of $\backslash arccos(\backslash xi)\backslash approx\; 18.785\backslash ,633\backslash ,958\backslash ,24^\{\backslash circ\}$ and one angle of $\backslash arccos(-\backslash phi^\{-1\}+\backslash phi^\{-2\}\backslash xi)\backslash approx\; 104.857\backslash ,464\backslash ,167\backslash ,03^\{\backslash circ\}$. Each face has three long and two short edges. The ratio $l$ between the lengths of the long and the short edges is given by

:$l\; =\; \backslash frac\{2-4\backslash xi^2\}\{1-2\backslash xi\}\backslash approx\; 1.774\backslash ,215\backslash ,864\backslash ,94$.

The dihedral angle equals $\backslash arccos(\backslash xi/(\backslash xi+1))\backslash approx\; 60.901\backslash ,133\backslash ,713\backslash ,21^\{\backslash circ\}$. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial $P$ play a similar role in the description of the great pentagonal hexecontahedron and the great 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 = GreatPentagrammicHexecontahedron| title =Great pentagrammic hexecontahedron}}

Category:Dual uniform polyhedra

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