Great pentagrammic hexecontahedron

Template:Short description Template:Uniform polyhedra db In geometry, the great pentagrammic hexecontahedron (or great dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its 60 faces are irregular pentagrams. File:Great pentagrammic hexecontahedron.stl

Proportions

Denote the golden ratio by $\varphi$. Let $\xi \approx 0.94673003356$ be the largest positive zero of the polynomial $P=8x^3-8x^2+{\varphi }^{-2}$. Then each pentagrammic face has four equal angles of $\arccos (\xi )\approx 18.78563395824^{\circ }$ and one angle of $\arccos (-{\varphi }^{-1}+{\varphi }^{-2}\xi )\approx 104.85746416703^{\circ }$. 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

$l=\frac{2-4{\xi }^2}{1-2\xi }\approx 1.77421586494$.

The dihedral angle equals $\arccos (\xi /(\xi +1))\approx 60.90113371321^{\circ }$. 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.

References

• Template:Citation

External links

• Template:Mathworld