Great dodecacronic hexecontahedron

In geometry, the great dodecacronic hexecontahedron (or great lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great dodecicosidodecahedron. Its 60 intersecting quadrilateral faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.

Great dodecacronic hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 44 (χ = −16)
Symmetry group	Ih, [5,3], *532
Index references	DU61
dual polyhedron	Great dodecicosidodecahedron

3D model of a great dodecacronic hexecontahedron

Proportions

Each kite has two angles of ${\displaystyle \arccos({\frac {5}{8}}-{\frac {1}{8}}{\sqrt {5}})\approx 69.788\,198\,194\,11^{\circ }}$, one of ${\displaystyle \arccos(-{\frac {1}{4}}+{\frac {1}{10}}{\sqrt {5}})\approx 91.512\,394\,720\,74^{\circ }}$ and one of ${\displaystyle \arccos(-{\frac {1}{8}}-{\frac {9}{40}}{\sqrt {5}})\approx 128.911\,208\,891\,04^{\circ }}$. The dihedral angle equals ${\displaystyle \arccos({\frac {-19+8{\sqrt {5}}}{41}})\approx 91.553\,403\,672\,16^{\circ }}$. The ratio between the lengths of the long and short edges is ${\displaystyle {\frac {21+3{\sqrt {5}}}{22}}\approx 1.259\,463\,815\,11}$.

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

External links

• Weisstein, Eric W. "Great dodecacronic hexecontahedron". MathWorld.