Great rhombidodecacron

In geometry, the great rhombidodecacron (or Great dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great rhombidodecahedron. It is visually identical to the great deltoidal hexecontahedron. Its faces are antiparallelograms.

Great rhombidodecacron

Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 42 (χ = −18)
Symmetry group	I_h, [5,3], *532
Index references	DU₇₃
dual polyhedron	Great rhombidodecahedron

3D model of a great rhombidodecacron

Proportions

Each antiparallelogram has two angles of $\arccos({\frac {1}{2}}+{\frac {1}{5}}{\sqrt {5}})\approx 18.699\,407\,085\,15^{\circ }$ and two angles of $\arccos(-{\frac {5}{8}}+{\frac {1}{8}}{\sqrt {5}})\approx 110.211\,801\,805\,89^{\circ }$ . The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{8}}+{\frac {9{\sqrt {5}}}{40}})\approx 51.088\,791\,108\,96^{\circ }$ . The dihedral angle equals $\arccos({\frac {-19+8{\sqrt {5}}}{41}})\approx 91.553\,403\,672\,16^{\circ }$ . The ratio between the lengths of the long edges and the short ones equals ${\frac {1}{2}}+{\frac {1}{2}}{\sqrt {5}}$ , which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.

References

Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 88

External links

Weisstein, Eric W. "Great pentagrammic hexecontahedron". MathWorld.

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