Great deltoidal icositetrahedron

In geometry, the great deltoidal icositetrahedron (or great sagittal disdodecahedron) is the dual of the nonconvex great rhombicuboctahedron. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models.

Great deltoidal icositetrahedron

Type	Star polyhedron
Face
Elements	F = 24, E = 48 V = 26 (χ = 2)
Symmetry group	O_h, [4,3], *432
Index references	DU₁₇
dual polyhedron	Nonconvex great rhombicuboctahedron

One of its halves can be rotated by 45 degrees to form the pseudo great deltoidal icositetrahedron, analogous to the pseudo-deltoidal icositetrahedron.

Proportions

Faces have three angles of $\arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }$ and one of $360^{\circ }-\arccos(-{\frac {1}{4}}+{\frac {1}{8}}{\sqrt {2}})\approx 265.800\,855\,570\,24^{\circ }$ . Its dihedral angles equal $\arccos({\frac {-7+4{\sqrt {2}}}{17}})\approx 94.531\,580\,798\,20^{\circ }$ . The ratio between the lengths of the long edges and the short ones equals $2+{\frac {1}{2}}{\sqrt {2}}\approx 2.707\,106\,781\,19$ .

References

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

External links

Weisstein, Eric W. "Great Deltoidal Icositetrahedron". MathWorld.

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