Great hexacronic icositetrahedron

Polyhedron with 24 faces
Great hexacronic icositetrahedron
Type Star polyhedron
Face
Elements F = 24, E = 48
V = 20 (χ = −4)
Symmetry group Oh, [4,3], *432
Index references DU14
dual polyhedron Great cubicuboctahedron

In geometry, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.

Proportions

The kites have two angles of arccos ⁡ ( 1 4 − 1 2 2 ) ≈ 117.200 570 380 16 ∘ {\displaystyle \arccos({\frac {1}{4}}-{\frac {1}{2}}{\sqrt {2}})\approx 117.200\,570\,380\,16^{\circ }} , one of arccos ⁡ ( − 1 4 + 1 8 2 ) ≈ 94.199 144 429 76 ∘ {\displaystyle \arccos(-{\frac {1}{4}}+{\frac {1}{8}}{\sqrt {2}})\approx 94.199\,144\,429\,76^{\circ }} and one of arccos ⁡ ( 1 2 + 1 4 2 ) ≈ 31.399 714 809 92 ∘ {\displaystyle \arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }} . The dihedral angle equals arccos ⁡ ( − 7 + 4 2 17 ) ≈ 94.531 580 798 20 ∘ {\displaystyle \arccos({\frac {-7+4{\sqrt {2}}}{17}})\approx 94.531\,580\,798\,20^{\circ }} . The ratio between the lengths of the long and short edges is 2 + 1 2 2 ≈ 2.70710678118655 {\displaystyle 2+{\frac {1}{2}}{\sqrt {2}}\approx 2.70710678118655} .

References

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

External links

  • Weisstein, Eric W. "Great Hexacronic Icositetrahedron". MathWorld.


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