Small hexacronic icositetrahedron

In geometry, the small hexacronic icositetrahedron is the dual of the small cubicuboctahedron. It is visually identical to the small rhombihexacron. A part of each dart lies inside the solid, hence is invisible in solid models.

Small hexacronic icositetrahedron
Type	Star polyhedron
Face
Elements	F = 24, E = 48 V = 20 (χ = −4)
Symmetry group	Oh, [4,3], *432
Index references	DU13
dual polyhedron	Small cubicuboctahedron

3D model of a small hexacronic icositetrahedron

Proportions

Its faces are darts, having two angles of ${\displaystyle \arccos({\frac {1}{4}}+{\frac {1}{2}}{\sqrt {2}})\approx 16.842\,116\,236\,30^{\circ }}$, one of ${\displaystyle \arccos({\frac {1}{2}}-{\frac {1}{4}}{\sqrt {2}})\approx 81.578\,941\,881\,85^{\circ }}$ and one of ${\displaystyle 360^{\circ }-\arccos(-{\frac {1}{4}}-{\frac {1}{8}}{\sqrt {2}})\approx 244.736\,825\,645\,55^{\circ }}$. Its dihedral angles equal ${\displaystyle \arccos({\frac {-7-4{\sqrt {2}}}{17}})\approx 138.117\,959\,055\,51^{\circ }}$. The ratio between the lengths of the long edges and the short ones equals ${\displaystyle 2-{\frac {1}{2}}{\sqrt {2}}\approx 1.292\,893\,218\,81}$.

References

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

External links

• Weisstein, Eric W., "Small hexacronic icositetrahedron" ("Uniform polyhedron") at MathWorld.