Great pentakis dodecahedron

In geometry, the great pentakis dodecahedron is a nonconvex isohedral polyhedron.

Great pentakis dodecahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 90 V = 24 (χ = −6)
Symmetry group	Ih, [5,3], *532
Index references	DU58
dual polyhedron	Small stellated truncated dodecahedron

3D model of a great pentakis dodecahedron

It is the dual of the uniform small stellated truncated dodecahedron. The pentagonal faces pass close to the center in the uniform polyhedron, causing this dual to be very spikey. It has 60 intersecting isosceles triangle faces. Part of each triangle lies within the solid, hence is invisible in solid models.

Proportions

The triangles have one very acute angle of ${\displaystyle \arccos({\frac {1}{10}}+{\frac {2}{5}}{\sqrt {5}})\approx 6.051\,689\,017\,91^{\circ }}$ and two of ${\displaystyle \arccos({\frac {1}{2}}-{\frac {1}{5}}{\sqrt {5}})\approx 86.974\,155\,491\,04^{\circ }}$. The dihedral angle equals ${\displaystyle \arccos({\frac {-24+5{\sqrt {5}}}{41}})\approx 108.220\,490\,680\,83^{\circ }}$.

References

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

External links

• Weisstein, Eric W. "Great Pentakis Dodecahedron". MathWorld.