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

Great pentakis dodecahedron
DU58 great pentakisdodecahedron.png
Type Star polyhedron
Face DU58 facets.png
Elements F = 60, E = 90
V = 24 (χ = −6)
Symmetry group Ih, [5,3], *532
Index references DU58
dual polyhedron Small stellated truncated 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 a \( {\displaystyle \arccos({\frac {-24+5{\sqrt {5}}}{41}})\approx 108.220\,490\,680\,83^{\circ }} \).
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.


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