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 |
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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