Medial pentagonal hexecontahedron

Template:Short description Template:Uniform polyhedra db In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

Proportions

Denote the golden ratio by Template:Mvar, and let $\xi \approx -0.40903778801442$ be the smallest (most negative) real zero of the polynomial $P=8x^4-12x^3+5x+1.$ Then each face has three equal angles of $\arccos (\xi )\approx 114.14440447043^{\circ },$ one of $\arccos ({\phi }^2\xi +\phi )\approx 56.82766328094^{\circ }$ and one of $\arccos ({\phi }^{-2}\xi -{\phi }^{-1})\approx 140.73912330776^{\circ }.$ Each face has one medium length edge, two short and two long ones. If the medium length is 2, then the short edges have length $$1+\sqrt{\frac{1-\xi }{{\phi }^3-\xi }}\approx 1.55076142720,$$ and the long edges have length $$1+\sqrt{\frac{1-\xi }{-{\phi }^{-3}-\xi }}\approx 3.85414587008.$$ The dihedral angle equals $\arccos \left(\frac{\xi }{\xi +1}\right)\approx 133.80098423353^{\circ }.$ The other real zero of the polynomial Template:Mvar plays a similar role for the medial inverted pentagonal hexecontahedron.

References

• Template:Citation

External links

• Template:Mathworld
• Uniform polyhedra and duals

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