Английская Википедия:Great pentagrammic hexecontahedron

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Шаблон:Short description Шаблон:Uniform polyhedra db In geometry, the great pentagrammic hexecontahedron (or great dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its 60 faces are irregular pentagrams.

Файл:Great pentagrammic hexecontahedron.stl

3D model of a great pentagrammic hexecontahedron

Proportions

Denote the golden ratio by <math>\phi</math>. Let <math>\xi\approx 0.946\,730\,033\,56</math> be the largest positive zero of the polynomial <math>P = 8x^3-8x^2+\phi^{-2}</math>. Then each pentagrammic face has four equal angles of <math>\arccos(\xi)\approx 18.785\,633\,958\,24^{\circ}</math> and one angle of <math>\arccos(-\phi^{-1}+\phi^{-2}\xi)\approx 104.857\,464\,167\,03^{\circ}</math>. Each face has three long and two short edges. The ratio <math>l</math> between the lengths of the long and the short edges is given by

<math>l = \frac{2-4\xi^2}{1-2\xi}\approx 1.774\,215\,864\,94</math>.

The dihedral angle equals <math>\arccos(\xi/(\xi+1))\approx 60.901\,133\,713\,21^{\circ}</math>. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial <math>P</math> play a similar role in the description of the great pentagonal hexecontahedron and the great inverted pentagonal hexecontahedron.

References

• Шаблон:Citation

External links

• Шаблон:Mathworld