Function Repository Resource:

Pyritohedron

Source Notebook

Create a pyritohedron

Contributed by: Jan Mangaldan

ResourceFunction["Pyritohedron"][h]

returns a pyritohedron with displacement parameter h, represented as a Polyhedron object.

Details

h is usually taken to be between 0 and 1.

A pyritohedron is a dodecahedron with pyritohedral symmetry (point group T_h in Schönflies notation).

ResourceFunction["Pyritohedron"][] is equivalent to ResourceFunction["Pyritohedron"][1/2]. This corresponds to an idealized version of a crystal of the mineral pyrite.

Examples

Basic Examples (1)

A pyritohedron:

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Applications (2)

Surface area and volume of a pyritohedron:

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Moment of inertia matrix for a pyritohedron:

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Properties and Relations (3)

Pyritohedron[] is equivalent to Pyritohedron[1/2]:

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Pyritohedron[1/GoldenRatio] is equivalent to a regular dodecahedron:

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${Graphics3D[ResourceFunction["Pyritohedron"][1/GoldenRatio]], Graphics3D[ Quiet@TransformedRegion[Dodecahedron[], AffineTransform[ EulerMatrix[{\[Pi]/2, -ArcCos[2/Sqrt[5] Sin[(2 \[Pi])/5]], 0}]/ GoldenRatio]]]}$

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A pyritohedron is self-dual:

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$With[{h = 1/2}, Graphics3D[{ResourceFunction["Pyritohedron"][h], TransformedRegion[ResourceFunction["Pyritohedron"][h], RotationTransform[\[Pi]/2, {0, 0, 1}]]}]]$