Dual polyhedron

This diagram shows how the cube and octahedron are dual. Each vertex of the octahedron matches with a face of the cube, and each vertex of the cube matches with a face of the octahedron

In geometry, every polyhedron is related to a dual polyhedron. The vertices (points) of one polyhedron match with the faces (flat surfaces) of the other. The edges connecting vertices in one polyhedron match with the edges connecting faces of the other. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

Of the five platonic polyhedra, only the tetrahedron is dual to itself. The cube and octahedron are dual, and the dodecahedron and icosahedron are dual.

Duals of Platonic polyhedra

Polyhedron	Vertices	Edges	Faces	Dual
Tetrahedron	4	6	4	Tetrahedron
Cube	8	12	6	Octahedron
Octahedron	6	12	8	Cube
Dodecahedron	20	30	12	Icosahedron
Icosahedron	12	30	20	Dodecahedron

Dual Polyhedron Media

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  The dual of a cube is an octahedron. Vertices of one correspond to faces of the other, and edges correspond to each other.

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  The dual of a Platonic solid can be constructed by connecting the face centers. In general this creates only a topological dual.Images from Kepler's Harmonices Mundi (1619)

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  Canonical dual compound of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common midsphere.

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  A Tetrahedron. A regular polyhedron.

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  Squarepyramid

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  Pentagonalpyramid

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  Hexagonal_pyramid transparent image

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  Elongated triangular pyramid

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  Elongatedsquarepyramid

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  Elongated pentagonal pyramid

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, p. 1, ISBN 0-521-54325-8, MR 0730208.