Dual polyhedron facts for kids

A dual polyhedron is a special partner shape in geometry. Imagine you have a 3D shape, like a cube. Its dual is another shape, like an octahedron. What makes them partners? The points (called vertices) of one shape match up with the flat surfaces (called faces) of the other. Also, the lines (called edges) connecting vertices in one shape match the edges connecting faces in the other. If you find the dual of a dual shape, you get back to the original shape!

This diagram shows how the cube and octahedron are dual. Each point (vertex) of the octahedron matches a flat side (face) of the cube. And each point of the cube matches a flat side of the octahedron.

Platonic Solids and Their Duals

Platonic solids are five very special 3D shapes. They are perfectly symmetrical. For these five shapes, only the tetrahedron is its own dual. This means if you find the dual of a tetrahedron, you get another tetrahedron!

Other Platonic solids have different shapes as their duals:

• The cube and the octahedron are duals of each other.
• The dodecahedron and the icosahedron are duals of each other.

This table shows the Platonic solids and their duals, along with how many vertices, edges, and faces each has:

Images for kids

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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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  The square tiling, {4,4}, is self-dual, as shown by these red and blue tilings

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  The Infinite-order apeirogonal tiling, {∞,∞} in red, and its dual position in blue

See also

In Spanish: Poliedro conjugado para niños