Overview | The Math | The Lesson

There happens to be only 5 solids built using faces of a given shape, and put together in a regular manner. For example, one solid is the cube, constructed using square faces; another is the pyramid of triangular base, which requires triangular faces. See the teacher page about regular polyhedra for a more detailed explanation.

These solids have many nice properties of symmetry. Other properties are harder to discover and describe. This lesson concentrates on the following three:

1. For each solid, it is true that:

(number of faces) - (number of edges) + (number of vertices) = 2

Curious about this fact? Read more about Euler's number.

2. Some of the Platonic solids can be paired with another Platonic solid that has complementary properties: Each member of a pair has the same number of faces as its partner has vertices. These polyhedra are called duals of each other. The hexahedron is the dual of the octahedron; the dodecahedron is the dual of the icosahedron; and the tetrahedron is its own dual.

3. Among the 5 solids, the octahedron is the only one for which one is able to place a fingertip on a vertex and "redraw" all the edges without lifting the finger and without going over an edge twice. (Or, equivalently, it can be constructed by just bending a long piece of wire.) This is part of the very interesting topic of networks.

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