174 // Many Knees, Many Seats

king's square, without passing through any square twice ­ in the smallest number of moves?

...........................................

Answer on page 288

Many Knees, Many Seats A polyhedron is a solid with finitely many flat (that is, planar) faces. Faces meet along lines called edges; edges meet at points called vertices. The climax of Euclid's Elements is a proof that there are precisely five regular polyhedrons, meaning that every face is a regular polygon (equal sides, equal angles), all faces are identical, and each vertex is surrounded by exactly the same arrangement of faces. The five regular polyhedrons (also called regular solids) are:

. the tetrahedron, with 4 triangular faces, 4 vertices and 6 edges . the cube or hexahedron, with 6 square faces, 8 vertices and

12 edges . the octahedron, with 8 triangular faces, 6 vertices and 12 edges . the dodecahedron, with 12 pentagonal faces, 20 vertices and

30 edges . the icosahedron, with 20 triangular faces, 12 vertices and

30 edges.

The five regular solids. Many Knees, Many Seats // 175

The names start with the Greek word for the number of faces, and `hedron' means `face'. Originally it meant `seat', which isn't quite the same thing. While we're discussing linguistics, the `-gon' in `polygon' originally meant `knee' and later acquired the technical meaning of `angle'. So a polygon has many knees, and a polyhedron has many seats.

The regular solids arise in nature ­ in particular, they all occur in tiny organisms known as radiolarians. The first three also occur in crystals; the dodecahedron and icosahedron don't, although irregular dodecahedra are sometimes found.

Radiolarians shaped like the regular solids.

It's quite easy to make models of polyhedrons out of card, by cutting out a connected set of faces ­ called the net of the solid ­ folding along edges, and gluing or taping appropriate pairs of edges together. It helps to add flaps to one edge of each such pair, as shown. 176 // Many Knees, Many Seats

Nets of the regular solids.

Here's a bit of arcane lore: if the edges are of unit length, then the volumes of these solids (in cubic units) are:

p

2 . Tetrahedron: 12 *0:117 851 . Cube: 1 p . Octahedron: 32 *0:471 405

p . Dodecahedron: 25 f4 *7:663 12

p . Icosahedron: 65 f2 *2:181 69

Here f is the golden number (page 96), which turns up whenever you have pentagons around ­ just as p turns up whenever you have spheres or circles. And * means `approxi- mately equals'.

Analogues of the regular polyhedrons can be defined in spaces of 4 or more dimensions, and are called polytopes. There are six regular polytopes in 4 dimensions, but only three regular

........................................... polytopes in 5 dimensions or more.

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