the film
Geometric ideas and concepts

The Cube

from Curious and Interesting Geometry by David Wells

The cube is the best known of the Platonic or regular solids. It has 6 faces, 8 vertices and 12 edges; and 13 axes of symmetry, 3 through the centers of opposite faces, 4 through opposite vertices and 6 through the mid-points of pairs of opposite edges. It is also a zonohedron.

Identical cubes fill space most naturally when each cube meets each of its neighbors across a whole face. However, they can fill space in an infinite number of ways. Not only will layers of cubes slide against each other, but cubes can be arranged in each layer in an infinite number of ways. No other space-filling solid has this flexibility.

Take a cube and delete the edges through a pair of opposite vertices. The mid-points of the remaining edges are the vertices of a plane regular hexagon. If some cubes are stacked to fill space in the natural way, the same plane cut which creates this regular hexagon in one cube will cut the stack in the semiregular tessellation of regular hexagons and equilateral triangles.

There are four ways of bisecting the cube by a cut forming a regular hexagon. The edges of all the hexagons are the twenty-four edges of a cuboctahedron.

The dual of the cube, formed by joining the center of each face to the centers of the adjacent faces, is a regular octahedron.

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