MAM2000 (Essays/SciAm)

# Dimensional Analogies

Dimensional analogies are valuable tools in constructing and understanding four-dimensional phenomena. The hypercube, for example, is derived from the cube just as the cube is derived from the square. To get the cube from the square lift the square in a direction perpendicular to its plane, up to a height equal to its side. The new Cube has eight vertexes, twice as many as the initial square, and 12 edges, four from the initial square, four from the final square that is lifted away from the initial square and four that arise when vertexes in the initial square are connected to their Counterparts in the final square. The cube also has six square faces, one coincident with the initial square, one coincident with the final square and one erected between each of the four pairs of edges that make up the initial and final squares.

If one pretends for the moment that an additional dimension is available, the same operation can be repeated with the cube: "lift" the cube away from ordinary space in the direction of the extra dimension, out to a distance equal to the side of the cube. The result is a hypercube. But in what direction does the extra dimension lie? I cannot explain that. Even a photograph of me pointing into the fourth dimension would be utterly useless. My arm would simply appear to be missing.

How a plane generates a cube and a cube generates a hypercube.

Nevertheless, the number of vertexes, edges, faces and hyperfaces (ordinary cubes) that make up the hypercube can readily be counted. The number of vertexes is just the number of vertexes in the initial cube plus the number in the final cube, or 16. Each of the eight vertexes in the initial Cube is joined by an edge to one of the eight vertexes in the final cube, and there are also 12 edges in each of the two cubes. Hence there are 9 + 12 + 12, or 32, edges in the hypercube. One can also show that the hypercube has 24 ordinary faces and eight hyperfaces. [an error occurred while processing this directive]