Hyperspheres in Hypercubes

By Joel Noche

Let a (circular) pizza have a radius of 1 unit (that is, a diameter of 2 units). Four of these pizzas would fit into a (square) pizza box where each side is 4 units long, with enough space in the center of the box to put in another pizza. (See the picture below.)

It can be seen that the extra pizza in the center has a smaller radius than the other pizzas. The distance from the center

of the box to the center of one of the four pizzas is units. The radius of each of the four

pizzas is 1 unit. Thus, the radius of the extra pizza is units.

Now let a (spherical) basketball have a radius of 1 unit. Eight of these basketballs would fit into a (cubic) box where each side is 4 units long, with enough space in the center of the box to put in another basketball.

The distance from the center of the box to the center of one of the eight basketballs is

units. The radius of each of the eight basketballs is 1 unit. Thus, the radius of the

extra basketball is units.

Note that when we added a dimension to the problem, the radius of the extra object became bigger. For two dimensions, we talk about circles in squares. For three dimensions, we talk about spheres in cubes. For larger dimensions, we will talk about hyperspheres in hypercubes.

For a four-dimensional hypercube, the extra hypersphere in the center of the hypercube has a radius of unit. The extra hypersphere is now as large as the other hyperspheres in the hypercube.

For a nine-dimensional hypercube, the extra hypersphere in the center of the hypercube has a radius of units (that is, a diameter of 4 units). The extra hypersphere now fits snuggly inside the hypercube (where each side is 4 units long).

For a ten-dimensional hypercube, the extra hypersphere in the center of the hypercube has a radius of units. It is now larger than the hypercube and yet is still in between other hyperspheres which are inside the hypercube.

It can be shown that the volume of this extra hypersphere becomes bigger than the volume of the hypercube as the number of dimensions increases. See Hamming (1980, pp. 168–171) for the (messy) details.

Reference

Hamming, R. W. (1980). Coding and Information Theory. Englewood Cliffs, New Jersey: Prentice Hall.

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