Recall that n dimensional cube has 2ⁿ vertices. One dimensional cube is the abscissa, two dimensional cube is the square, and so one.

The elements of the distance matrix D of a cube are distances between its vertices. The longest distance corresponds to the dimensionality of the cube. For example, the distance matrix of the unit square is

Finding the eigenvalues of the distance matrices D of lowest cubes, it was found that they have rather simple pattern of eigenvalues. The eigenvalues of the distance matrices D of cubes are tabulated, as follows

Always, the main eigenvalues are n2^n-1, the negative eigenvalues are 2^n-1. The number of nonzero eigenvalues corresponds to the dimensionality of the cubes.

The sum of distances from the root (zero vertex), otherwise the first moment of the distance distribution, is determined by the equation

The sum of squared distances from the root, otherwise the second moment, is determined by the equation

The proof of the first identity (1)

The proof of the second identity (2) by the full induction, dividing the cube into two parts, one formed by the lower dimensional cube, and the second by new moments, is

finding the square, we get

where Z_n-1 is the sum of the first moments.

For example, 240 = 2x80 + 2x32 + 16 = 6x5x8.

Both identities are well known. Here a new interpretation appears, the first moment is the main eigenvalue of the corresponding matrix, simultaneously.