Projections of the Hypercube on the Line and the Plane

Abstract

We investigate the images of the n-dimensional unit cube E ⁿunder projections (i.e., linear mappings) of all its vertices on the line or on the plane. Images are considered to be distinct only if the projections of the vertices of E ⁿ are distinctly ordered on the axes of coordinates. Such an image of E ⁿ on the line is called a line order and on the plane, a plane order. An order induced by invertible projections is called complete. If all vertices of E ⁿ with the same number of Is have the same projection on one of the axes, then a plane order is called a layer order. It is shown that the number of line orders and the number of complete layer orders are not less than 3^{n (n-o(n))/2} and not more than 3^{n (n-o(n))}. The exact values are presented of the number of line orders for n ≤ 4 and of the number of layer orders for n ≤ 7.

This research was partially supported by the Russian Foundation for Fundamental Research (Grant 93–01–01484).