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Projections of the hypercube on the line and the plane. (English. Russian original) Zbl 0838.51012

Summary: We investigate the images of the \(n\)-dimensional unit cube \(E^n\) under 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 1s 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 \(n \leq 4\) and of the number of layer orders for \(n \leq 7\).

MSC:

51M05 Euclidean geometries (general) and generalizations
90C05 Linear programming