Recursivity and geometry of the hypercube Original Research Article

It is a difficult theoretical and computational problem to describe explicitly the list of hyper-planes spanned by the vertices of the n-cube. In this paper we describe a procedure to generate hyper-planes of the n-cube from hyper-planes of the (n − 1)-cube. Our main theorem says that starting with the hyper-planes of the 1-cube and iterating this procedure we obtain the complete list of hyper-planes of the n-cube up till n = 6. For n = 7, with this procedure, not all the hyper-planes are obtained. An explicit description of the hyper-planes of the 7-cube is given, followed by a brief analysis of some further consequences of the results obtained.