Inspired by the recent sums of the squares law obtained by Kovacs-Fang-Sadler-Irwin we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.
Authors: Vincenzo Nardozza
Comments: 19 Pages. I know that 9 versions is the maximum allowed and this is the 10th. However, I have found several errors in the notation that make the paper unreadable in same parts.
A method for dealing with the product of step discontinuous and delta functions is proposed. A standard method for applying the above defined product of distributions to polyhedron vertices is analysed and the method is applied to a special case where the well known angle defect formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus. The angle defect formula is the discrete version of the curvature for vertices of polyhedra. Among other things, this paper is basically the formal proof of the above statement.
We’ll prove now that there is a similar relation for the isometric cevians as Steiner's relation for the isogonal cevians.