On Multidimensional Pythagorean Numbers

Abstract

To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for integers to spaces with more than three dimensions. Thus, taking up a definition of polygonal numbers given by Diophantus and by Nicomachus, and generalizing the Pythagorean concept of gnomon, one is led through quite elementary means to a single, unified definition of multidimensional number formations henceforth called hypersolids.