On Multidimensional Pythagorean Numbers

TL;DR

This paper explores extending the concept of Pythagorean numbers from two and three dimensions to higher-dimensional spaces, proposing a unified framework for multidimensional number formations called hypersolids.

Contribution

It introduces a novel, unified definition of multidimensional number formations, generalizing classical polygonal and Pythagorean numbers to higher dimensions.

Findings

Defined hypersolids as multidimensional number formations

Extended polygonal number concepts to higher dimensions

Provided elementary methods for multidimensional representations

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.