Generalization of Multi-dimensional Polygonal Numbers

Abstract

In this study, the construction of polygonal numbers in higher dimensional spaces was studied, based on the creation of triangular, tetragonal, pentagonal, hexagonal and more generally polygonal numbers in two- three- and four dimensional spaces. Polygonal numbers in multidimensional space are geometrically associated by taking their projections into three-dimensional space. Furthermore, the general term of the number sequence was calculated by constructing polygonal geometric numbers in k-dimensional space, with k being a natural number. This generalization method was expressed in a theorem and proved by using the induction method. In addition, a program developed with JavaScript language was created using the method obtained by using the generalization rule to draw each term of each polygonal number in different dimensions. Projections of polygonal numbers in 4-dimensional and higher-dimensional spaces into 3-dimensional space were drawn through program. As a result, polygonal number sequences in multidimensional spaces were generalized by an original method using the Pascal triangle, and each term in each dimensional space was expressed algorithmically and examined in terms of the number of points added in dimension increase. Through this method, showing the number of points that are different (added) in each dimensional space in the figures with a table, an appropriate relationship with the generalization method was obtained.