A Geometrical Way to Sum Powers by Means of Tetrahedrons and Eulerian Numbers

TL;DR

This paper introduces a geometric approach to summing powers using tetrahedrons and Eulerian numbers, providing a visual and combinatorial method to compute sums of powers within a cube.

Contribution

It presents a novel geometric proof linking tetrahedrons tessellating a cube to Eulerian numbers for summing powers.

Findings

Number of tetrahedrons in a cube relates to Eulerian numbers.

Sum of powers can be computed via tetrahedral tessellations.

Geometric proof simplifies calculations of power sums.

Abstract

We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the sums of the tetrahedrons, whose calculation is trivial.