# n-Balls: A Multi-Dimensional Analysis of Spherical Structures with Volume

**Abstract**

## This project is an applied analysis of objects in multi-dimension space. The goal is to demonstrate the effects on surface area and volume of *n*-spheres (unit spheres of dimension *n*) when dimension shifts.

## Counter-intuitively, this paper formally demonstrates (by way of theoretical computation) that both the surface area and volume of *n*-spheres begin to decrease in higher dimensions. Furthermore, and even more counter to intuition, the volume of a given *n*-sphere begins to decrease after reaching the 5th-dimension while the surface area of a given *n*-sphere begins to decrease after reaching the 7th-dimension.

### Author(s)

## Jesse Hanson

Wyatt "Miller" Miller

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