A New Take on Classic 'Pen Problems'

TL;DR

This paper generalizes the classic farm pen optimization problem to higher dimensions and various geometric shapes, providing new solutions and insights into these extended scenarios.

Contribution

It introduces novel generalizations of the farm pen problem to n-dimensional rectangles, regular polygons, and platonic solids, expanding the problem's scope.

Findings

Solutions for n-dimensional rectangular pens

Analysis of pens as regular polygons

Study of pens as platonic solids

Abstract

In this article we generalize the classic "farm pen" optimization problem from a first course in calculus in a handful of different ways. We describe the solution to an $n$-dimensional rectangular variant, and then study the situation when the pens are either regular polygons or platonic solids.