How Long a String to Light Your Holiday Tree?

Introduction

In the article preceding this one, Ellison [2013] calculates how long a string of lights you need to decorate a holiday tree if the windings are equally spaced.

We recount briefly his model and results and compare it to a simplistic model that has attracted widespread attention.

We then investigate the inverse question that my family addresses every year:

Given a length of lights, how far apart to space windings so as to cover the tree? We answer that question using Ellison's model but also investigate a different model for lighting the tree, in which the windings are not equally spaced.

Ellison uses a handy custom Python program to answer his question. We use a short Mathematica program to confirm his calculations. We arrive at a simple and useful approximation that lets us give a table of numerical values to help the tree decorator.