The Mathematics of String Art Nets

TL;DR

This paper explores the geometric properties of a specific string art pattern with pegs on diverging axes, revealing symmetry, segment length relations, and area invariance under certain conditions.

Contribution

It introduces a mathematical analysis of string art nets with pegs on diverging axes, identifying symmetry, segment length ratios, and area invariance properties.

Findings

Lines are divided into segments with all but one of equal length.

Quadrilaterals along a diagonal have equal areas despite being incongruent.

Properties hold when the angle varies but only with equidistant pegs.

Abstract

String art is an arrangement of pegs on a board with thread strung between these pegs to form beautiful geometric patterns. In this article, we consider a simple form of string art where pegs are placed on two diverging axes, and segments of string join the first peg on one axis to the last peg on the second axis, the second peg on the first axis to the second-to-last peg on the second axis, and so on. The resulting pattern is a peculiarly shaped net consisting of quadrilaterals and triangles that exhibits unexpected symmetry. Each line of the net is divided by other lines into segments, all but one of the same length, and one of twice the length of the others. Furthermore, quadrilaterals in the net that are arranged along a diagonal from the upper left to the lower right corner are in general incongruent but have equal areas, and the same is true of triangles formed along the upper-right border of the net. Finally, we show that these properties are preserved when the angle between the axes changes, but hold only when consecutive pegs on the axes are equidistant.