The Mathematics of String Art Nets

Chaz Lebouthillier; Mateja Šajna

Home/Archive/arXiv

arXiv · 2008.10693

The Mathematics of String Art Nets

Chaz Lebouthillier·Mateja Šajna

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Chaz Lebouthillier, Mateja Šajna. 2020-08-24. The Mathematics of String Art Nets. https://arxiv.org/abs/2008.10693

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗