The shape of a ponytail and the statistical physics of hair fiber bundles

TL;DR

This paper develops a continuum physics model for hair bundles, specifically ponytails, considering elasticity, gravity, and disorder, and derives an equation of state from laboratory measurements to describe their shape and internal forces.

Contribution

It introduces a novel continuum theory for hair fiber bundles, linking microscopic curvature randomness to macroscopic ponytail shape through a differential equation and an experimentally derived equation of state.

Findings

Derived a differential equation for ponytail shape

Established a simple equation of state from measurements

Linked internal pressure to hair curvature statistics

Abstract

A general continuum theory for the distribution of hairs in a bundle is developed, treating individual fibers as elastic filaments with random intrinsic curvatures. Applying this formalism to the iconic problem of the ponytail, the combined effects of bending elasticity, gravity, and orientational disorder are recast as a differential equation for the envelope of the bundle, in which the compressibility enters through an 'equation of state'. From this, we identify the balance of forces in various regions of the ponytail, extract a remarkably simple equation of state from laboratory measurements of human ponytails, and relate the pressure to the measured random curvatures of individual hairs.