The ponytail hairstyle is a synonym of comfort and simplicity, and what was once considered a traditional schoolgirl style, nowadays it has become popular again thanks to clever styling. But trying to work out what shape someone’s ponytail will be has puzzled scientists and artists since Leonardo da Vinci.
In 2012, scientists from the University of Cambridge and University of Warwick developed the ponytail shape equation (PSE) to unravel some of the mysteries of the ponytail. Their model takes into account the gravity ($g$), the elasticity of the hair, the presence of random curliness of hair, and an outward swelling pressure that arises from collisions between the component hairs (which explains how a bundle of hair is swelled).
This equation can be used to find $R$, the radius of the ponytail, in terms of $s$, the arc length along the ponytail. The length at which gravity bends the hair is $l$, $L$ is the length of the ponytail, $P$ is the pressure due to the hairband, $A$ is the bending modulus, and $\rho$ is the hair’s density.
The Rapunzel number, $\text{Ra}$, of a ponytail is the ratio $L/l$. This dimensionless number determines the effect of gravity on hair. When $\text{Ra}<1$, the hair doesn’t bend much, leading to a thin, straight ponytail. When $\text{Ra}>1$, the hair bends strongly under gravity leading to a wide, bushy ponytail.
The relevance of this equation is that it could help in understanding the structure of materials made up of fiber and depicting hair realistically in animation and video games. But most importantly, if you want to look good at a party or a maths conference, simply calculate your Rapunzel number and pop on a hairband that exerts the correct pressure.
References
- RE Goldstein, PB Warren, RC Ball, Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles, Physical Review Letters, 108, 078101, (2012).
- RE Goldstein, (2016, September 11), Leonardo, Rapunzel and the Mathematics of Hair.
More from Chalkdust
Christmas lights, trees and maths
Why do Christmas lights get tangled? And what's the perfect way to decorate a Christmas tree? Find the answers here.
How to make: a Möbius Surprise
Surprise yourself by making this thing
Hilbert’s hotel: the board game
Print it out and play against your friends!
Prize crossnumber, Issue 08
Win £100 of Maths Gear goodies by solving our famously fiendish crossnumber
Which number are you?
Are you perfect or imaginary? Are you a square?
Dear Dirichlet, Issue 08
Village fetes, Brexit and flowers are among the topics of discussion in this issue's Dear Dirichlet advice column