Editing Ponytail (section)

== Scientific studies ==
[[File:Street Parade 2013 033.jpg|thumb|A woman's ponytail from the side]]
The first equation of state for hair was developed by C. F. van Wyk in 1946.<ref>Wyk, C. M. V. (1946). "[https://doi.org/10.1080/19447024608659279 20—Note On The Compressibility Of Wool]." ''Journal of the Textile Institute Transactions'', 37(12).</ref>

Scientists in the UK have formulated a [[mathematical model]] that predicts the shape of a ponytail given the length and random curvature (or curliness) of a sample of individual hairs. The Ponytail Shape Equation provides an understanding of how a ponytail is swelled by the outward pressure which arises from interactions between the component hairs.<ref name=Scienceponytail>"[https://www.bbc.co.uk/news/science-environment-17012795 Science behind ponytail revealed]."(2012, 13 February)</ref>

The researchers developed a general [[Continuum (theory)|continuum theory]] for a bundle of hairs, treating each hair as an elastic filament with random intrinsic curvature. From this they created a [[differential equation]] for the shape of the bundle relating the elasticity, gravity, and orientational disorder and extracted a simple [[equation of state]] to relate the swelling pressure to the measured random curvatures of individual hairs.<ref name=Goldstein2012>Goldstein, R. E., Warren, P. B., & Ball, R. C. (2012). "[https://doi.org/10.1103/physrevlett.108.078101 Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles]." ''Physical Review Letters'', 108(7).</ref><ref>{{cite journal|title=Synopsis: Ponytail physics|url=http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.108.078101|journal=Physical Review Letters|date=13 February 2012|volume=5|doi=10.1103/PhysRevLett.108.078101|pmid=22401258|access-date=14 February 2012|last1=Goldstein|first1=R. E.|last2=Warren|first2=P. B.|last3=Ball|first3=R. C.|issue=7|page=078101|arxiv=1204.0371|s2cid=31964168}}</ref> The equation itself is a fourth order non linear differential equation.<ref name=Goldstein2012 />

The Rapunzel number is a ratio used in this equation to calculate the effects of gravity on hair relative to its length.<ref name=Goldstein2012 />

{{math|''Ra'' ≡ ''{{sfrac|L|l}}''}}

This number determines whether a ponytail looks like a fan or whether it arcs over and becomes nearly vertical at the bottom. A short ponytail of springy hair with a low Rapunzel number, fans outward. A long ponytail with a high Rapunzel number, hangs down, as the pull of gravity overwhelms the springiness.

It is now also known why jogger's ponytails swing side to side.<ref name="Keller2012">Keller, J. B. (2010). "[https://doi.org/10.1137/090760477 Ponytail Motion]." ''SIAM Journal on Applied Mathematics'', 70(7), 2667–2672.</ref> An up and down motion is too unstable: a ponytail cannot sway forward and backward because the jogger's head is in the way. Any slight jostling causes the up and down movement to become a side to side sway.

The research on the shape of the ponytail won the authors the [[Ig Nobel]] for Physics in 2012.<ref>{{cite web | url=http://www.improbable.com/ig/winners/#ig2012 | title=The 2012 Ig Nobel Prize Winners | work=Improbable Research | date=August 2006 | access-date=14 August 2016}}</ref>

The Rapunzel number is important for the [[computer graphics]] and [[animation]] industry, as it helps animators resolve challenges relating to the realistic digital representation of hair and hair movement.<ref name=Scienceponytail />