What determines the curl diameter of naturally curly hair

Settled in 2014 by J. T. Miller, A. Lazarus, B. Audoly, P. M. Reis (MIT / UPMC Paris).

Curly hair's characteristic curl diameter is an everyday observation, but predicting it from first principles requires modeling a thin elastic rod with built-in (intrinsic) curvature hanging and deforming under its own weight, a genuinely nonlinear mechanics problem. Building on 2002 work by McMillen and Goriely on related curved-rod ('tendril perversion') phenomena, Miller, Lazarus, Audoly and Reis combined desktop experiments with rods of controlled curvature, numerical simulation, and analytical theory to construct a full phase diagram of curl shapes (flat curls, localized helices, fully developed helices) as functions of a rod's intrinsic curvature, weight, and elastic stiffness, published in Physical Review Letters in 2014. This gives a quantitative, testable answer for how curl geometry follows from a hair fiber's own material properties, though it takes the strand's intrinsic curvature as a mechanical input rather than explaining its biological (keratin/follicle) origin.

The bounce and diameter of a single curl of hair, something hairdressers judge by eye, follows exactly from treating the strand as a simple elastic rod bent by its own curvature and weight.

Explanations that circulated

• Simple 'fixed helical spring' analogies that treat a curly hair strand as retaining one unchanging helix shape fail to capture the richer geometry (flat/planar curls, localized helices, and fully developed global helices) that the elastic-rod model shows actually occurs depending on the strand's length, weight, and intrinsic curvature

References

• Shapes of a Suspended Curly Hair, J. T. Miller, A. Lazarus, B. Audoly, P. M. Reis, Physical Review Letters, 2014