Many tough biological materials are composed of layers of aligned fibers, in which each layer rotates by a fixed angle relative to the adjacent layer. This angle is known as the pitch angle, and the resulting architecture is called the Bouligand structure. Crab shells, fish scales, and beetle cuticles are formed in this manner, and fiber-reinforced plastics share the same motif. Whereas most studies on such laminates focus on how they fracture, this study aims to answer a different question. If the fibers are rigid rods whose surfaces carry periodic corrugations that must nest into one another, which pitch angles enable such stacking patterns to repeat layer-by-layer?

As a minimal model, this study used Playstix, a construction toy of rods whose corrugations repeat at fixed periods. Three parameters describe a stack: the spacing between neighboring rods in a layer, their axial shift, and the pitch angle between layers. For the stack to repeat, two conditions must hold: every contact point must settle into a concavity, and each layer must reproduce the spacing and axial shift.

Consequently, the pitch angle is restricted to a discrete family of values fixed by two integers that count the periods along the rods. The two simulations confirmed this rule. First, the predicted spacing and shift bring the two layers closer together at each pitch angle. Second, rods settling under gravity hold their arrangement at large pitch angles but lose it as the angle decreases.

Stacking and toughness favor opposite angles. A large pitch angle makes it easy to build and maintain the stack. A small angle is better for toughness because a crack must then twist gradually through the thickness. The two need not conflict because the geometric conditions apply to rigid rods, whereas real fibrils bend and twist. However, this difference raises the question of how organisms assemble and maintain their small-pitch architectures. More broadly, the results provided a simple rule that links the surface pattern of a rod to the angles at which it will stack. Fibers or fiber bundles with corrugations designed based on this principle could lock into a prescribed pitch angle simply by stacking, offering a geometric route to helicoidal materials whose layer angles are encoded directly into the fibers themselves.

(Written by Tomohiro Mimura on behalf of all authors.)

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