From Fish to Composites: The Latent Structure of Fiber-Angle Optimization

The W-shaped muscle segmentation pattern visible in a salmon fillet is the solution to a constrained optimization problem: maximize total bending moment subject to a maximum fiber strain constraint. We prove that the solution — \(\cos\theta(r) = \varepsilon_{\max}/(r\kappa)\), where \(r\) is depth and \(\kappa\) is curvature — is unique and strictly optimal, with 19 formally verified theorems. This one-parameter family of fiber-angle distributions is identical to the optimal Variable Angle Tow (VAT) design for composite beams in pure bending, yet the biological and engineering literatures have not been connected. We introduce the Latent dimension of a structural optimization as the minimal number of parameters generating the full optimal geometry. For pure bending, the Latent dimension is 1, reducing the design space from \(\mathbb{R}^N\) to \(\mathbb{R}^1\) and computation from \(O(N^2 \times 10^3)\) to \(O(N)\). We provide the classification of loading types by Latent dimension and identify the boundary where closed-form solutions cease to exist.

Keywords: myomere, variable angle tow composites, fiber-angle optimization, uniform strain, Latent structure, formally verified