Decoding the Phenomenon of Light Refraction: Bridging Fermat's Principle and Huygens' Construction through "Linear Sums" and "Sum of Squares"

The phenomenon of light refraction serves as a cornerstone in physics education, yet traditional curricula frequently isolate Snell's Law (empirical kinematics), Fermat's Principle (calculus optimization), and Huygens' Construction (wavefront geometry). This fragmented approach imposes severe extraneous cognitive load on students, obscuring the profound conceptual unity underlying these three principles. To bridge this epistemological gap, this paper introduces a novel pedagogical framework termed the "Algebraic Order" (代数阶数). We propose a structural distinction between 0th-order linear scalar aggregation (the temporal optimization of Fermat's Principle) and 2nd-order quadratic metrics (the spatial geometry of Huygens' sum of squares). Through a constructivist narrative, we first present a rigorous calculus derivation, followed by an intuitive but ultimately flawed geometric attempt—the Y-axis spatial stretch—to highlight the destruction of the isotropic spatial metric. We then introduce Huygens' geometric construction and prove Theorem 1 (sinθ₂ = OB / OC), demonstrating how Euclidean right triangles can bypass complex calculus. Furthermore, we resolve the common "rigid body illusion" of the wavefront by introducing the "compressible matchstick" model, clarifying the microscopic lateral compression of the wavefront during the transitional phase. By framing this dual structure, the paper provides a structured pathway for physics educators to reduce cognitive load, transitioning students from mechanical calculation to deep conceptual understanding, while setting the stage for advanced topics such as Wave-Particle Duality.