Question

A few quick examples:

• Breeding bunnies $\rightarrow$ the Fibonacci sequence
• Projectile/planetary motion $\rightarrow$ conic sections
• Natural springs and sinks $\rightarrow$ Gauss' law
• Galilean relativity + constant speed of light $\rightarrow$ non-Euclidean geometry
• Relativistic gravity $\rightarrow$ intrinsic curvature
• Invariances of Maxwell equations $\rightarrow$ Conformal transformations
• Observables in quantum mechanics $\rightarrow$ Lie groups (a la Wigner)

EDIT:

• Symmetries of the electron $\rightarrow$ Quaternions