Mathematics → Physics → Cryptography. One shape, three worlds.
A point at angle θ on the unit circle is (cos θ, sin θ). Euler links rotation to complex exponentials. Summing rotations at 1×, 3×, 5× frequency traces a square wave — circles building any periodic signal.
Uniform circular motion projected onto a line is Simple Harmonic Motion: x = A cos(ωt). Velocity is tangent to the circle. Two phasors at ω₁ and ω₂ add; when close, their sum pulses — acoustic beats are vector addition in disguise.
Integers mod n form a discrete circle. Addition wraps around. Diffie-Hellman uses repeated multiplication: ga mod p hops around the circle. Easy forward, hard backward (discrete log). φ(n) counts numbers coprime to n — the size of the multiplicative group.