An interactive series

Imaginary Circles

Start with the angle where an arc equals its radius, then follow that one rotating point all the way to quantum mechanics — through waves, Fourier series, e, i, pi, and Feynman's spinning arrows.

01 The Radian Circle A radian is the angle where the arc equals the radius. Drag the point through the sixteen principal angles and watch it measured both ways. 02 Circle → Wave → Fourier → Schrödinger One rotating point generates every sine wave, builds arbitrary signals as a Fourier series, and carries a quantum state's phase through time. 03 e, i, and the Long Detour into Quantum Mechanics Compound interest and an unsolved cubic gave us e and i — four centuries later they turned out to be the native language of quantum theory. 04 The Ubiquity of Pi Pi begins as circumference over diameter, then surfaces in coin flips, factorials, needle drops, and the floor of quantum uncertainty. 05 Feynman's Rotating Clocks Feynman explained quantum electrodynamics with spinning arrows: sum a clock hand for every path, and its length gives the probability.