Measure a turn
A circle gives the first unit of rotation: the radian, where the arc and radius finally speak the same language.
An interactive series
From the first circle to the quantum world.
A story of mathematics told through one rotating point.
Begin with Euclid's clean geometry: a radius, an arc, and an angle. Follow that motion through Newton's heavens, Fourier's waves, Euler's exponential, and Feynman's clocks, where probability itself is drawn with rotating arrows.
Geometry as a sketchbook. The same circle keeps reappearing: first as an angle, then as an orbit, a wave, and a quantum phase.
Predict the sky
Rotating geometry becomes a way to model repeating motion, from epicycles to Newton's sweep of planets through space.
Build any wave
Sines, cosines, and complex exponentials turn circles into a toolkit for sound, light, heat, and quantum state changes.
Sum the clocks
Feynman's path amplitudes bring the rotating arrow back: add the little clocks, then square the result.
Eight doors into the same idea.
Each chapter is a small interactive step in the larger arc: circle, shadow, wave, spiral, phase, and probability.
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.
06
The Logarithmic Spiral
Give e^(iθ) a real exponent and the turning point starts to grow. One constant sets how tightly it winds — the same equiangular curve behind shells, galaxies, and sunflower heads.
07
Epicycles
Chain enough rotating circles, each spinning at its own steady rate, and their tip traces any shape at all — a square wave, a star, a signature. Fourier, drawn.
08
Eratosthenes and the First Cosmic Ruler
A noon shadow, two cities, and one circular arc turned Earth's curve into a measurement — the first step toward scaling the heavens.
Eight chapters to begin with — the series is still growing.