Drawing with Circles, Epicycles, and the Magic of Frequency Decomposition
Every closed curve can be drawn by spinning circles. The Discrete Fourier Transform decomposes any shape into a sum of rotating vectors—epicycles—each with its own frequency, amplitude, and phase. Add enough circles and you can trace anything: stars, hearts, portraits, or your own freehand drawings. These demos explore the beautiful intersection of mathematics and art, from classical waveform synthesis to shape morphing via coefficient interpolation.
Draw your own paths and watch the Fourier transform reconstruct them with spinning circles. The magic of the DFT in action.
Draw any path with your mouse or finger, then watch a cascade of spinning circles trace it back. Adjust the number of epicycles to see approximation quality improve.
Paint directly in the frequency domain by drawing amplitude bars, and watch the synthesized time-domain waveform update in real time above.
Build classic waveforms from sine harmonics and explore the mathematics of Fourier series convergence.
Watch spinning circles add their radii to construct square, sawtooth, and triangle waves. See each harmonic contribute to the composite signal.
Drag 12 harmonic sliders to sculpt custom waveforms. See individual sinusoids, the composite signal, and a live frequency spectrum simultaneously.
The ~9% overshoot that never goes away. Watch Fourier series converge at discontinuities, zoom into the ringing, and animate term-by-term growth.
Predefined shapes, spirographs, and artistic patterns all generated by systems of rotating circles.
Classic spirograph patterns generated by nested rotating circles. Adjust the three radii and watch mesmerizing geometric curves emerge with color-gradient trails.
Stars, hearts, trefoil knots, butterflies, and more—all traced by epicycles computed via DFT. Control the number of circles to see approximation in action.
Morph between shapes by interpolating their Fourier coefficients. Watch a square smoothly become a star, or a heart transform into a triangle.
Multiple harmonic systems creating interference patterns and collaborative visual compositions.
Multiple points orbit at different harmonic frequencies, connected by proximity lines. Watch constructive and destructive interference create evolving webs of light.
Two to eight independent epicycle systems drawing simultaneously, connected by luminous threads. A visual orchestra where each instrument is a chain of spinning circles.