Beautiful Mathematical Curves
A single equation—x(t), y(t)—can create infinite beauty. From the spirographs of childhood to the superformula that encodes every natural shape, parametric curves reveal the hidden geometry of mathematics. Watch roses bloom from sine waves, butterflies emerge from exponentials, and harmonographs paint patterns that mesmerized Victorian audiences. Every curve here is alive, interactive, and endlessly customizable.
Circles rolling on circles create hypnotic curves called trochoids—the mathematics behind the beloved Spirograph toy.
A circle rolling inside another. Adjust radii and pen position to create hypotrochoids—from simple ellipses to intricate star patterns.
A circle rolling OUTSIDE another. The Wankel engine's rotor traces an epitrochoid. Explore the surprisingly different family of outer-rolling curves.
The intricate interlocking wave patterns found on banknotes and security documents. Multiple spirographs woven together.
Two pendulums coupled to a pen create haunting, slowly-decaying Lissajous-like patterns. A Victorian parlor favorite, now digital.
Three independent frequencies create a dancing 3D curve. Rotate to see knots, pretzels, and impossibly smooth sculptures in space.
Polar curves that bloom into flowers, stars, and mandalas—simple trigonometry creating surprising complexity.
r = cos(kθ) creates roses with k petals (or 2k if k is even). Sweep k from 1 to 10 and watch flowers bloom from mathematics.
Connect points on a rose curve at angular steps of d degrees. Small changes in d create wildly different star-burst patterns.
Gielis's superformula generates circles, squares, triangles, stars, flowers, and natural shapes from a single equation with 6 parameters.
Temple Fay's butterfly: e^(sinθ) - 2cos(4θ) + sin&sup5;(θ/12). A single equation that draws a butterfly.
|x|^n + |y|^n = 1 morphs from diamond (n=1) to circle (n=2) to square (n=∞). The squircle lives at n≈4.
Curves that have fascinated mathematicians for centuries—each with a beautiful story and surprising properties.
Heart-shaped curves from rolling circles and reflected light. You see cardioids in coffee cups and nephroids in swimming pools.
The path traced by a point on a rolling wheel. Inverted, it's the fastest slide between two points—the brachistochrone curve.
Unwinding a string from a curve traces its involute. The center of curvature traces its evolute. Every curve has this dual pair.
A hanging chain forms a catenary, not a parabola. Flip it upside-down and you get the strongest arch—the secret of Gothic cathedrals.
Four dogs at the corners of a square, each chasing the next. Their spiral paths create a beautiful logarithmic vortex.
From the nautilus shell to the galaxy, spirals are everywhere. And some curves have properties so strange they changed mathematics.
Archimedean, logarithmic, Fermat, hyperbolic, and golden spirals. Each has unique growth properties and natural occurrences.
Drag control points to sculpt curves. Watch de Casteljau's algorithm construct the curve step by step. The foundation of all computer graphics.
Hypocycloids with 4 and 3 cusps: star-like curves from circles rolling inside circles. Surprisingly, a line segment rolls inside an astroid!
Any closed curve can be decomposed into rotating circles. Draw a shape and watch an army of epicycles reconstruct it.
Smoothly morph between all the curves in this gallery. Watch a circle become a butterfly, then a rose, then a spirograph.