Recursive Geometry
Shapes built by repeating a rule at shrinking scales
What it is
These shapes come from explicit geometric construction rather than iteration in the complex plane: a line segment replaced by a bent version of itself, a triangle subdivided into smaller triangles, a curve traced by a circle rolling inside another circle. Apply the same rule to every new piece, recursively, and the result is a shape with detail at every scale.
History
Helge von Koch introduced his snowflake curve in 1904 as an example of a continuous curve with no tangent anywhere - a deliberately "pathological" object meant to challenge intuitions about curves, not a piece of art. Wacław Sierpiński described his triangle and carpet around 1915-1916 in the context of set theory and topology. Both constructions predate the word "fractal" by half a century; they were curiosities for pure mathematicians long before Benoit Mandelbrot reframed them as fractals with a shared vocabulary - self-similarity, fractal dimension.
Main Contributors
Helge von Koch, Wacław Sierpiński, Georg Cantor (whose 1883 "dust" set is an ancestor of the idea), and Benoit Mandelbrot, who unified these separate constructions under fractal geometry.
Latest Trends
Procedural generation in games and film leans heavily on recursive/L-system rules to grow trees, terrain, and coastlines on demand. Generative-art tooling increasingly animates the construction process itself - watching a curve grow through recursion depth - rather than only showing the finished shape. And tiling or packing variations turn a single recursive curve into a repeating surface pattern, which is the direction this site's own Koch and Sierpiński pieces take.
Koch Code
A grid tiled edge-to-edge, where every line segment is itself a Koch curve, animated with organically drifting recursion depth.
Koch Snowflake
The classic closed six-pointed Koch curve, hex-packed across the page so neighboring snowflakes nest into each other's gaps.
Sierpinski Fractal
A genuine edge-to-edge triangular tessellation where every tile is its own independently animated Sierpinski subdivision.
Spirograph
The classic rolling-circle curve, fully parametric and animated - outer/inner radius, line count, and detail all live-adjustable.