Concept
Self-similar shapes at every scale — revealing nature's jagged geometry beyond smooth curves.
Understand it in one breath
A structure with similar roughness or pattern across changes of scale. The exactly self-similar Koch snowflake has finite area and infinite perimeter. Fractal models can describe coastlines, clouds, or vascular networks over limited scale ranges, but natural objects need not be infinitely or exactly self-similar.
Zoom it yourself — Mandelbrot sandbox
The same patterns reappear wherever you zoom. Click the canvas to zoom ×2 at that point, then copy a share link so a friend sees the same view.
At a glance
Key formula
Key moments
Weierstrass — a continuous but nowhere-differentiable function
Weierstrass presented a function that is unbroken everywhere yet smooth nowhere. Contemporaries regarded such examples as mathematical monsters.
The Koch curve — infinite perimeter, finite area
Helge von Koch introduced the snowflake curve: iteration drives its perimeter to infinity while the enclosed area remains finite.
Mandelbrot coins the word fractal
Working at IBM, Benoit Mandelbrot formed the term from the Latin fractus, meaning broken or fragmented, and unified a family of irregular self-similar shapes.
The Mandelbrot set — an icon of computer visualization
The simple iteration z ↦ z²+c generates a boundary of inexhaustible complexity, becoming a defining image of the computer-graphics era.
Modern applications
Natural scenery in computer graphics, compact antenna design, communication-signal compression, and analysis of tumor boundaries in medical images.
Beyond MathVoyage
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