Fractal Geometry

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Lore & Background

Starting with 17th-century recursion notions, fractals moved through rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass. On July 18, 1872, Karl Weierstrass presented the first definition of a function with a graph considered a fractal at the Royal Prussian Academy of Sciences. In 1883, Georg Cantor published examples known as Cantor sets. Felix Klein and Henri Poincaré introduced "self-inverse" fractals later that century. Helge von Koch gave a geometric definition in 1904 including hand-drawn images now called the Koch snowflake. Wacław Sierpiński constructed his famous triangle in 1915, followed by his carpet one year later. Benoît Mandelbrot coined the term "fractal" in 1975 based on the Latin frāctus, meaning "broken" or "fractured," extending fractional dimensions to natural patterns.

In Their Own Story

Analytically, many fractals are nowhere differentiable. An infinite fractal curve winds through space differently from an ordinary line; although topologically 1-dimensional, its fractal dimension indicates it locally fills space more efficiently. Consider the Koch curve: rep-tiled into four sub-copies each scaled by a factor of 1/3, satisfying 3^D = 4 for its unique real number dimension D. Doubling edge lengths of a filled polygon multiplies area by two squared, but doubling a fractal's one-dimensional lengths scales spatial content by a non-integer power greater than its conventional dimension. Attempting to measure an infinitely "wiggly" curve like the Koch snowflake with straight segments fails; the jagged pattern reappears at arbitrarily small scales, requiring infinite tape to cover the entire curve.

Reader's Guide

There is disagreement among mathematicians on a formal definition. Mandelbrot summarized it as "beautiful, damn hard, increasingly useful." In 1982, he defined a fractal as "a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." He later simplified this to "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," eventually proposing using "fractal" without pedantic definition. The consensus holds that theoretical fractals are infinitely self-similar iterated constructs. Fractals describe processes in time and appear in nature, technology, art, and architecture. They are particularly relevant in chaos theory, showing up as attractors or boundaries between basins of attraction in geometric depictions of chaotic processes.

Did You Know?

• The term 'fractal' was coined by Benoît Mandelbrot in 1975 from the Latin frāctus, meaning 'broken' or 'fractured'.
• Karl Weierstrass presented the first definition of a fractal-like function (continuous but nowhere differentiable) on July 18, 1872.
• If you try to measure the perimeter of the Koch snowflake with straight segments, you will need infinite tape because the pattern repeats at arbitrarily small scales.
• Gottfried Leibniz pondered recursive self-similarity in the 17th century but mistakenly believed only a straight line was self-similar in that sense.
• Mandelbrot later expanded his definition to describe fractals as shapes where parts are (at least approximately) reduced-size copies of the whole.