Shapes whose parts resemble the whole
The word fractal was coined in 1975 by the Polish-born French-American mathematician Benoit Mandelbrot from the Latin fractus, "broken". A fern frond is made of smaller fronds shaped like the whole, which are themselves made of still smaller ones; this property, where zooming in reveals structures that resemble the whole, is called self-similarity. Mathematical fractals are built by repeating a simple rule forever. The Koch curve, made by repeatedly dividing each segment into thirds and raising a triangular bump on the middle third, encloses a finite area yet has an infinitely long boundary. Mandelbrot's 1967 paper "How long is the coast of Britain?" popularised the idea through the coastline paradox: the shorter your measuring stick, the longer the coastline becomes.
Fractal dimension
A line is one-dimensional and a plane two-dimensional, but a fractal can sit in between. If a shape can be split exactly into N copies scaled by 1/s, its dimension is D = log N / log s. A square splits into nine copies at one-third scale, giving log 9 / log 3 = 2; the Koch curve splits into four copies at one-third scale, giving about 1.26. The higher the dimension, the more densely the shape fills space.
| Shape | Calculation | Dimension |
|---|---|---|
| Koch curve | log 4 / log 3 | ≈ 1.262 |
| Sierpiński triangle | log 3 / log 2 | ≈ 1.585 |
| Menger sponge | log 20 / log 3 | ≈ 2.727 |
| West coast of Britain (Richardson) | measured | ≈ 1.25 |
The Mandelbrot set and fractals in nature and technology
The Mandelbrot set is the set of complex numbers c for which repeating z → z² + c does not run off to infinity. Computers first drew it around 1980, and because zooming into its boundary reveals endlessly new detail it became the emblem of fractals. The strange attractors of chaos theory also have fractal structure. In nature, Romanesco broccoli, tree branches, river networks, lightning, snowflakes and the airways and blood vessels of the lungs are approximate fractals; branching packs an enormous surface area into a small volume, which is how the lungs fit a surface of roughly 70 m², about half a tennis court, inside the chest. Technology uses fractals in compact mobile-phone antennas, procedurally generated mountains and clouds in films and games, and image compression.
FAQ
- Do fractals really repeat forever?
- Mathematical fractals do, but natural ones stop at limits such as the size of cells or atoms, so natural objects are described as fractal over a certain range of scales.
- What does a fractal dimension of 1.26 mean?
- The shape fills more space than a line (dimension 1) but less than a plane (dimension 2). The more wiggly and complex a shape is, the higher its dimension.
Sources reviewed: 2026-10-26
References
- Mandelbrot, B. B. (1967). How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156(3775), 636–638.
- Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman.
- Falconer, K. (2014). Fractal Geometry: Mathematical Foundations and Applications (3rd ed.). Wiley.
- Gleick, J. (1987). Chaos: Making a New Science. Viking.