Every term the site uses, in plain English, each linked to the chapter that explains it properly. Type in the box to filter; it searches the definitions too, so "Julia" finds the Fatou set and "1982" finds Douady and Hubbard. Below the list: the dimensions, constants and dates worth knowing by heart.
Common mix-up: self-similar does not mean fractal, and fractal does not mean exactly self-similar. A straight line is perfectly self-similar and has no fractal detail at all. The Mandelbrot set is the most famous fractal there is, yet no part of it is an exact copy of the whole: the mini-brots are always slightly distorted and wrapped in different filaments.
A fractal dimension says how fast detail piles up as you zoom in. For a shape built from N copies of itself, each shrunk by a factor s, it is log N / log s. Chapter 2 explains why.
| Shape | Dimension | Where it comes from |
|---|---|---|
| Cantor set | ≈ 0.6309 | 2 copies at ⅓ size: log 2 / log 3 |
| Koch curve | ≈ 1.2619 | 4 copies at ⅓ size: log 4 / log 3 |
| Heighway dragon boundary | ≈ 1.5236 | The edge of the paper-folding dragon; the dragon itself fills area |
| Sierpinski triangle | ≈ 1.585 | 3 copies at ½ size: log 3 / log 2 |
| Sierpinski carpet | ≈ 1.8928 | 8 copies at ⅓ size: log 8 / log 3 |
| Mandelbrot set boundary | 2 | Hausdorff dimension, proved by Shishikura (1998) |
| Menger sponge | ≈ 2.7268 | 20 copies at ⅓ size: log 20 / log 3 |
| Name | Value | What it means |
|---|---|---|
| Feigenbaum δ | 4.669201609… | Ratio of successive period-doubling intervals (Chapter 11) |
| Feigenbaum α | 2.502907875… | The scaling of the doubling branches' widths (Chapter 11) |
| Escape radius | 2 | Once |z| > 2, an orbit of z² + c is certain to escape (Chapter 8) |
| Smooth count | ν = n + 1 − log₂(log|zₙ|) | Continuous escape time; natural log inside, base-2 log outside (Chapter 15) |
| Distance estimate | |z| log|z| / |z′| | Distance to the Mandelbrot set, good to a factor of two (Chapter 17) |
| Mandelbrot set area | ≈ 1.5065918849 | A numerical estimate; the exact value is unknown |
The short version; History & the people tells the stories.
| Year | What happened |
|---|---|
| 1883 | Cantor publishes the set now named after him |
| 1890 · 1891 | Peano's and then Hilbert's space-filling curves |
| 1904 | Koch's curve |
| 1915 · 1916 | Sierpinski's triangle, then his carpet |
| 1917–1919 | Fatou and Julia study the iteration of rational maps |
| 1926 | Menger's sponge |
| 1961 · 1967 | Richardson's coastline measurements; Mandelbrot's "How Long Is the Coast of Britain?" in Science |
| 1963 | Lorenz's butterfly attractor |
| 1968 | Lindenmayer introduces L-systems |
| 1975 | Mandelbrot coins "fractal" |
| 1976 | May's logistic-map paper in Nature |
| 1978–1980 | Brooks and Matelski's first picture of the set; Mandelbrot's 1980 images |
| 1981 | Hutchinson's theory of IFS; Witten and Sander's diffusion-limited aggregation |
| 1982 | The Fractal Geometry of Nature; Douady and Hubbard prove the Mandelbrot set connected |
| 1986 · 1988 | Peitgen and Richter's The Beauty of Fractals; Barnsley's Fractals Everywhere |
| 1989 · 1992 | Markus–Lyapunov fractals; the Burning Ship (Michelitsch and Rössler) |
| 1998 | Shishikura: the boundary of the Mandelbrot set has dimension 2 |
| 2003 | Draves and Reckase, "The Fractal Flame Algorithm" |
| 2009 · 2010 | The Mandelbulb (White and Nylander); the Mandelbox (Lowe) |
| 2013 · 2020 | K. I. Martin's perturbation method for deep zooms; Ottosson's OKLab color space |