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Ask how long a coastline is and the answer depends on the ruler you use: a shorter ruler follows more bays and rocks and gives a longer answer. Benoit Mandelbrot turned that annoyance into a branch of geometry. This post explains who he was, what a fractal dimension measures, how to calculate one for a famous curve, the mistakes students make, and what his way of working can teach you.
Who was Benoit Mandelbrot?
According to his MacTutor biography at the University of St Andrews, Mandelbrot was born in Warsaw on 20 November 1924 and died in Cambridge, Massachusetts, on 14 October 2010. His family moved to France in 1936, and his uncle Szolem Mandelbrojt, a professor at the Collège de France, took charge of his education.
The Second World War broke up his schooling, and MacTutor describes him as largely self-taught, with a strongly geometric way of seeing problems. He studied at the École Polytechnique from 1944, then moved to the United States permanently in 1958 and spent decades as a researcher at IBM, which gave him the freedom to choose his own problems. He later became Sterling Professor of Mathematical Sciences at Yale. MacTutor lists honours including the Wolf Prize for physics in 1993 and the Japan Prize in 2003.
Why does a coastline have no single length?
In 1967 Mandelbrot published How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension in Science. Its abstract says geographical curves are so involved in their detail that their lengths are often infinite or, rather, undefinable. Many are statistically self-similar: each portion looks like a reduced-scale image of the whole. For such curves, a quantity D that behaves like a dimension, though it is fractional and greater than 1, describes how complicated they are.
The MacTutor history of fractal geometry explains the background. Mathematicians such as Georg Cantor and Helge von Koch had already built strange self-similar objects, which many treated as curiosities. Mandelbrot linked them to the real world and defined fractals in 1975.
How do you calculate a fractal dimension? A worked example
Start with a simple rule for self-similar shapes. If a shape is made of N copies of itself, each scaled down by a factor r, then
N = (1/r)^D, so D = log N / log(1/r)
Check it on shapes you know:
A line segment cut into 2 halves: N = 2, r = 1/2, so D = log 2 / log 2 = 1.
A square cut into 4 smaller squares, each with half the side: N = 4, r = 1/2, so D = log 4 / log 2 = 2.
Now the Koch curve, described by von Koch in 1904. Take a straight segment, remove its middle third and replace it with two sides of an equilateral triangle. You now have 4 pieces, each 1/3 the length of the original. Repeat on every piece, forever.
So N = 4 and r = 1/3:
D = log 4 / log 3 = 0.6021 / 0.4771 ≈ 1.26
Now the length. Each step multiplies the length by 4/3. After 10 steps a segment of length 1 has length (4/3)^10 ≈ 17.76. Every further step makes it longer still, so the finished curve has no finite length, even though it fits inside a small box. That is Mandelbrot's coastline in clean mathematical form: the answer to "how long?" depends on how finely you look, and D is the better question to ask.
What mistakes do students make with fractals?
Using the wrong r. In the Koch curve each piece is 1/3 of the original, so log(1/r) = log 3, not log 4.
Confusing number of pieces with length. N counts copies; the length factor N × r = 4/3 is a different quantity.
Thinking any pretty pattern is a fractal. The key test is self-similarity across scales, exact or statistical.
Expecting an exact D for real coastlines. Mandelbrot's coastlines are statistically self-similar, so their D is estimated from measurements, not derived from a rule.
If you want to see how small changes can grow into complex patterns over time, read our companion post on Edward Lorenz and the butterfly effect.
What can a student learn from how Mandelbrot worked?
Mandelbrot looked at objects other mathematicians had called pathological and asked whether nature was full of them. He also drew pictures: the fractal history notes that he, like von Koch, preferred visual representations. When a formula feels abstract, sketch it, and when a "strange case" appears in your textbook, ask where it might show up in the real world.
Logarithms do most of the work here. If you need to revise them, the Fermi Concept Library has a lesson for every concept in mathematics.
Try the same calculation for the Cantor set, where each segment becomes 2 pieces of 1/3 the length, and see whether D comes out between 0 and 1.
