# Did fractals already exist in the 17th century?

The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s, but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975.

I am always fascinated by fractal dimensions. Infinity in a finite plane. An infinite line (plane?) fit into a small area (volume?). It is always said that Mandelbrot invented them in 1975. Now I read that already in the 17th century they were given thought. But where and who gave it thought?

• @AlexandreEremenko The non-integer value maked it so strange. Can you say iits the part of the plane covered by the line plus one? Aug 8 at 5:43
• A Platonist would insist that fractals have always existed. Aug 8 at 20:20
• If "existed" means "were pondered by people" (at least unwittingly) then no. What Wikipedia means, apparently, is "the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them". First examples date to late 19th century. Aug 9 at 7:45

The term "fractal" was coined to B. Mandelbrot (1970th). But these objects were known to mathematicians for long time. The first definition of dimension which can take non-integer values is due to Hausdorff,

Hausdorff, F. Dimension und äußeres Maß. (German) JFM 46.0292.01 Math. Ann. 79, 157-179 (1918).

Some first mathematical examples of what is called fractals nowadays are Weierstrass continuous nowhere differentiable function, which was discovered in 1872 but not published (Weierstrass rarely published his results, preferring to communicate them in lectures and talks).

Another early example was a Cantor-like set occuring in the work of Schottky (some time in 1880s, before Cantor). Julia sets were introduced independently by Fatou and Julia approximately in 1918. Of course nobody knew that these examples have fractional dimension until Hausdorff introduced his notion of dimension.

By the way, exact value of dimension of the graph of Weierstrass function is still unknown.

I do not think that any sets of fractional dimension occured in mathematics before this example of Weierstrass. And to my knowledge no mathematician discussed fractional dimensions before Hausdorff. Of course some fractal-like patterns that exist in the nature drew attention of physicists, see Lichtenberg figure and philosophers, for example there is a famous description of an (imaginary) fractal in an old Buddust text.

• How does one know about Weierstraße's fractal if he didnt publish it? Has somebody else published it? Aug 8 at 8:49
• Bolzano had an earlier continuous nowhere-differentiable function (1820), also not published until 1920. Aug 8 at 13:30
• 1. The word on Weierstrass function spread quickly, new examples were constructedand (Koch's snowflakes, and others), and before long it was in textbooks. The first Calculus textbook with this example was probably E. Landau. 2. No. Their mathematical definition is independent of chaos, though these things frequently occur together. Aug 8 at 13:32
• And to my knowledge no mathematician discussed fractional dimensions before Hausdorff. --- Possibly Borel's classification of measure zero sets qualifies (JFM 42.0089.12 and 43.0485.01 and 44.0090.01 44.0556.02 and 47.0181.02 and many others). The earliest published definitive speculation about defining such a notion I know of is due to William Henry Young (continued) Aug 8 at 16:55
• on p. 462 of his 1905 paper Linear content of a plane set of points: It is clear that the mere fact that the plane content of a certain closed set $G$ is zero is not of itself sufficient to justify our putting all such sets in one class without further sub-divisions. In fact, denoting by $F(d)$ the content of the small regions corresponding to the norm $d,$ to say that the plane content of $G$ is zero is merely to assert that $F(d)$ has the limit zero when $d$ is indefinitely decreased. (continued) Aug 8 at 17:05

Maybe this is what Wikipedia meant by "1600s".

B. B. Mandelbrot, in his book The Fractal Geometry of Nature, has some historical remarks in the last chapter, including:

The idea of fractional differentiaton had occurred to Leibniz as soon as he had developed his version of calculus and invented the notations $$\frac{d^k F}{dx^k}$$ and $$(\frac{d}{dx})^kF$$ ... Leibniz's letter to de l'Hospital dated September 30, 1693 ... “John Bernoulli seems to have told you of my having mentioned to him a marvelous analogy which makes it possible to say in a way that successive differentials are in geometric progression. One can ask what would be a differential having as its exponent a fraction. You see that the result can be expressed by an infinite series. Although this seems removed from Geometry, which does not yet know of such fractional exponents, it appears that one day these paradoxes will yield useful consequences, since there is hardly a paradox without utility. Thoughts that mattered little in themselves may give occasion to more beautiful ones.”

• And consequences it had. Early in last century, as can be read in the other answer. Or were fractals developed in chaos theory only? Aug 8 at 17:03
• What does fractional differentiation has to do with fractals? Aug 9 at 13:25