A History of Fractal Geometry

 Any mathematical concept now well-known to school children has gone through decades, if not centuries of refinement. A typical student will, at various points in her mathematical career -- however long or brief that may be -- encounter the concepts of dimension, complex numbers, and "geometry". If the field of mathematics does not particularly interest her, this student might see these concepts as distinct and unrelated and, in particular, she might make the mistake of thinking that the Euclidean geometry taught to her in school encompasses the whole of the field of geometry. However, if she were to pursue mathematics at the university level, she might discover an exciting and relatively new field of study that links the aforementioned ideas in addition to many others: fractal geometry.

in 1883 Georg Cantor, who attended lectures by Weierstrass during his time as a student at the University of Berlin  and who is to set theory what Mandelbrot is to fractal geometry, introduced a new function, ψ , for which ψ' = 0 except on the set of points, {z}. This set, {z}, is what became known as the Cantor set.

The function ψ is singular, monotone, non-constant and ψ' = 0 almost everywhere. It also has the property that

ψ(1) - ψ(0) = 1 however ψ' (x) dx = 0 

The Cantor set has a Lebesgue measure of zero; however, it is also countably infinite.What is more, it has the property of being self-similar, meaning that if one magnifies a section of the set, one obtains the whole set again. Looking at Figure 4, one can easily see that each horizontal line is one third the size of the horizontal line directly above it. In fact, self-similarity is a feature of fractals, and the Cantor set is an early example of a fractal, though self-similarity was not defined until 1905 (by Cesàro, who was analysing the paper by Helge von Koch discussed below) and fractals were not defined until Mandelbrot in 1975, thus Cantor would not have thought of it in those terms.

In a paper published in 1904, Swedish mathematician Helge von Koch constructed using geometrical means the now-famous von Koch curve and hence the Koch snowflake, which is three von Koch curves joined together. In the introduction to his paper he stated the following about Weierstrass's 1872 essay :

... it seems to me that his [Weierstrass's] example is not satisfactory from the geometrical point of view since the function is defined by an analytic expression that hides the geometrical nature of the corresponding curve and so from this point of view one does not see why the curve has no tangent. Rather it seems that the appearance is actually in contradiction with the factual reality established by Weierstrass in a purely analytic way.

Von Koch's curve, like the Cantor set, has the property of self-similarity. It, too, is a fractal, though, like Cantor, von Koch was not thinking in such terms. He merely aimed to provide an alternative way of proving that functions that were non-differentiable (i.e. functions that "have no tangents" in geometric parlance) could exist -- a way that involved using "elementary geometry" (reference 's title translates to On a Continuous Curve without Tangent Constructible from Elementary Geometry). In doing so, von Koch expressed a link between these non-differentiable "monsters" of analysis and geometry.

Von Koch himself was a fairly unremarkable mathematician. Many of his other results were derived from those of Henri Poincaré, from whom he knew it was possible to obtain "pathological" results -- i.e. these so-called "monsters" -- but never really explored them, outside of the aforementioned essay. Poincaré, it should be noted, studied non-linear dynamics in the later 19th century, which eventually led to chaos theory, a field closely related to fractal geometry, though beyond the scope of this paper. It is therefore fitting that a mathematician whose work followed that of Poincaré so closely would turn out to be one of the forefathers of a field that is closely related to the area of study for which Poincaré himself helped lay the foundations.

An absolutely key concept in the study of fractals, aside from the aforementioned self-similarity and non-differentiability, is that of Hausdorff dimension, a concept introduced by Felix Hausdorff in March of 1918. Hausdorff's results from the same paper were important to the field of topology, as well; however that his definition of dimension extended the previous definition to allow for sets to have a dimension that is an arbitrary, non-zero value (unlike topological dimension) ended up being integral to the definition of a fractal, as Mandelbrot defined fractals "a set having Hausdorff dimension strictly greater than its topological dimension.