As the years went by, the mathematical community refined the definition of fractal to include and exclude many aspects of Lvy's original theory and expanded it beyond the plane to include topological spaces in general, and even noncommutative semigroups. For this reason, Lvy's original fractal curve is now sometimes referred to as the Lvy fractal.
In 1988, Tom Van Flandern and George Fenstad proposed a definition of ten-dimensional "fractals" that was somewhat different than those mentioned previously. Flandern and Fenstad proposed that a fractal was a recursive function of a function, established a few years earlier by Robert Brooks: a box on a square box on a square box on a square box, etc. The square was finitely iterated in terms of a simple recursive base function. Such a recursive base in the sense of recursive dimensional nesting functions allowed them to define the base functions as fractals, and therefore, to talk about ten-dimensional spaces that were fractals in the traditional sense. (Brooks's Recursive function theory was not popular because his theory was based on the concept of Cantor sets.)
Another developer in the field, Pierre Voronoi, was jointly awarded a 1932 academy prize with M. Ferdinand de S./J. Rupert (who had performed primary research on functional groups and applications of stereochemistry) for proving that fractal forms occur in nature.[notes 2][notes 3][notes 4][notes 5][notes 6] On the other hand, the Italian mathematician Giuseppe Peano had already published his 1936 paper, "The numerical determination of the dimension of a curve," in which he defined such a curve as a set of points forming a curve lying on a plane, along with a function of a metric that gave the degree of closeness of two points on the curve to a point in the plane. Aaron Joaquin Cobham had precisely the same idea in 1934 when he proved an iterated function with two variable parameters that was one of the first examples of a fractalfor which, in 1937, he described as a Mandelbrot set, which was recognized as a generic pattern by Mandelbrot. d2c66b5586