What Is a Fractal?
By: Bryan Dijkhuizen
Tags: Science, Math, Society, World

SCIENCE & MATH

What Is a Fractal?

A fractal, sometimes also referred to as a fractal, is a geometric figure that is self-similar. It is made up of parts that are more or less similar to the figure itself.

Photo by Danilo Batista on Unsplash

Fractals have an infinite amount of detail, and some fractals feature motifs that repeat themselves on an increasingly smaller scale. Typically, fractals can be generated by repeatedly applying a particular operation. The term fractal was introduced in 1975 by Benoît Mandelbrot and is derived from the Latin fractus (broken).

Mathematical objects with fractal properties were discovered in the late 19th and early 20th century by mathematicians such as Karl Weierstrass, Helge von Koch, Georg Cantor, Henri Poincaré, and Gaston Julia.

Fractal geometry is the branch of mathematics that deals with the properties of fractals. It complements classical geometry, with applications in science, technology, and computer art.

The most famous fractals are the Mandelbrot set and the Julia set.

Fractal dimension

A straight line is one-dimensional in ordinary geometry, a plane is two-dimensional, and a spatial shape is three-dimensional. For fractals, the dimension cannot be stated so easily.

In the iterative process where a line figure approximates a fractal, sometimes a two-dimensional area can be filled, and the one-dimensional shape approximates a two-dimensional one. Mandelbrot used the generalized Hausdorff concept of dimension and found that most fractals have a non-integer dimension, called a fractal. That led to the definition:

A fractal is a set whose Hausdorff dimension is greater than the Lebesgue overcover dimension.

So any set with a non-integer dimension is a fractal. However, the reverse does not apply: there are also fractals with an integer dimension, such as the Brownian movement.

Measure dimensions

The dimensionality of some figures is so obvious that it does not seem necessary to have a method at hand to determine the dimension.

For example, a straight line is ‘clear’ one-dimensional and a flat surface two-dimensional. We could determine that — if there was any doubt — as follows:

Consider a bounded object. We choose a radius R and determine how many spheres with radius R are needed to cover the object completely. We now look at what happens to the number of spheres needed when we make the radius smaller. If we have a segment and halve the radius, then we need twice as many spheres.

For a bounded area, we need four times as many spheres (4 = 22). In general, we define that if we divide the radius by S, the number of spheres required increases by Sd, then d is the dimensionality of the object.

This seems a bit silly for lines and surfaces, but not if the set of points resembles a cloud or a coastline, for example.

In that case, it is possible to define sets in which the number of spheres increases by a factor of S2,324 or S1,324. These kinds of figures for which the dimension is not an integer are called fractals.

Applications

Fractal mathematics was too popular among scientists in the 1980s — 1990s. Fractals were thought to be recognized everywhere and in everything, and these mathematics was applied inappropriately and inappropriately, so much so that in 2004 fractals are somewhat discredited in science.

This is all the more curious because a fractal, like a sphere or a triangle, is a mathematical concept that is neither true nor false but simply created by definition.

Yet, there are various applications of fractals that can no longer be ignored. The description of chaos, for example, is inconceivable without the background of fractals.

The Poincaré image of a chaotic system is a fractal. The characterization of apparently very messy structures, for example, particles with a particularly rough surface or the characterization of the leaf shape of ferns or the branch structure of trees, makes grateful use of fractal mathematics.

With the aid of scattering at small angles of both X-ray and neutron radiation (SAXS or SANS), fractal dimensions of, for example, colloidally suspended small particles can be measured directly.