Fractals: Monsters or Wonders?

by Stephan Lebohec & Clément Vidal

In the late 19th century mathematicians such as du Bois-Reymond, Weierstrass, Cantor, von Koch, or Riemann had found non-differentiable structures. The curves representing such structures are so irregular that they generally have no tangents. One of the central mathematical tool for physics, differential calculus, could not be applied to these curves — at least not in a straightforward manner. To mathematician Charles Hermite (1822–1901), such fractal objects were mathematical monsters. On May 20th 1893, he wrote to his colleague Thomas Stieljtes: “I turn away with fright and horror of this terrible scourge of continuous functions without derivative.”

However, Hermite’s judgement was based purely on the puzzling properties of such functions. He might have considered them not as monsters, but as wonders had he the chance and the tools to visualize them. In the 1970s, Benoît Mandelbrot was able to use computers to produce graphical representations of mathematical fractals. At first, the scientific community failed to recognize the value of Mandelbrot’s visual exploration. To reply to this skepticism, Mandelbrot published The Fractal Geometry of Nature, showing that fractals are ubiquitous in nature, and that such geometries can be used in science and technology. Today, we know that fractal structures are everywhere, from clusters of galaxies, to clouds, lungs, artery networks and to the antenna in our mobile phones. To appreciate the breadth of fractal applications, we recommend the science documentary Fractals — Hunting the Hidden Dimension. In the spirit of Mandelbrot’s emphasis on experimentation and visualization, let’s first have a look at a few mathematical fractals.

A particularly popular fractal curve is one due to Helge von Koch (fig 1).

Fig. 1 Construction of the von Koch curve. At each iteration, the middle third of each interval is replaced by the other two sides of an equilateral triangle. Source: (Falconer 2003, xix)

It is constructed from the trisection of a segment, followed by the replacement of the central third by an equilateral triangle. The procedure is then repeated iteratively and indefinitely with all the segments of the construction to ultimately result in the von Koch curve. Zooming on the von Koch curve keeps on revealing structures identical to the full curve. This property of some fractals is referred to as self-similarity. Self-similarity prevents one from being able to identify the actual scale of a given region (see Fig. 2). :

Fig. 2. von Koch curve. What is striking — and what may make you fractal-sick — is that despite the zooming on the curve, there are always self-similar structures appearing, ad infinitum. Scaling is like nothing! Source: Wikipedia

Alternatively, we could construct a curve like von Koch’s, except that each triangle is made to point toward one side of the curve or the other, depending on a coin flip. Figure 3 shows a comparison of the regular von Koch curve with a curve obtained with a random choice for the orientation of each triangle. Fractals generated using self-similar fractal rules, peppered with some randomness are sometimes referred to as being statistically self-similar. This type of fractals is commonly used in computer generated images for the natural looking character they bring to landscapes. It was made famous by the first fractal-generated planet, in Star Trek II: The Wrath of Khan.

Fig. 3. Top: Original von Koch curve; 
Bottom: Random von Koch curve.
Source: Logan Axon, Random Fractals, http://web02.gonzaga.edu/faculty/axon/talks/gonzaga.pdf

Different fractals display different wealths of structures appearing from one iteration to the next. This is characterized by the fractal dimension, which we will discuss in a later post. There are many examples of different fractals classified by their fractal dimension. The wealth of structures (or fractal dimension) can also vary with position, time or even scale.

The example of Fig. 4 is also a variation of the von Koch curve, where the fractal dimension varies with position. The angles of the triangles vary from 180 degrees on the left to 0 degree on the right. The curve thus goes from a smooth line to a plane-filling curve.

Fig. 4 — A fractal curve whose dimension varies according to position. At the utmost left, we have a straight line, and at utmost right, the curve is so rough that it becomes plane-filling, like a surface. Source: Adapted from (Nottale 1993, 80)

In this case, the property of self-similarity is lost globally. But locally, zooming on a specific region of the curve, the self similarity is recovered: for example, the fine inspection of the region where the angle is 60 degrees will reveal just the familiar von Koch curve.

Fractal dimension can be a function of time as well. For example the surface of our faces has a fractal dimension close to 2 when we are young while as we age, with the appearance of wrinkles, the fractal dimension of our faces tend to increase (See Figure 5).

Figure 5: The more wrinkles the higher the fractal dimension.

A variation of the fractal dimension with scale is more difficult to represent graphically as it implies to visualize an object at different resolutions. This is not only a mathematical game, since objects may look very different depending on the resolution with which we look at them. The optical illusion of Figure 6 exploits this very fact.

Figure 6: When observed closely, this picture seems to represent Albert Einstein. However, at larger distances from which fine details are lost due to the limited resolution of our eyes, Marilyn Monroe appears instead.

It is possible to imagine more complicated structures in which a pattern may appear between different resolution scales and locations. The nature of the object then does not depend on any single location or resolution scale. Instead the nature of the object really is in the relation between its different constituents and the different resolution scales making them appear. The riddle of Figure 7 represents just this.

Fig. 7: Will you be able to guess the great mathematician and the great musician in the image above? (Hofstadter 1979, 331). Tip: there is different information at different scales, and you need to exploit them all.

To be understood from a physics approach, the universe must be considered in a similar way (See Figure 8). Different structures appear at different scales and different locations while subtended by similar physics principles (see this dynamic illustration). Scale Relativity is the proposal of a relativistic approach to account for the interplay between different scales at the most fundamental level in the formulations of the laws of nature.

Fig 8. Planet Earth from space: a smooth world in-between roughness. Looking at the surface of the Earth from space, it looks smooth, like a sphere. However, as you zoom in, you’ll see mountains and thus rough structures (Right: such mountain in the Himalayas). If you zoom out at very large scales, you’ll discover clusters of galaxies that also have fractal features (left: simulation of galaxy filaments). This illustrates the point that the fractal dimension in nature changes as we change our resolution.

References

Falconer, Kenneth. 2003. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons.

Hofstadter, Douglas R. 1979. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books.

Nottale, L. 1993. Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific