Fractals

Romanesco Broccoli.

Relevant examples from Escher's work:

One of Escher's recurring themes is his desire to represent infinity in an artwork. His tessellations are potentially infinite, but disappointing in that only a finite portion can be shown at once. After 1955 Escher switched from experimenting with congruent shapes to experimenting with similar shapes to better represent the infinite.

Smaller and Smaller was an early attempt, and one of his first to show infinitely many motifs on the page. We can see infinity in the center of the print, but at the outside edge there would be room to add increasingly larger images of lizards. Unsatisfied with this, Escher created Square Limit, in which the infinite motifs collect at the edge of the print rather than in the center. Both Smaller and Smaller and Square Limit are excellent examples of fractals.

Explorations

Begin learning about self-similarity and fractals with:

Self-Similarity

This square pattern is self-similar
Smaller and Smaller, with outside lizards removed, showing a copy of itself at smaller scale.

A figure is self-similar if it contains copies of itself at smaller scales.

All figures with dilation symmetries (as in Similarity Transformations) are self-similar. For example, the pattern of squares contains a copy of itself, turned at 45° and scaled by ${\frac {1}{{\sqrt {2}}}}$.

The outside ring of lizards in Escher's Smaller and Smaller are different from the rest, making a clean frame for the print. Ignoring those, the inner pattern is self similar. It contains a copy of itself at half scale. Because each smaller copy must also contain a smaller copy of itself, any self-similar figure must contain infinitely many repetitions of itself, at smaller and smaller scales.

In a looser sense, any figure which contains some sort of copy of itself is often referred to as self-similar. For example, the head of Romanesco broccoli at the top of the page is self similar, because each whorl looks like a tiny copy of the entire brocolli head. A close inspection reveals that each whorl is itself made up of smaller whorls, each of which looks like the original head.

 Example: Escher's ink drawing Self-Portrait in Spherical Mirror (Ink) shows an image of a reflecting sphere, and in that sphere we see M.C. Escher. He is drawing the picture Self-Portrait in Spherical Mirror (Ink), so we see a smaller image of the drawing inside, and inside of that drawing is another small image. The overall image is self-similar (in a loose sense, because the smaller images are distorted).
 Example: Coastlines are self similar[1], in that they look roughly the same at different scales. The four images on the right show the Pacific coast of North America, of California, of San Diego county, and of Ocean and Mission Beaches in San Diego. As the scales vary from about 3000 miles long to about 10 miles long, the visible jaggedness remains about the same.

Fractals

The term "fractal" is less precise than most mathematical terms, and is used loosely to cover a wide range of related ideas.

One possible definition is that a fractal is an irregular object which displays some level of self-similarity.

Benoît Mandelbrot, who was the first to use the term (in 1975), said that a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole." [2]

 The Mandelbrot set. A magnification of the Mandelbrot set.

One of the most well known fractals is the Mandelbrot set (image on the left). The Mandelbrot set is created by iterating the function $f(z)=z^{{2}}+c$ for various values of $c$. For some values of $c$, the iteration process will produce larger and larger results, and for others the process stays small. The Mandelbrot set corresponds to the values of $c$ for which iteration stays small, and it is shown in black. The other points are colored depending on how quickly the iteration process gets large. Using complex numbers ($c{=}a+bi$), the possible values of $c$ correspond to the entire plane.

The Mandelbrot set is self similar. Upon magnification we can see several smaller copies of this set appear. The complexity and beauty of this set, and the related Julia sets, is best explored through a computer program. Some suggestions are given in the Related Sites section.

Fractals show up in the work of artists as varied as M.C. Escher, Salvador Dali, Jackson Pollock[3], and Max Ernst (via a process called Decalcomania).

Fractals are common in nature as well, showing up in plant life, diffusion, lightning, and other chaotic processes. In computer science, there are image compression algorithms that take advantage of self-similarity for more efficient data storage.

Fractal Dimension

Sierpinski like structures on an African mask. Photo courtesy of B. van Assche

The Sierpinski triangle is a well known fractal created with iteration. The initiator is an equilateral triangle, and the three transformations are dilations (by 1/2) towards each corner. Alternately, begin with an equilateral triangle, find the midpoints of all the sides, and use those to create four smaller triangles. Throw away the central triangle, and iterate the process with each smaller triangle.

...

Above you see iterations 0-3 of the process. First the triangle is divided into four smaller triangles and then the middle triangle is removed. In the next step, the remaining three triangles are each divided into four triangles, discarding the middle ones. The final image shows the sixth iteration.

The real fractal is the result of infinitely many iterations. Practically, one can only draw stages of the fractal as it is being produced, but the sixth iteration is indistinguishable from the fractal Sierpinski triangle itself.

It is remarkable that the Sierpinski triangle contains three copies of itself, each scaled by a factor of 1/2. This is very strange behavior, though it requires some thought to understand why.

Consider a line segment. It can be divided into two copies of itself, each scaled by a factor of 1/2, or three copies of itself, each scaled by a factor of 1/3, or in general, $b$ copies of itself, each scaled by a factor of $1/b$:

Next, divide a rectangle. It can be divided into four copies of itself, each scaled by a factor of 1/2, or nine copies of itself, each scaled by a factor of 1/3, or in general, $b^{2}$ copies of itself, each scaled by a factor of $1/b$.

Finally, look at a rectangular parallelepiped (a block). It can be divided into 8 copies of itself, each scaled by a factor of 1/2, or 27 copies of itself, each scaled by a factor of 1/3, or in general, $b^{3}$ copies of itself, each scaled by a factor of $1/b$.

There is a pattern here. An $N$ dimensional object can be covered by $b^{N}$ copies of itself at $1/b$ scale. Turning this idea around, if an object can be covered by $C$ copies of itself at $1/b$ scale, then $C=b^{N}$, and we can solve this equation using logarithms to arrive at the following definition:

Fractal Dimension
The fractal dimension of a self-similar set is ${\frac {\log(C)}{\log(b)}}$, where the set is covered by $C$ copies of itself, each scaled by a factor of $1/b$.

For the line segment, solid rectangle, and block, the fractal dimensions are 1, 2, and 3, respectively. This is exactly what we would expect - for ordinary things, the fractal dimension is just the usual dimension.

The Sierpinski triangle contains three scale copies of itself, each scaled by 1/2 from the original, so the fractal dimension of the Sierpinski triangle is ${\frac {\log(3)}{\log(2)}}\approx 1.58$. The Sierpinski triangle is not a one dimensional object, nor a two dimensional object, but something in between, a fractional dimension. It's got more weight than a line, but doesn't cover any solid portion of the plane, either.

Fractals typically, but not always, have dimensions which are fractional. The name 'fractal' was chosen to suggest this.

Example: The Koch Snowflake

The middle third of a segment is replaced by an equilateral “bump” consisting of two new segments. Iterating on these new segments (and the remaining two segment thirds of the initial segment) results in the "Koch edge".

The Koch Snowflake

The Koch Snowflake refers to the object you get if you apply the iteration to all three sides of a triangle. In the image on the right, only the fourth iteration is shown, but it clearly shows the fractal nature of the Koch Snowflake.

The Koch edge consists of four scale copies of itself, each scaled by a factor of 1/3. Thus it, and the Koch snowflake, have fractal dimension ${\frac {\log(4)}{\log(3)}}\approx 1.26$.

At each step of the construction of the Koch edge, one segment is removed, and replaced by two - you are adding more line segment than you are removing. This means that the length of the object is steadily growing. The real fractal is the result of iterating the process infinitely often, and for the Koch snowflake this results in a curve with infinite length. Also note that the area bounded by the snowflake remains finite. The Koch snowflake is remarkable: it is an infinitely long curve that surrounds only a finite area.

Notes

1. Benoît Mandelbrot, 1967, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, New Series, Vol. 156, No. 3775. (May 5, 1967), pp. 636-638. doi:10.1126/science.156.3775.636
2. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman and Company.
3. The Case of Pollock's Fractals Focuses on Physics, The New York Times, December 2, 2006