Fractals Aisha Torres 10/19/14

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. Self-similarity is found by zooming in on digital images to see previously invisible, new structures. If this is done on fractals, however, no new detail appears because the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over.

The mathematics behind fractals began to develop in the 17th century when the mathematician and philosopher Gottfried Leibniz studied recursive self- similarity. Leibniz used the term “fractional exponents”, but lamented that geometry did not yet know of them. For two centuries few mathematicians attempted to tackle fractals until Karl Weierstrass in 1872. He presented the first definition of a function with a graph that would be considered a fractal today, which is that it has the property of being everywhere continuous but nowhere differentiable. Soon after Weierstrass, in 1883, Georg Cantor published examples of subsets of the real line known as Cantor sets, which are now recognized as fractals. Later in the same century Felix Klein and Henri Poincar´eintroduced the category of “self-inverse” fractals.

Even more development in the study of fractals came in the 20th century. In 1904 Helge von Koch gave a more geometric definition of fractals and created the famous Koch curve. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely.

Koch curve:

1 In 1915, Waclaw Sierpi´nskicreated the Sierpi´nskitriangle and carpet. In- terestingly the Sierpi´nskitriangle is given by Pascal?s triangle if you color all the odd even numbers.

Sierpi´nskitriangle:

Pascal’s triangle:

Sierpi´nskicarpet:

In 1918, two French mathematicians, Pierre Fatou and Gaston Julia ar- rived simultaneously at results describing fractal behavior associated with mapping complex numbers and iterative functions and further ideas about points that attract or repel other points (attractors and repellors). Also in 1918, Felix Hausdorff expanded the definition of “dimension” so that sets could have noninteger dimensions. In 1938, Paul L´evydescribed the L´evy C curve.

2 In 1975, BenoˆıtMandelbrot coined the word “fractal” and illustrated his definition with computer-constructed visualization. Consequently, the frac- tal studies today are almost all computer-based. The Mandelbrot set is the 2 set obtained from the quadratic recurrence equation zn+1 = zn + C. With z0 = C , where points C in the complex plane for which the orbit of zn does not tend to infinity are in the set. Setting z0 equal to any point in the set that is not a periodic point gives the same result.

Some other famous fractals include:

Barnsley’s Fern:

3 Box Fractal:

Gosper Island:

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