(E) Fractal Geometry establishes the repetition of geometric patterns at different scales so common in nature around us. Stones look like rocks and rocks like mountains. Small branches look like bigger ones and bigger ones look like trees. Fractal Geometry, from the Latin fractus, the origin of fracture in French or fractured in English, describes the discontinuity and irregularity of Mother Nature geometry: continents, lands, mountains, stones… The fundamental assumption of Fractal Geometry is that there is no qualitative change when the scale of an object changes.  Fractals or objects that kept the same shape or properties over different scales have “scale invariance”. Or said in a reverse way, invariance in a Fractal relates the whole to its parts.

The shape of a Fractal can be produced by repeating patterns that can be deterministic or random. The first step in the construction of a Fractal is the initiator which is a simple line, triangle or solid ball. The second step is the generator which is the geometric pattern that will model the Fractal such as a zigzag or crinkly curve. The process of building a Fractal is a rule of recursion which applies a generator to an initiator.

There are multiple types of Fractals. Self-similar Fractals scale the same way in all directions. Self-affine Fractals scale more in one direction than in another direction. And, Multifractals scale in many different ways at different points.

A great property of Fractals is that its mathematics can be leveraged to study any type of patterns in space or time that remains the same even as the scale of the observed changes.

Another beauty of Fractal Geometry is that it enables us to redefine more precisely our perception of dimension. We are used to considering, a point as a zero dimension, a line as one dimension, a plane as two, a sphere as three and with Einstein’s Relativity our world as four dimensions (space plus time). Or the fundamental assumption of Fractal Geometry as not better expressed than by Mandelbrot himself is that: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” An object can have multiple dimensions. If you look at a star far away, it looks like a point. If you look at it closer, it looks like a circle (like the moon), but if you get very close to it, it looks like a sphere. So a star can have three dimensions!

And of course, dimensions can be fractals. The Pacific coasts of North and South America looks more like a line while the coasts of Western Africa or Brazil are more curly. Therefore, the Pacific coasts of North and South America have a Fractal dimension closer to one while the coasts of Western Africa and Brazil have much higher dimensions closer to two.

Fractal Geometry can be applied to appreciate the beauty of Mother Nature and to the work of man in both the arts (music, poetry, paintings, and architecture) and the sciences (biology, meteorology, seismology, finance…).

Musicians and poets have extensively used Fractals in their arts to wake up, nurture and expand our emotions: Bach in his Fifth Brandenburg Concerto, Beethoven in his Moonlight piano sonata, Rachmaninoff in his Third piano concerto, Baudelaire in the Flowers of Evil, Apollinaire in Alcools and Prevert in Paroles, to name a few examples.

And, mathematicians have created many fascinating Fractals such the Cantor Dust which is constructed by recursively eliminating the middle third of intervals, or the Koch Flacke which is constructed by recursively transforming segments into tent-shaped curves, or the Mandelbrot Set which is constructed by recursively calculated complex numbers z = z₀ ² + c with zero as the starting point (Surrounding pictures are representations of the patterns of the Mandelbrot Set).

Some of those Fractals, in particular, the Mandelbrot Set are interesting applications of Chaos Theory. Changes to the input of z in the Mandelbrot Set can lead to very different outputs as illustrated by the changes in shapes and colors as you zoom in any portion of the set.

Reference
Mandelbrot Benoit (1982), The Fractal Geometry of Nature, W.H. Freeman.

Note 1: An excellent tutorial about Fractal Geometry can be found at classes.yale.edu/fractals

Note 2: The picture above is Hilary Hahn one of the best classical music violinists. She plays beautifully the concertos from the Fractal music of Bach. And on November 26, 2008, in San Francisco, she played marvelously the Tchaikovsky violin concerto with the San Francisco Symphony Orchestra.

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