More on Benoit Mandelbrot: Hunting the Hidden Dimensions

Introduction to the Mandelbrot Set

The Mandelbrot set, named after Benoit Mandelbrot, is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers.

Understanding complex numbers

The Mandelbrot set is a mathematical set, a collection of numbers. These numbers are different than the real numbers that you use in everyday life. They are complex numbers. A complex number consists of a real number plus an imaginary number. The real number is an ordinary number, for example, -2. The imaginary number is a real number times a special number called i, for example, 3i. An example of a complex number would be -2 + 3i.

The number i was invented because no real number can be squared (multiplied by itself) and result in a negative number. This means that you can not take the square root of a negative number and get a real number. By definition, when you take the square root of a number, you find a number that can be squared to get that number. The number i is defined to be the square root of -1. This means that i squared is equal to -1. So when you square an imaginary number you can get a negative number. For example, 3i squared is -9.

Real numbers can be represented on a one dimensional line called the real number line. Negative numbers like -2 are plotted to the left of zero and positive numbers like 2 are plotted to the right of zero. Any real number can be graphed on the real number line.

Since complex numbers have two parts, a real one and an imaginary one, we need a second dimension to graph them. We simply add a vertical dimension to the real number line for the imaginary part. Since our graph is now two-dimensional, it is a plane, the complex number plane. We can graph any complex number on this plane. The colored dots on this graph represent the complex numbers [2 + 1i], [-1.5 + 0.5i], [2 - 2i], [-0.5 - 0.5i], [0 + 1i], and [2 + 0i].

Graphing the Mandelbrot set

The Mandelbrot set is a set of complex numbers, so we graph it on the complex number plane. However, first we have to find many numbers that are part of the set. To do this we need a test that will determine if a given number is inside the set or outside the set. The test is based on the equation Z = Z² +  C.C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing, the point on the complex plane that will be plotted when testing is complete. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.

We're not really interested in the actual value of Z as it changes, we just look at its magnitude. The magnitude of a number is its distance from zero. For example, the number -9 is a distance of 9 from zero, so it has a magnitude of 9. The magnitude of a complex number is harder to measure. To calculate it, we add the square of the number's distance from the x-axis (the horizontal real axis) to the square of the number's distance from the y-axis (the imaginary vertical axis) and take the square root of the result. In this illustration, a is the distance from the y-axis, b is the distance from the x-axis, and d is the magnitude, the distance from zero.

As we iterate our equation, Z changes and the magnitude of Z also changes. The magnitude of Z will do one of two things. It will either stay equal to or below 2 forever, or it will eventually surpass two. Once the magnitude of Z surpasses 2, it will increase forever. In the first case, where the magnitude of Z stays small, the number we are testing is part of the Mandelbrot set. If the magnitude of Z eventually surpasses 2, the number is not part of the Mandelbrot set.

As we test many complex numbers we can graph the ones that are part of the Mandelbrot set on the complex number plane. If we plot thousands of points, an image of the set will appear:

The Mandelbrot set is an incredible object. It's really amazing that the simple iterated equation Z = Z² +  C can produce such beautiful works of mathematical art.

Fractal Art

Mathematical Paintings

On the image above we see artwork "The Great Wave off Kanagawa" by Japanese artist Hokusai, which was published in 1832 as the first in Hokusai's series 36 Views of Mount Fuji. It depicts an enormous wave threatening boats near the Japanese prefecture of Kanagawa; Mount Fuji can be seen in the background. The main reason of publishing this artwork here is highly detailed painted wave. As we know, some artworks, which are close to fractal images by detailed elaboration, were created long before the inventing fractals by Benoît Mandelbrot. Sea waves can be represented by many types of fractals, as you can see below.

Fractal artworks with waves are created by modern artists too. For example, Robert Fathauer depicted a wave fractal with Escher-like fish tiling in his artwork "Fractal Fish - Grouped Groupers" (2001).

Also, a wave with horses figures are depicted on the cover of English rock group Keane "Under The Iron Sea" (2006).

Fractals in Nature

"The Universe is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth." - Galileo (1623)

Trees

Mountains

Coral

Plants

Clouds

Coast Lines

Broccoli 

Lighting

Flowers

Peacock feathers

Fern leaves

Snow flake

Flowers

Universe

Exploring Fractals

Fractal geometry and chaos theory are providing us with a new perspective to view the world. For centuries we've used the line as a basic building block to understand the objects around us. Chaos science uses a different geometry called fractal geometry. Fractal geometry is a new language used to describe, model and analyze complex forms found in nature.
A few things that fractals can model are:

plants, weather, fluid flow, geologic activity, planetary orbits, human body rhythms, animal group behavior, socioeconomic patterns, music, etc.

This is how nature creates a magnificent tree from a very small seed the size of a pea.

Fractal dimension can measure the texture and complexity of everything from coastlines to mountains to storm clouds. We can now use fractals to store photographic quality images in a tiny fraction of the space ordinarily needed.
Fractals win prizes at graphics shows and appear on Tshirts and calendars  Their chaotic patterns appear in many branches of science. Physicists find them on their plotters. Strange attractors with Fractal turbulence appear in celestial mechanics. Biologists diagnose dynamical diseases. Even pure mathematicians such as Bob Devaney, Heinz-Otto Peitgen and Richard Voss go on tour with slide shows and videos of their research.
Fractals provide a different way of observing and modeling complex phenomena than Euclidean Geometry or the Calculus developed by Leibnitz and Newton. An arising cross disciplinary science of complexity coupled with the power of desktop computers brings new tools and techniques for studying real world systems.

Hilbert Curve

Koch Snowflake

Fractals

A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales.
Illustrated above are the fractals known as the Gosper island, Koch snowflake, Box fractal and Sierpiński triangle.

Platonic Solids: Beautiful Generative Art

Polyhedra and Art

Through history, polyhedra have been closely associated with the world of art. The peak of this relationship was certainly in the Renaissance. For some Renaissance artists, polyhedra simply provided challenging models to demonstrate their mastery of perspective.  For others, polyhedra were symbolic of deep religious or philosophical truths.  For example, Plato's association in the Timaeus between the Platonic solids and the elements of fire, earth, air, and water (and the universe) was of great import in the Renaissance.  This was tied to the mastery of geometry necessary for perspective, and suggested a mathematical foundation for rationalizing artistry and understanding sight, just as renaissance science explored mathematical and visual foundations for understanding the physical world, astronomy, and anatomy.  For other artists, polyhedra simply provide inspiration and a storehouse of forms with various symmetries from which to draw on.  This is especially so in twentieth century sculpture, free of the material and representational constraints of earlier conceptions of sculpture.

Patterns in the Void - Platonic Solids in Nature

Platonic solids, or regular convex polyhedra, are named after the Greek philosopher Plato who theorized that the five classical elements (Empedocles’ wind, water, fire, and earth, with an added element for spirit) were actually comprised of regular polyhedra. They are five in number and named for the number of faces they exhibit. They are the tetrahedron, the hexahedron, the octahedron, the dodecahedron, and the icosahedron. Platonic solids have been the metaphysical and aesthetic inspiration of geometers for thousands of years. 

Plato wrote about these polyhedra in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one’s hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. Moreover, the solidity of the Earth was believed to be due to the fact that the cube is the only regular solid that tesselates Euclidean space. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, “…the god used for arranging the constellations on the whole heaven”. Aristotle added a fifth element, aithêr (aether in Latin, “ether” in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato’s fifth solid.

Platonic solids occur frequently in nature. Their forms are the complex crystalizations of minerals and appear as the skeletal remains of several species of amoebic sea creatures in the Radiolarian phylum. These creatures were beautifully illustrated by the Victorian-era biologist Ernst Haeckel in his Kunstformen der Nature.

Platonic Solids can be found in atoms in crystals, it commonly occurs in organic and inorganic chemistry, viruses and radiolarian skeletons.

These platonic solids have been found living in the sea.

Platonic Solids

The Platonic Solids are the five shapes that define the symmetry of points in space and are named after Plato. They are the tetrahedron, cube (or hexahedron), octahedron, dodecahedron and icosahedron. These are considered ‘perfect’ shapes because they look the same from every corner point, every surface is identical, and all sides are made of the same shape.
Their striking beauty is derived from the symmetries and equalities in their relations.
The greeks associated four shapes of the Platonic solids with the four classical elements: Cube/Earth, Octahedron/Air, Icosahedron/Water; and Tetrahedron/Fire. Plato wrote that the fifth solid, the dodecahedron, “…the god used for arranging the constellations on the whole heaven.” /Universe.

More Pythagorean Trees