THE ALLURE OF FRACTALS: PART I
The topic of this post will be continuation of the last, since the topic is art, particularly a contrast between two forms of art: Euclidean art and non-Euclidean art (of which belongs fractal art). Art can take many forms, but in this post, two dimensional, three dimensional, and even "four dimensional" images will be presented, as well as forms of music. The clovis point was a work of art, a three dimensional sculpture in a medium of stone, and the Wisdom Seeker IDE will be a work of art, sculpted partially in logic and partially in the nonlinear system dynamics of multi agent system simulation software executing in a medium of silicon, gallium arsenide, or some other exotic material (compromising the base for integrated computer circuitry). How can I make such a bold statement? The answer is that I have faith in complexity theory, an area of mathematics that was introduced to me somewhat surreptitiously during my early college years through constructs known as fractals. However, my acquaintance with geometric form actually goes back farther than that - so, as the jedi master Yoda implied to his apprentice Luke Skywalker in the movie Star Wars: The Empire Strikes Back" - I have had "much to unlearn". For the duration of this post, please take an opportunity to let your mind wander as you "unlearn" many of the things you have been taught in school (unless you are already in the field of complexity theory) and become my "apprentices", as we contemplate the "forceful" effect that Euclidean geometry has had especially on the western mind and the rebel challenge that fractals represent. Did you know that there is actually a religion based on the Star Wars movies called Jediism? From the above, you may get the impression that I practice it... or that Yoda has a sense of humor :-).
I took geometry as a freshman in high school. I found it to be so much different than algebra, because the emphasis was not on equations, but shapes. Rene Descartes, a figure that frequently comes up in both mathematical and philosophical discussion, was responsible for linking the two fields back in the 1600's (see Descartes links one, two, and three) It was strange not to work as much with numbers as with proofs. I can remember sitting in study hall laboring away on geometric proofs while a senior star of the football team (the panthers) kept kicking the back of my chair from his seat. This continued for about two weeks until I said for the room to hear, "G****** it, cut it out!". This is how the topics of geometry and high school athletics became unalterably intertwined in my mind :-) For an introduction to the wide world of high school geometry, check out the mathforum.org web site. I am being facetious, of course, but what I will not be facetious about is that the Euclidean geometric concepts introduced in high school are NOT REPRESENTATIVE of REAL WORLD PHENOMENA - they are just mathematical abstractions. This, however, is not what the western world thought for thousands of years. Not much is known about the life of Euclid, but among is writings is a famous book called The Elements, which present five basic postulates that are the framework of Euclidean geometry. Since good things sometimes come in groupings greater than three, chapter 13 of the elements, Euclid gives constructions for the five "Platonic Solids". Not only are the Platonic solids just mathematical constructions, but they are the basis for some of Plato's philosophy. Euclid was a Platonist. If you care to look at the Wisdom Seeker IDE open source software development web site, it will be very evident that Plato is my "main man" in providing direction for this venture. For an illustration of how the Platonic Solids apply to art, plus the impact of other geometric notions, check out this web site. Ironically, for all their simplistic beauty AND the fact that Plato is involved, I am going to debunk their utility to model nature in the paragraphs to follow.
Have you ever heard the saying "running around like a chicken with its head chopped off?". Most certainly the picture to left brings this verbal imagery to mind. The picture, however, is actually art based upon "non-Euclidean geometry". In the early 1800's alternatives to Euclidean geometry, non-Euclidean geometries, were discovered. Non-Euclidean geometries have since been proven to have many real world applications, perhaps most importantly in Albert Einstein's general theory of relativity. Yet, Euclidean geometry still takes precedence in our secondary education systems, a trend that is challenged by the following article (which includes a link to some non-Euclidean art). I encourage you to play around with the software in the article, because you will learn, as any directionally challenged person knows, that the shortest distance between two points... is not always a straight line :-) An example of an early twentieth century art form that was influenced by non-Euclidean geometry was cubism, and perhaps its most famous painter was Pablo Picasso. I found a web site for a most interesting college course in aesthetics whose goal for the first week is to compare Plato and Picasso. This web site says: "Philosophy has always been interested in the question concerning art. Plato is famous for his position on the relation between art and social harmony. As you will discern in the reading, Plato does not hold a high opinion of artists. While his criticisms of the poets are mainly directed at various cultural interpretations of Homer's Iliad and Odyssey, there is a strong indication that Plato did not believe that the arts (as a mode of free expression) are as valuable as the analytic disciplines of mathematics and philosophy. In the Republic we are told that the arts will be subordinate and controlled (insofar as they are employed to educate) by the higher discipline of philosophy." As evidenced by the clovis point story, I have a high opinions of artists, and I highly disagree. I do agree that if I had to choose between a picture of a pyramid and the chicken painting, I would choose the pyramid as having a higher degree of aesthetics. But then again, I might change my mind about that closer to Thanksgiving... :-)
To use a double negative, until my early college years, I had "not disliked" art. I had always liked music since starting band in middle grade school, but I had not been into pictures. At best, my notion of aesthetics had been a starving artist's landscape painting advertised during a public television fund raising drive. But I had always been a math nut, and had I known more about the relationship between mathematics and art, I may have had a different persausion. However, out of the blue, pictures (links one and two) of objects called fractals (links one and two) began to permeate the popular culture (links one and two). Fractals are manifestations of non-euclidean geometry, but they actually have fractional dimension (as opposed to being 2D, 3D, 4D, etc.). Software actually exists to determine the fractional dimension of a given image. Roughly speaking, an object has a fractional dimension based upon its degree of "jaggedness". By manipulating the applets in this web page, you can see how jaggedness "builds up" in fractals as more detail is added. Java applet technology has made possible such K-12 classroom applications as the "Fractal Microscope". Such educational experiences are important since shapes in the real world are fractal. So is the structure of music, either man made like Bach's Art of the Fugue or synthesized from fractal images (links one and two). Watch out, starving artists! :-)
To be continued...
