WARNING: the following is a post about Math. Side effects include, but are not limited to; headache, nausea, migraine, eye strain, and emotional, psychological and physical stress. Lie down if needed.
Let’s face it, those who actually like math, and are even really good at it, can be counted by hand. I will try to explain this as slow and as simple as possible, bear with me.
During the fall semester of 2010 at University of Balamand (UoB), the SMA club, in partnership with the Office of Public Relations, organized an exhibition called “Math Visions.” The aim of the exhibition was, first and foremost, to bridge the gap between science and art, and display complex concepts of mathematics through art.
The reason I wanted to bring this back on the table is because it needs to be remembered every once in a while, and because we were proud to say that it was the first of its kind in Lebanon.
The SMA team contacted the math professor behind this genius concept. Dr. Abdulrahaman Abdulaziz.
Born on September 13, 1965, in Bakhoun, North Lebanon, Dr. Abdulaziz moved to the United States in 1989. He studied at California State University, where, by the year 1991, he earned a BS in computer sciences and applied math, and an MS in applied math. That same year, he migrated to Australia, where he undertook pure math courses between the years 1993 and 1998. Dr. Abdulaziz was awarded his Ph.D. early in 1999 from the University of Sydney, after which he returned to Lebanon and started teaching at UoB.
Dr. Abdulaziz’s passion was in the chaos theory. His Ph.D. dissertation was on attractors in the chaotic dynamical systems.
An attractor is a concept in dynamical systems whereby it consists of numerical values that tend to “evolve” in a system, that system ultimately becoming a 1D, 2D, or 3D graphical representation in space and over time. Let me help you visualize this, remember in sine, cosine, tangent, lin, log functions and their graphical representation? Sort of like that but on a much, much -MUCH- bigger scale, so much so that a graph will represent multiple points in space as well as their trajectories as they vary with each defined value.
Now that you have a very vague notion of what I’m saying – I hope – let’s up it a notch. First, you have this equation/s, correlating x, y, and z, then you have parameters that may take any value, and thereby change your points’ trajectories in space and time. There can be as many equations as needed to define the system and points, and just as many or more parameters. With each variation, you basically get a different “graph.” And of course, there’s multiple types of attractors depending on how the equation looks like, behaves, and evolves. I will not go into detail about this but feel free to search for the different types all by yourselves – maybe take some painkillers before.
NOW, the art part; I will be presenting a couple of the pieces we showcased in the exhibition along with their equations. You can also recreate, or dabble in these things, using a very simple program called Chaoscope (free for download here). Pieces were made by Dr. Abdulaziz, and all names were given by him and reflect his personal view. (If you see something else in them, then that’s a Rorschach thing, and subject to a separate post I guess).
Don’t think about them too much, just enjoy.
E.
Image type: Lorenz-84 — Image Name: CUP
First introduced in Lorenz’s paper entitled “Irregularity: a fundamental property of the atmosphere”, this equation is a low-dimensional model for long-term atmospheric circulation. Rather than a graphical representation of atmospheric currents, the orbit coordinate are the three variables of the model. Parameters: A, B, F, G, and dT.
Image type: Icon — Image Name: BUBBLE
The ICON equation was used by Michael Field and Martin Golubitsky to demonstrate how Chaos could yield symmetry. This attractor is an example of a two dimensional attractor converted to a three dimensional one: values of z are not used for x and y calculations.
Parameters: Degree, Alpha, Beta, Lambda, Gamma, and Omega.
Image type: Lorenz-84 — Image Name: TUNNEL
First introduced in Lorenz’s paper entitled “Irregularity: a fundamental property of the atmosphere”, this equation is a low-dimensional model for long-term atmospheric circulation. Rather than a graphical representation of atmospheric currents, the orbit coordinate are the three variables of the model. Parameters: A, B, F, G, and dT.
Image type: Lorenz-84 — Image Name: TRUMPET
First introduced in Lorenz’s paper entitled “Irregularity: a fundamental property of the atmosphere”, this equation is a low-dimensional model for long-term atmospheric circulation. Rather than a graphical representation of atmospheric currents, the orbit coordinate are the three variables of the model. Parameters: A, B, F, G, and dT.
Image type: Icon — Image Name: JELLYFISH
The ICON equation was used by Michael Field and Martin Golubitsky to demonstrate how Chaos could yield symmetry. This attractor is an example of a two dimensional attractor converted to a three dimensional one: values of z are not used for x and y calculations.
Parameters: Degree, Alpha, Beta, Lambda, Gamma, and Omega.
Image type: Lorenz — Image Name: KNOT
A beautifully simple equation created by Edward Lorenz to demonstrate the chaotic behavior of dynamic systems. This attractor is also historically important because Lorenz discovered, while working on weather patterns simulation, one of the fundamental laws of the Chaos Theory: “the sensitive dependence on initial conditions” he himself dubbed “the butterfly effect”. Although moving the initial orbit won’t affect the shape of a strange attractor, the position of the orbit on the attractor after several iterations will vary considerably from one initial position to the other. Interesting attractors can be found with a relatively high value for dT. Parameters: A, B, C and dT.
Image type: Julia — Image Name: MIRROR
Alan Norton was the first to render Quaternions in the early eighties. He was followed later by John C. Hart, whose code was used for Chaoscope. The inverse of the equation yields a slightly different result than the regular z = z2+ c: the set is perfectly symmetrical. Also, because iterating the equation like a normal attractor proved not very effective, the depth first tracing method has been implemented. The Level parameter defines the depth of the square roots tree. A high level (i.e. > 16) will produce more detail and won’t slow down the rendering. Parameters: Level, Creal, Cimag, and Phi
Spreading The Word 