The concept of beauty in math is often reserved for describing the elegance of a proof or the importance of a theorem. The beauty of math is usually of a more esoteric nature, appreciated only by professors or theoretical mathematicians. However, fractals – geometric, self- similar shapes found both in the natural world and the theoretical world – exemplify a visual beauty that everyone can appreciate. Though the concept of fractals dates back to the 17th century when Gottfried Leibniz proposed recursive self-similarity and lamented that this did not really exist in geometry, it was not until the 1970s, when Benoit Mandelbrot used computer imagery to create what is now known as the Mandelbrot set, that the study of fractals formally took root and these infinitely repetitive geometric shapes gained wide popularity for their intricate patterns and winding curves. Rapid advances in technology and the increase of computer processing power took fractals from the purely theoretical world of abstract definitions to a world where mathematically accurate visualizations could be produced, studied, and, of course, admired. For my DCC Capstone project, I plan to create an exhibition that explores the concept of math as a form of art through a series of computer generated fractals and fractal inspired art pieces.
The focus of my exhibition will be drawing attention to the ubiquity of fractals in nature. Fractal-like forms appear in a myriad of places, from tree and fungus growth to coastlines to clouds. Today, computer generated fractals are often used to model these natural forms in a range of fields such as video game design and geographical modeling. I plan to generate a series of fractals that model or are inspired by fractal like forms in nature, emphasizing both the beauty of the fractal and the natural form. Then, for every mathematically accurate fractal I generate, I will create an art piece inspired by the fractal. For the final exhibition, I will display poster printouts of the computer-generated fractals next to their complementary art pieces.
To create the mathematically accurate fractals, I plan to use Java and the computational software Mathematica. I am currently not sure what the advantages are to using either Java or Mathematica, so this is something I will need to explore as I learn more about computer generated fractals. With Java, I hope to write my own code to generate various fractals. I have experience using Java, but I will still face quite a learning curve when attempting to write code for fractals. However, in my classes I have seen examples of code for the Mandelbrot set and the Koch Curve (perhaps the two most well known fractals), and there are many textbooks and online resources that I can use to figure out how to accomplish this. I do not have any experience yet using Mathematica, though I am familiar with MATLAB (a similar software program) and have used it in my math classes. Mathematica is known for having very strong visualization and animation tools, so it has the potential to be a very useful tool for my project. I am also currently doing research with the Experimental Geometry Lab (part of the Math Department here at UMD). The professors and graduate students at the EGL explore non-Euclidean geometry using Mathematica and have programmed various toolkits in Mathematica for generating different transformations in hyperbolic space. I am currently learning about hyperbolic space, the tools they have created, and how they use Mathematica, so I will be able to apply what I learn at the EGL to my project. Also, depending on what I am able to accomplish with Java or Mathematica, I hope to be able to create animations of some of the fractals I generate to emphasize the fact that fractals continue their patterns of self similarity infinitely. In addition to having poster sized printouts of all the fractals, having a monitor up displaying an animation that zooms in and out of the fractal would be the best way to emphasize one of the properties that make fractals such an incredible (and beautiful) mathematical discovery.
To create the complementary art pieces, I plan to use a range of mediums, depending on the natural form I am using as inspiration. The medium I choose will be purposely selected to emphasize certain aspects of the natural form. For example, to create a piece based on the self- similarity of Romanesco Broccoli, I will use clay in order to emphasize its definite, yet still fractal-like form; on the other hand, to create a piece based on a fractal modeling the behavior of a coastline, I might print onto fabric to emphasize the fluidity of the fractal. I have experience with acrylic and oil painting, drawing, and screen printing, and some experience using clay. However, I would like to explore mediums that I have not tried before and create mixed media pieces. The artwork I create will not be mathematically accurate; this is not only impossible (due to the infinite nature of fractals), but would not have any purpose because this can be done on the computer. The artwork will be more loosely based on the shape or patterns that stand out in the fractal model. Furthermore, fractal art often tends towards the psychedelic (think dizzyingly colorful 70s posters), but this will be something I will avoid. I hope to create fractal based art that has an organic and elegant nature.
There is a whole world of math that people often do not get to experience because they are quickly turned off by the seeming pointlessness of solving for x in high school math classes. My Capstone project will be an investigation into how the digital age has opened new channels of creativity and has allowed the word ‘beauty’ to be redefined in the context of math.
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