||Mary Katherine Battles
||Exploring Polynomiography and Its Algorithmic and Mathematical Applications
Polynomiography is, as Dr. Bahman Kalantari states, "the art and science of visualization in approximation of zeros of complex polynomials, via fractal and non-fractal images created using the mathematical convergence properties of iteration functions." It is through his software that people are able to graph polynomials, including complex roots, to create beautiful images. This summer, I will explore the infuences of specific aspects of polynomials on their graphs such as coefficients, degree, and roots. What trends do they set? What degree of symmetry is created with each change? Can they be organized according to these characteristics? As a Chemistry and Art double major, I am also curious as to the contrast between the images projected via the computer and on physical media. Could a 3D model or painting shed light to a new way of viewing polynomiographs? Before the close of REU at DIMACS, I hope to grasp a better understanding of the algorithm, modulus, and beauty behind complex polynomials.
- Week 1:
- I settled down on Sunday and began working on research Polynomiography. After orientation and talking with my roommates and apartment neighbors, I became even more excited to see the result of this program come mid July. On Monday I spoke with my mentor and discussed some plans to help in my research. Friday I gave my presentation.
- Week 2:
- After emailing my mentor, and discussing Polynomiography with Colin, another researcher who is studying Polynomiography, I was able to acquire the software that would enable me to play around with the program. Since then, I have worked in the office, taking note of trends. At first, I began inputting simple polynomials such as x^2+1, x^2+242x-435 and so on while recording the resulting images. Thus far, the images are simple and do not stray far from each other. While experimenting with more complicated polynomials I have discovered some images that I have saved and would like to draw or paint. In relation to my drawings, I have worked on finding a clearer way to show ascension and descention
- Week 3:
- My mentor returned from his two week conference in South Korea on Tuesday. On the following days, we discussed in more depth modulus and what we plan to do with the physical art aspect of polynomiography. I also was invited to join the International Symposium of Voronoi Diagrams that is held June 27, 28, and 29. After talking, I plan to buy a plethora of paints, paper, and brushes. I have focused more on drawing polynomiographs this week and plan to return to the computer this weekend. I was able to draw multiple versions of circles and an eye influenced by polynomiography. The artwork is moving slowly but as expected when there are so many possibilities of pictures as well as the fact that art takes time.
- Week 4:
- Now at the close of the midway point in this research program, I am eager to fully implement this newly acquired paint and explore more with the software. However, I am finding it is harder to pinpoint aspects of the polynomial that affect the picture so much as to alter the appearance to a great extent. I have graphed many random polynomials already and found it to be more difficult than expected to create something new and different. I also attended the ISVD conference and listen to some amazing speakers and learned a few new things about Voronoi Diagrams, and I ate delicious food.
- Week 5:
- This week I further sketched and painted my interpretations of polynomiographs. I met with my mentor once and worked on my painting for the rest of the week.
- Week 6:
- For the most part, I have finished painting one of the sketches I have made and started sketching another to be painted. My mentor and I have met three times this week and discussed the trends and some aspects of polynomiography. I hope to finish three more sketches before the end of this Sunday and then paint as much as I can before the program ends.
- Week 7:
- This week I spent meeting with my mentor and discussing the material for my presentation. The rest of the week was time used to paint.