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Polynomiography is a mathematically-inspired computer medium based on algorithmic visualizations of one of the most basic and fundamental tasks in sciences and math: solving a polynomial equation. Polynomiography serves as a powerful medium for creativity, discovery, and learning, with numerous applications in education, math, science, art and design. Interested students will select a particular problem and explore related theoretical and/or practical applications of polynomiography. In particular, some specific problems will be described from computational geometry, numerical analysis, discrete math, education, or algorithmic mathematical art.
My problem of study is the Algebraic Art Gallery Problem: Given a set of n Euclidean points, which point in their convex hull maximizes the product of the distances to the n given points?
Presentation 1 (ppt)
Presentation 2 (ppt)
Week 1: I met Professor Kalantari and we spoke about his research and my goals.I started reading his book - Polynomial Root-Finding and Polynomiography - to get a feel for the subject matter.
Week 2: I had a ton of questions about what I had read in his book, so Professor Kalantari worked through them with me. We talked about what my specific question of study will be - the Algebraic Art Gallery Problem. I also worked on my website and prepared for the first presentaiton (link above).
Week 3: Now that I had some background knowlede and knew what I will be looking at, I worked on solving the Art Gallery Problem for a simple specific case. To do this, I learned how to use Mathematica. I met with Professor Kalantari again and he gave me some more ideas and problems to explore.
Week 4: I began working on the Art Gallery Problem in 3D. I read Dr. Kalantari's proof that the Maximum Modulus Principle would hold in 3D, and worked on some examples to check this. Meanwhile, I continued exploring the triangular case and wrote a program that would calculate the "optimal points" given an input of three vertices.
Week 5: I got really interested in finding some patterns for the triangular case. I started observing what happens to isosceles triangles of various sizes. I had to modify and debug my program to be able to work on more triangles and to analyze the results for me. My explorations went further when Dr. Kalantari suggested that there may be a relationship between the optimal points and the Gauss-Lucas points.
Week 6: I had a lot of data regarding isosceles triangles, using which I was trying to figure out when the AG point would be at the midpoint of the base vs. when it would be on the legs of the triangle. I started creating graphs to better understand my data. Again, this required some debugging and improving my Mathematica skills. At the end of the week, I met Dr. Kalantari in Princeton and we worked on these problems. We also worked on the proof for the maximum modulus principle for a triangle.
Week 7: After our Princeton meeting, I had a lot of ideas and approaches to explore in dealing with the isosceles triangles. I proved that for an equilateral triangle, the optimal point is at the midpoint of each side of the triangle. I also discovered a small deviation between the 2D and 3D cases of the Art Gallery Problem. Finally, I put everything together for our final presentations.
Week 8: I spent a little more time trying to understand the isosceles triangles. I found some new patterns but still no concrete explanation. I learned how to use LaTeX and started typing up my work. Dr. Kalantari and I hope to produce a paper about our findings. Thanks for everything to Dr. Kalantari and to DIMACS!
Links and Resources
Kalantari, Bahman. Polynomial Root-Finding and Polynomiography. World Scientific: New Jersey, 2009.
Kalantari, Bahman, Iraj Kalantari, and Fedor Andreev. "Animation of Mathematical Concepts Using Polynomiography", 2004.
Kalantari, Bahman. "Polynomiography and Applications in Art, Education, and Science", 2004.