A Geometric Exploration of Generalized Mandelbrot Sets
Project by Polygence alum Rohan
Project's result
I formally proved the escape criterion of a generalized Mandelbrot set of degree j. Following that, I ran simulations for the creation of Mandelbrot sets for different j values using the escape-time algorithm, and plotted coordinates of the form (j, area) and generalized a curve. The data helped me to derive conjectures regarding generalized trends for the area and escape radius. Furthermore, my computational evidence helped me to observe patterns in symmetry, and I was able to prove reflectional and rotational symmetries using explicit algebraic formulae. Finally, I discuss petal-like properties of the generalized Mandelbrot set, which has a strong and clear connection to the symmetries I proved.
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Summary
The exploration of Mandelbrot sets offers readers an insightful look into the deep and elegant connections between complex numbers and geometry. By plotting Mandelbrot sets and running simulations using tools like matplotlib and googleColab, the data has be useful for aiding me in deriving conjectures, and the tools I gained over the course of the project by continuously solving problems, has been invaluable for proving these conjectures. Interesting properties analyzed from the data include graphical relationships between jth degree polynomials of the form f(z) = z^j + c and area approximations, along with the investigation of relationships between degree and petals, and various symmetries. By running simulations, and deriving and proving various conjectures, my research contributes to the field of complex dynamics by tackling open problems regarding mandelbrot sets.
Jason
Polygence mentor
PhD Doctor of Philosophy
Subjects
Computer Science, Quantitative
Expertise
Mathematics (number theory, algebraic geometry, numerical methods, classical analysis), Data Science (tree-based models, data and graphs, neural networks)
Rohan
Student
Hello! My name is Rohan Senapati and I worked with Polygence to explore the depths of fractal geometry. From a young age, I have been interested in competitive mathematics. Recently, as I was working through a problem set, I came across the field of complex numbers, and its intricacies truly fascinated me. After working through many problems regarding complex numbers and gaining intution, I narrowed my interest down to the Mandelbrot set. My passion drove me to derive and prove various properties of the Mandelbrot set, some of which have yet to be documented in literature. This project not only deepened my understanding of complex dynamics and fractals, but it also fueled my passion for mathematical discovery. Besides theoretical problems, fractals have real-world applications today, such as cryptography. Their complex patterns can be used to create more secure encryption systems, which is central to cybersecurity. As I move forward, I am hoping to apply my knowledge to practical applications in the field of dynamical systems, and contribute to the broader field of computational mathematics.
Graduation Year
2027
Project review
“My mentor was great and our discussions were insightful and fun.”
About my mentor
“I highly recommend Jason Liang for future students. He encourages you to think through ideas and fosters problem-solving necessary for research.”
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