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Bahman Kalantari

High School Students’ Feedback on Polynomiography Summer Course, at "Governor’s Summer School of Engineering and Technology", Rutgers University, 2011.

Taking Polynomiography has been an absolutely amazing exprrience. Even if I did not like anything else about the Governor’s School, simply taking Polynomiography would have made up for it all. As a general rule, I never really enjoyed mathematics, but my first year in calculus helped change that. Now Dr. Kalantari has through Polynomiography opened my eyes to yet another world of mathematics. He has shown me that math (specifically polynomials) can be beautiful. I am amazed at the amount and degree of advanced algorithms the Polynomiography software entails. I also consider myself so lucky to have taken a class that many people have never heard of, let alone learned about. I am really excited to see where Polynomiography wil lead in the future. It has so many applications, from a learning software to encoding messages to simply being used for art. I am confident that wherever it goes it will be successful and receive the admiration it deserves. Lastly, I would like to thank Dr. Kalantari for taking the time to teach me and share his creation with me. I have thoroughly enjoyed his class. The experience has been unbelievable.
Aaron Crasner

I find the subject of Polynomiography very interesting and versatile. It was really helpful to me because it showed me the various applications of math to other fields. As a student whose passion and forte is math, I enjoyed applying the calculus concepts I learned in school to this subject and software. Furthermore I was able to appreciate the value and qualities of art through math. This helped expand my knowledge and appreciation of other fields that I had earlier considered to be unrelated to math. I hope a commercial version of the software will soon be available since it will be a very useful and effective tool for teaching math, art, etc. Thank you very much for devoting your time to teaching us this class over the summer.
Sreya Basuroy

Math has always been one of my favorite subjects, but there were some parts that were hard to visualize at timed. This new Poynomiography software helps visualize these objects. Complex numbers and higher degree polynomials are now visible and I believe easy to understand. But what makes the software so valuable is that more math concepts can be taught than just polynomials, such as Newton’s method and convex hulls.
Lauren Tragesser

Most people probably have never seen anything like polynomiography. I am no exception. Our vision of polynomials is often limited to a single line in the plane of real numbers. The polynomiography software expanded this vision into the complex plane, where every single point contributes to the overall visual representation. Through this course, I better understood complex numbers, especially ways in which we manipulate and look at them. Learning about different ways of expressing complex numbers, such as the polar and exponential approach is definitely a useful tool in higher level math. This class in polynomiography also taught me more advanced topics I would not learn in a school setting, such as Voronoi regions, convex hulls, and the Gauss-Lucas theorem, but the software's visuals made the concepts much easier to grasp. I also found myself exploring the algorithms behind the software and how it uses iterations of Newton's method to portion the complex plane. I would lastly like to thank Dr. Kalantari for exposing us to this creative software that allows users to observe polynomials in a different light, as well as for enriching our understanding of the math behind these polynomials.
Elizabeth Yang

I personally find Polynomiography to be absolutely fascinating. It turns a boring intangible school subject into a fabulous work of art. The software is incredible and the polynomiographs that one can create are infinite in style, color, and design. No one polynomiograph is identical to another. The masterpieces that I have generated in this course I will be ones that I save forever. Polynomiography is a fascinating fun and interactive way to use math and I really hope that I will be seeing this software in the immediate learning environment in the near future.
Daniellle Wehner

Dr Bahman Kalantari, Polynomiography class is the only class I have that does not bore me when the teacher speaks. It has sparked my interest not only because I am obsessed with higher mathematics, but also because it is an ingenious idea. Combining math and art using polynomials has never crossed my mind, but now that I see it in action, I realize that it is elegance at its extreme. When I first fiddled with the software, and saw polynomials transform into gorgeous images, I knew there was a pattern, an idea, an equation behind the transformation, and I was instantly interested in what it was. Now I finally know, and realize that it is indeed absolute ingenuity. Your lectures never bored me, and I am sad to see the class end. Thank you for your time, I really enjoyed it.
Xiaotian Shi

Polynomiography at first seemed like a random art generator, where one plugs in ant mathematical equation and the software spits back a work of art. Soon after I realized it was not simple; with each work of art generated by the mathematical equation, there exists a subtlety, a bridge between math and art. After exploring a little more I no longer thought that the work of art was random. From the complex plane to Newton’s method, I now can say how each graph is generated. By combining both math and art, even those who dread mathematics can explore a new perspective through Polynomiography.

Polynomiography exposed me to an entirely new way of looking at math. It was a refreshing change from the usual school math curriculum, which has a rigid structure. I learned many interesting concepts and was able to exercise a lot of creativity as well. I thought that the way in which complex and intricate patterns came up from some basic rules and math was very cool. I think that polynomiography is a great way to reveal a relatively unknown side of math to people.
Omar Rizwan

Polynomiography has allowed me to experience math in a way that I never have before. I've always enjoyed math, but never saw it as something with the potential to be really beautiful. I love making pieces of art which beautify the equation which I've inputted. At the same time, I've been intrigued by learning the basics of the the higher-level math of polynomial root finding under Professor Kalantari. The software and corresponding course are a perfect solution for those who want to fulfill their love of academia, art, or both. I think this software has amazing potential to be used in schools, especially in urban schools that are suffering from low test scores, low income, and resulting lack of student motivation, in order to foster a love for math.
Liz Kantor,
NJ Governor's School of Engineering & Technology

I like polynomiography because it converts the finite equations of mathematics into visible and tangible artwork. The beauty that can be found in everyday polynomials is astounding. This class has definitely given me a new appreciation for the math I use every day. I would recommend this software to everyone who has any interest in the artistic aspect of mathematics.
Eliza Cricco-Lizza

I like math, I won't pretend that I don't. I feel that the major reason most people don't like math is because they don't understand it, and the main reason they don't understand it is because they can't visualize what's actually happening in an equation. That's why Polynomiography is so great; one can take a math nerd such as myself, as well as someone who hasn't even taken any sort of advanced math, and show them both how beautiful math can be. People learn better when they know and enjoy what's happening behind the scenes, and that makes Polynomiography invaluable.
Shannon Sabinov

Polynomiography is a great way to mesh art with math. Before using the software, I considered myself inartistic because I was never interested enough in art. Not only did the software make art more fun, but it also offered a method of actually seeing what equations look like. Polynomiography is a cool way to visualize math while creating works of art. I’m sure any person who tries this software will have a new sense of artistic ability and a source for further inspiration. The experience of working with and learning from Dr. Kalantari was priceless and enriched my understanding of math. I will surely recommend it to my teachers as a tool to reinvigorate students’ passion in math and art.
Tom R.

Polynomiography is a course unlike any other that I’ve ever experienced. It was also very interesting and informative. I was able to learn very much about polynomials. I never would have expected to learn so much about them previously. Learning about Fourier transforms and convex hulls, among other things, was something I could never have dreamed of doing in my high school. Furthermore, through this class, I was able to view polynomials in a fun, bold and creative way. I am very glad that I had the opportunity to take this course, and will definitely recommend it to my peers and teachers.
Pallavi Koppol

I have truly enjoyed learning polynomiography for the past month. This visual aid has shown me that there is more to math than mere numbers; math may be an art form, controlled and manipulated by the ideas and numbers we input and provide. I had only thought of polynomials as numbers and simple graphs. Now, however, I am able to think of polynomials as an art. Furthermore, the mathematical ideas present in polynomiography also interest me. I took AP Calculus BC this past year, and it has been fulfilling seeing various concepts that we discussed being implemented in polynomiography (e.g. Newton’s Method, discussion about complex numbers, etc.). I have also enjoyed learning about more advanced mathematical ideas as part of our discussion about polynomiography, most prominently the more advanced concepts about complex numbers (like the modulus of a complex point) and Voronoi diagrams.
Jason Qin

I was never artistic. In elementary and middle school, art class was hell for me, so when high school finally rolled around, I didn’t bother to enroll in art classes at all. However, I was good with numbers from the beginning. While other kids were outside skipping around playing hopscotch, I stayed inside studying so I could skip a few math levels. I never thought, however, that my allegiance to math would someday give me artistic support and inspiration. In polynomiography, I’ve learned that it’s truly possible to connect both disciplines- art and math- into creations of complex beauty. With polynomiography, all you need to do is generate numbers and functions, something that everyone, including myself, can easily accomplish. Thus, polynomiography is not only a software full of fun and innovation, but also an enabler of creativity for those whom have the most difficulty with drawing, painting, or simply creating by
Wesley Yiin

When I first saw the course description for polynomigraphy, I did not quite understand what the subject consisted of. I had no clue about “visualizing polynomials” or creating art from mere mathematical expressions—I hardly believed that it could be done! However, after just one class, my mind was completely changed: I was intrigued by what I had discovered, and I was feeling extremely lucky to have been placed in this course. I believe I can speak for everyone in the class when I say that we were all awestruck by what we learned in the first few days. The images that we created from such commonplace expressions as “z3 – 1” astounded us.
In addition to having fun with the software, we were educated about various topics in mathematics, topics that we would likely have discussed in a high school course. I found some topics fascinating, like convex-hulls and the Gauss-Lucas theorem. I doubt I would have touched upon such ideas until my third or fourth year of college, and even that might be a little too optimistic.
With regards to the software, I have only words of praise and satisfaction. I was extremely impressed by the capabilities of the program, especially because it was a watered-down demo version. I think that the art that can be created from polynomials is utterly amazing and inspiring. In addition, polynomiography undoubtedly has many undiscovered uses: I recall one of the students suggesting a code based on the images. It can also serve as a powerful teaching tool for middle- and high-school students. If the software ever goes on sale, I plan to be one of the first people to buy it.
Basith Fahumy

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