INTERDISCIPLINARY HONORS SEMINARFALL SEMESTER 2019 – 3 Credits – Course Number: 01:090:294:H5 – Index # 10964

POLYNOMIALS & POLYNOMIOGRAPHY

Ask not what mathematics can do with polynomials - Ask what polynomiography can do forSTEM and Art

Three Facets of a Polynomial: A Polynomiograph, Rangoli, and Marble Tabletop

Bahman KalantariProfessor of Computer Science, Rutgers University

Email: kalantari@cs.rutgers.edu, Web: www.polynomiography.com

Meeting day & time: W 02:50-05:50 PMLocation: HC S126 College Ave Campus

Bahman Kalantari has created a beautiful new genre of mathematical visual art, that is quite distinct from Fractal

Art, and is just as beautiful. Not only is the art beautiful, but the mathematics and the elegant algorithms that generate

it. This book can be read on quite a few levels, all very rewarding, and will inspire lots of future research and new

gorgeous art. Professor Doron Zeilberger, Rutgers University, Winner of the Steele Prize

This interdisciplinary seminar is designed for the SAS Honors students, including students from the finearts, introducing Polynomiography, the fine art and science of visualizing polynomials equations, and muchmore. Polynomiography is a medium of creativity, where working with its software is similar to learningto work with a sophisticated camera: one needs to learn the basics, the rest is up to the photographer.Metaphorically, polynomiography is a game of hide-and-seek with a bunch dots on a canvas - you hide thedots with a polynomial equation, then paint the canvas while searching for them algorithmically. It can thusbe considered as minimalist art. Through its software, apps and by learning its techniques, you can visualizepolynomial equations, turning them into colorful images of tremendous beauty, diversity and complexity.Polynomials, these building blocks of STEM, through polynomiography, also become building blocks of artand design. What can you do with polynomiography? create a 2D print, a 3D artwork, invent a game, inventa dance, turn a polynomiograph into a painting, learn mathematics, learn algorithms, design and implementa related algorithm, make a conjecture, prove a conjecture, produce a scientific or artistic animation, makeand compare polynomiographs with traditional human art and design, combine it with photography, exploreit in cryptography, mix it with calligraphy, make an app, create a musical piece, produce a textile, use it asfashion, develop a K-16 lesson plan, inspire others and children, promote STEM and Art, and much more.Examples of such existing projects will be given.

Course DescriptionThis seminar will study polynomials and polynomiography. Polynomials are the most fundamental enti-

ties in all of STEM. Polynomiography unravels a new but beautiful facet of these mathematical abstractionsthrough their algorithmic visualization in the search for their roots. Could this lead us to the roots of artand design? Through polynomiography you learn about many mathematical and algorithmic concepts, notall about polynomials. You learn about classic and modern concepts, such as solving a polynomial equation,complex numbers, fractals, dynamical systems, Newton’s method, techniques for solving polynomials equa-tions, geometric concepts, and modern applications of polynomials. However, the goals in this seminar areinterdisciplinary. You will also learn to create art and design by turning the polynomial root-finding problemupside down. That is, through the ease of polynomiography software you will be able to experiment withpolynomials and root-finding algorithms as the basis for creating intricate designs and patterns, or anima-tions. Not only polynomiography and individual student’s creativity could result in art and design analogousto the most sophisticated human creations, but artwork of a degree of complexity and sophistication notpossible without the use of its algorithms and software. Polynomiographer does virtual painting using thecomputer screen as the canvas.

GRADING and PREREQUISITES: (1) Classroom participation: 20 %. (2) Presentation of a course project,to be carried out individually or in small groups: 20 %. (3) Paper version of the project (10 - 15 pages),including supplementary material: 60 %. Other options will be discussed. Hopefully the projects would bewell-done so that they could be showcased online, maybe some eventually find their way into a publication.By mid-semester students will make a short presentation describing the goals of their projects. Prerequisiteis mathematical maturity, or artistic maturity, or the consent of the instructor. Reading material to be re-viewed, including optional textbook, are listed below. Student will be able to use polynomiography software.They can also download Poly-z-vizion apps (Search “Polynomiography” on Instagram).

• Polynomial Root-Finding and Polynomiography, World Scientific, Hackensack, NJ, 2008 , B. Kalantari.• The Fundamental Theorem of Algebra for Artists, Math Horizon, 2013. Selected for publication in: The BestWriting On Mathematics 2014, Princeton University Press. B. Kalantari and B. Torrence.• Polynomiography: From the Fundamental Theorem of Algebra to Art, Leonardo, Vol. 38, No. 3, 233-238, 2005. B.Kalantari.• Polynomiography and Applications in Art, Education, and Science, Computers & Graphics, 28, 417-430, 2004. B.Kalantari.• How Many Real Attractive Fixed Points Can a Polynomial Have?, Mathematical Gazette, 2019. T. Coelho and B.Kalantari.• Newton-Ellipsoid Algorithm Polynomiography,” Journal of Mathematics and the Arts, 2019. B. Kalantari and E.Lee.• A Geometric Modulus Principle for Polynomials, The American Mathematical Monthly, 2011. Volume 118, No 10,(2011) 931-935. B. Kalantari.• An Invitation to Polynomiography via Exponential Series, https://arxiv.org/pdf/1707.09417.pdf, 2017. B. Kalantari.

Bahman Kalantari is a professor of computer science at Rutgers University. He received his Ph.D in ComputerScience from the University of Minnesota, as well as Masters degrees in Mathematics and in Operations Research.His main research areas are theory and algorithms in optimization, computational geometry and polynomial root-finding. He has introduced the term polynomiography for algorithmic visualization of polynomial equations and holdsa U.S. patent for its technology. Polynomiography has been featured in several national and international media.He has delivered over 100 presentation on polynomiography in USA, as well as in Argentina, Brazil, Canada, Korea,Japan, India, Iran, Turkey, Puerto Rico, Denmark, Austria, Spain, Poland, Italy, France, Germany and Belgium.Audiences have included, middle and high school students and K-12 teachers, university students and professors,artists, educators and the general public. He is also interested in algorithmic mathematical art. He is the author ofthe book Polynomial Root-Finding and Polynomiography. His article, “The Fundamental Theorem of Algebra forArtists” was selected for inclusion in Princeton University Press book, The Best Writing On Mathematics 2014. Hemaintains the website www.polynomiography.com.