According to www.polynomiography.com, polynomiography is the “fusion of art, mathematics, and computer science.” Touted as a visual method for finding the zeros of polynomials, it is based off both fractal and non-fractal images. While we learned in class that fractals can indeed arise from iterations of Newton’s method, the art of polynomiography can create images from non-fractal iterations. Bahman Kalantari of Rutgers University (who coined the term polynomiography himself as a mix between “polynomial” and the word root -graphy) claims that an endless number of images can be made by just using one polynomial by using a wide number of iteration functions. In other words, Newton’s method, Secant method, and others would all result in different images. Another point of interest he brings up is the idea of “reverse root finding,” that is, given a certain image, what iteration function would you need to use on a polynomial to generate that image?

Kalantari introduces something that he calls the “basic family,” which is represented by some mathematical notation that I’ve never seen before. But he goes on to define it as a summation of terms including the determinant of a special matrix based on terms in the polynomial’s Taylor Series, and then gets into some complicated complex algebra that’s over my head. If you’re interested in the specific mathematic formulation, go here: http://www.polynomiography.com/images/artmath.pdf. The result is a set of points on the plane. By bounding them in their respective Voronoi Diagrams (http://en.wikipedia.org/wiki/Voronoi), the desired patterns appear.

The outcome of the picture, while dependent on the polynomial, is also very dependent on the iterative function. The coloration in these images are personal, but by looking at “Life” and “Death” we see two distinct images coming from the same polynomial. Symmetric images can be made by using polynomials whose roots are the nth root of a real number r. Multiplying and manipulating these functions, says Kalantari, can result in very beautiful and complex images. (Below “Death” is on left; “Life” on right)

Lastly, Kalantari talks of some applications of Polynomiography. By representing numbers as polynomials (for instance, a0a1a2a3…a9 maps to a9x^9 + a8x^8 + … + a0) and then applying polynomiography to that polynomial, we get an image that is essentially an encryption of the number. If the iterative function is known, theoretically it could be possible to then take that image and convert it back to the original polynomial, returning the number. He also sees it as a tool for visualization in high school education, as well as a method for letting people understand the advantages and disadvantages of various root finding methods. Truthfully I think he’s really reaching with that one, and the encryption seems a little farfetched when you have something much simpler and feasible like RSA, but in theory, they’re at least original ideas.

In the end, I’m still not convinced polynomiography should be considered a real word. That doesn’t make it any less cool.

Sources:

www.polynomiography.com

http://dimacs.rutgers.edu/Workshops/CompGeom/abstracts/005.pdf

http://en.wikipedia.org/wiki/Voronoi

http://www.polynomiography.com/images/artmath.pdf