General Information

Student: Katherine Garzarella
Office: CoRE 536
School: Rutgers University
Major: Cell Biology and Neuroscience
E-mail: k.garzarella@gmail.com
Project: Biological Applications of Polynomiography

Project Description

Polynomiography software was developed by Dr. Bahman Kalantari, a professor and leading researcher of Computer Science at Rutgers. The software utilizes an algorithm to create unique graphs of polynomial equations, termed polynomiographs, that in turn visualize the approximation of zeros. The aesthetic value of polynomiographs can be easily recognized upon a quick glance at one, but the other areas of application are still being developed. Throughout the DIMACS-REU program, I aim to explore its potential applications to the living world of biology.

Weekly Log

Week 1:
The mathematical basis for the Polynomiography, including iteration functions, fractal and non-fractal images, and root-finding methods, was studied. Familiarity with the program's inputs, outputs, and limitations were gained.
Week 2:
Literature research on biological applications of polynomiography was conducted. Fractals were found to be very prevalent in nature and an expanding focus of research in the field of disease characterization and diagnosis. An introductory presentation was given detailing our aims to explore this topic more and link it to polynomiography.
Week 3:
The foundational methods of fractal geometry analysis were surveyed.
Week 4:
The use of fractal geometry methods for cancer quantification were investigated. The technique was found to have value as well as limitations. The majority of cancer is diagnosed via qualatative methods, highlighting the value of developing a highly reproducible, computerized, quantitative measure.
Week 5:
Fractal analysis in other structures and systems were researched. Efforts were focused primarily on the various structural facets of DNA, brain organization, connectivity of thought pattern, pulmonary composition and branching, cardiovascular system constituent patterns and strutural aspects of the cardiovascular system as a whole. The Cancer Institute of New Jersey was visited and lectures regarding the use of mathematical modelling and programming in the field were attended.
Week 6:
Chaos theory and its relevance in nature and polynomiography was studied.
Week 7:
Linear programming methods and its role in polynomiography were examined to expand background knowledge.
Week 8:
The culmination of this work was written and submitted to faculty, and presented orally to faculty and peers. A guest lecture on this work was given to a summer course at Rugters University. Plans for publication were discussed and planned with Dr. Kalantari


References and Relevant Literature

Ary L. Goldberger, Luis. A. N. Amaral, Jeffrey M. Hausdorff, Plamen Ch. Ivanov, C.-K. Peng, and H. Eugene Stanley. ?Fractal dynamics in physiology: Alterations with disease and aging.? PNAS Feb 19, 2002 vol 99.
Baish, James W. and Rakesh K. Jain. "Fractals and Cancer." Perspectives in Cancer Research. Cancer Research: 60, 3683?3688, July 15, 2000
Cipra, Barry. “A Healthy Heart is a Fractal Heart.” SIAM News, Volume 36, Number 7, September 2003
Felsenfeld, Gary and Mark Groudine. “Controlling the double helix” Nature 421, 448-453(23 January 2003) doi:10.1038/nature01411 http://www.nature.com/nature/journal/v421/n6921/fig_tab/nature01411_F1.html
“Fractal Globules in DNA” Nov 5, 2009: Fractals in Nature. http://fractalfoundation.org/2009/11/fractal-globules-in-dna/
Kalantari, Bahman. ?Polynomiography: A New Intersection between Mathematics and Art.? Technical report DCS-TR 506, Rutgers University, New Brunswick, New Jersey, 2002
Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company. ISBN 0-7167-1186-9
May, Robert. “The chaotic rhythms of life.” http://members.fortunecity.com/templarser/rhythm.html
Pincus, David. “Fractal Brains: Fractal Thoughts.” The Chaotic Life. Sept 4, 2009. http://www.psychologytoday.com/blog/the-chaotic-life/200909/fractal-brains-fractal-thoughts
Peterson, Ivars. ?The Colors of An Equation.? Science News. http://www.sciencenews.org/view/generic/id/3771/title/Math_Trek__The_Colors_of_an_Equations_Roots
Renuka Uppalauri, Eric A Hoffman, Milan Sonka, Patrick G. Hartley, Gary W. Hunninghake, and Geoffrey Mclennan. “Computer Recognition of Regional Disease Patterns.” Am. J. Respir. Crit. Care Med., Volume 160, Number 2, August 1999, 648-654

Additional Information