Divisor Plot

This web page is an expanded version of a presentation given in Boston at the 7th International Conference on Complex Systems

Prime Numbers are the Holes Behind Complex Composite Patterns

Visualizing Number Theory with the Divisor Plot
JJ Ventrella - Jeffrey@Ventrella.com ...(www.Ventrella.com)

8 Conclusion

Numbers are often described as having only one attribute: size. But in everyday experience, and in computer representations, unless they are very small, numbers must be expressed as sequences of digits, or in algorithmic steps. In other words, they have structure. And the bigger they are, the more structure they have (unless they are prime).

This structure is recursive and hierarchical in the larger composite numbers. The divisor plot is a portrait of the number line in all its structure – at least in terms of integer divisibility (there are other kinds of structure as well, such as partitioning)

This exploration may not provide immediate clues to prime number distribution. But it does reveal a variety of intriguing structures among the composite numbers, and so it may be used to enhance mathematical intuition - to let the visual brain be a lever for mathematical understanding. Composite numbers are metaphors for the structure of the universe. Seen in this light, the primes simply become a background to this beautiful, endless complexity.

Personal Note
In high school I never got past the level of basic algebra. And I made poor grades in general. In undergraduate college I failed the lowest possible math course. But I was very accomplished in all my art classes. I graduated from Virginia Commonwealth University with a degree in Art Education, and lots of course work in Aesthetics and Art History. My potential teaching career was abruptly derailed by my first experience in writing a program to generate a fractal using BASIC. Within months I enrolled in a Masters program at Syracuse University where I studied programming for making dynamic computer art, under the direction of Ed Zajec. This resulted in an intermediary career as a budding computer artist, and specialist in data visualization for university research in many scientific disciplines. By the time I was invited to study for my second Masters at MIT, I had built up enough of a portfolio of work that no one bothered to ask about my math credentials. I am glad no one asked.

I am not an advocate of learning math by rote methods. Traditional math is taught without emotion, aesthetics, discovery, analogy, or metaphor. Students are asked to learn rules and highly-abstracted equations and expressions, without knowing about the emotional and intellectual journeys that resulted in these distilled bits of symbolic language. The undergraduate math course that I failed at VCU was taught by....a tape recorder! Each student had to work in a small cubicle with headphones, a tape recorder, and a workbook. There was no real teacher - only a teaching assistant on hand to answer questions. There was always a long line of students waiting to talk to him. I believe that they were lined up because they yearned for interaction with a real human.

I agree with Lakoff and Nunez: Mathematics arises from the embodied mind. Humans are a highly visual species. Our language is based on grounding metaphors. And math is the ultimate precise distillation of our language. But while a mathematical gem can be a beautiful thing, it is meaningless without an understanding of how it came about. To ignore the experiential and exploratory aspects of math, and to only teach the rules and equations that are the result of this deeply-human process - that is not good way to teach math to young people. That is an arrogant way to teach math.

The computer is a programmable microscope for exploring the deep, dense fabric of numbers. That makes the mathematical experience much different than it was before computers. In addition to that, the internet is helping people of many ages and many backgrounds to participate in the learning process, and to collaborate in the creation of new mathematical understanding. With the Java applet that I created for exploring the divisor plot, I want to offer a new tool that lets people discover the beauty of composite pattern on the number line, and to arrive at their own personal appreciation of number theory - in the same way that I have.

Acknowledgments
I would like to thank Dr. David Burton, Professor of Art Education at Virginia Commonwealth University, where I failed Math. Dr. Burton encouraged me to pursue my love for patterns and geometry, which lead to an appreciation for math. Like Dr. Burton, I want to inspire young people to appreciate math through art and visual studies.

I also want to thank Robert Sacks, for making the number spiral which influenced my approach, and after contacting him, for his many ideas and suggestions concerning relationships of the divisor plot to the number spiral - and for new discoveries in terms of parabolas.

Thanks to David Espinosa, for help with math, and for general brilliance.

Thanks to Hector Sabelli for inspirational conversations, and a healthy viewpoint on how mathematics relates to art, phychology, and philosophy.

And finally, thanks to Beth O'Sullivan, for her great mind and heart, and observations on prime number patterns.

Bibliography

[1] Chaitin, G. 2005 Meta Math! The Quest for Omega. Pantheon Books. page 16

[2] Sacks, R. Number Spiral. 2003. web site published online at:
http://www.numberspiral.com/ © 2003 – 2007 by Robert Sacks

[3] Sabbagh, K. quote from the book, What We Believe but Cannot Prove. Ed. Brockman, J. Harper Perennial, 2006. page 21

[4] Stein, M. L., S. M. Ulam, and M. B. Wells. 1964, A visual display of some properties of the distribution of primes. AMA 71(May):516-520.

[5] Thomasson, D., From Knight Moves to Primes. 2001. web site published at http://www.BordersChess.org/KTprimes.htm © 2001 by Dan Thomasson

[6] Wolfram, S. 2002. A New Kind of Science. Wolfram Media, Inc. page 132