Divisor Drips and Square Root Waves
by Jeffrey Ventrella
9. A New Appreciation of Number
Numbers are often described as having only one attribute: size. But in everyday experience, and in computer representations, unless they are very small (i.e., subitizable), 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.
This structure becomes intricately hierarchical in the larger, highly composite numbers. The divisor plot is a portrait of the number line in expanded form, showing its beautiful 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. Like trying to see the forest but having difficulty because there are so many trees in the way, understanding the distribution of the primes might require shifting the focus to the composites. 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.
A Personal Note
I am an artist, visual language evangelist, and computer programmer who enjoys discovering patterns. Of course mathematicians discover patterns as well. And so do economists, biologists, music theorists, historians, and psychologists. Maybe it doesn't really matter what we call ourselves.
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 very human emotional and intellectual journeys that resulted in these distilled bits of language. Not only did I barely make it past Algebra in High School, but I failed the only Math course I ever took in college. Why? Here's one possible reason: it was taught by...a tape recorder! Each student had to work in a small cubicle with headphones, a tape recorder, and a rather gruesome workbook. There was no actual 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. While a mathematical gem can be a beautiful thing, it is meaningless without an understanding of how it came about - the story behind its discovery and its distillation. To ignore the experiential and exploratory aspects of math, and to only teach the rules and equations that are the end 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.
To be perfectly honest, I am thankful that I failed mathematics in college, and that I feared and hated math for half of my life. Because when I decided to pursue computer programming as a way to explore visual language, I suddenly had a real reason to learn math. Having even the slightest amount of dyslexia can be a barrier to learning math using traditional approaches. But visual imagery can be used to stimulate the
nonlinear, visual, spatial-oriented thinkers
and to make math a joyful subject.
Following Buckminster Fuller's advice: "dare to be naive", I set myself free to discover math in my own way, on my own terms, avoiding the narrow ruts and cliches, simply because I never learned them. With this fresh agenda, I choose to navigate clear of the cobwebs of history and academia. The ladder that I had constructed to climb out of ignorance is the same ladder that I use to climb ever higher, and grow more informed of traditional mathematics - now with a reason to care. As I develop from child-explorer to adult-child-explorer, I have found that a large community of like-minded people have also come out of the woodwork, thanks to the internet. They are communicating with me regularly on this subject. They range from young novices to (yes) professional mathematicians, and that is very exciting. We are all sharing in the joy of pattern-finding.
The computer is a programmable telescope for exploring the vast, dense fabric of numbers. That makes the mathematical experience much different than it was before computers were invented. Even more exciting: the internet is allowing people of many ages, races, cultures, and backgrounds to participate in the learning process, and to collaborate in the creation of new mathematical understanding.
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