"For Monet, on this occasion, water lilies were the measure of water lilies; and so he painted them."
Aldous Huxley - The Doors of Perception & Heaven and Hell
Aldous Huxley - The Doors of Perception & Heaven and Hell
Perceiving the New in the Old
A few years ago while I was thinking about primes’ patterns - a hopeless pursuit, but a great exercise – I started thinking about the special places on the number line where primes show up. Below is the image of primes on the number line as often shown in middle and high school mathematics textbooks; it’s an artifact of a sort that I’m familiar with:
Why aren't primes in other places?, is the obvious next question. The familiar image doesn't give visual hints to an answer. The image is static, the dots are randomly placed; there is no rhythm, no symmetry, no distinctive shape, or much aesthetic quality in this depiction. Primes are notoriously hard to handle, though the volumes of their mathematical analyses could fill libraries we still haven’t captured what’s most intriguing about them: what’s the pattern? Could we reimage them in the form of art; in a representation that could lead the viewers to insights intuitively graspable by virtue of aesthetic quality instead of rational analysis?
If we imagine that numbers have their own will, then we may say that primes don’t like other primes. Fourteen’s right to be prime was taken away by Two and Seven, and these two have deprived every other existing even and seventh positive integer from being a prime. Everywhere they pass primes rule out other potential candidates to primehood. This narrative ignited a spark in my mind in the form of a moving image; I drew a still version of it on paper a few days later. It looked like this:
If we imagine that numbers have their own will, then we may say that primes don’t like other primes. Fourteen’s right to be prime was taken away by Two and Seven, and these two have deprived every other existing even and seventh positive integer from being a prime. Everywhere they pass primes rule out other potential candidates to primehood. This narrative ignited a spark in my mind in the form of a moving image; I drew a still version of it on paper a few days later. It looked like this:
Primes wave their way through the number line, marking it everywhere they cross it, leaving only random and increasingly infrequent gaps behind. Grounded on the number line, the image looks like this:
The gaps left in between the waves are the magic spots where new prime waves sprout and start their own path down into infinity. Could a pattern be visualized in the balanced aesthetics of this representation? I haven’t found one then or since (you would know about it, guaranteed!).
Observation – Imaging – Reinterpretation
Perception is the observer's transcription of sensory input into his own personal conception of the essence of the object he observes. We may perceive the same old object differently every day, all we need are new eyes to see with. This exercise in perception didn't start with a physical sensation, it started with numbers. Numbers are abstractions whose physical representations are everywhere; two birds, two beeps of the microwave oven, four thuds of four wheels over a speed bump, seven people, ten pages of a book… But what could smell, taste, look, sound, or feel, like the randomness of primes? In this particular case the idea is more abstract than simple numbers are, and readily available real life examples of it are very rare (though reproductive cycles of some plants and insects show prime patterns but we can’t readily perceive them because they are too spread out in time).
What are the stages of perception in this example? The object to be perceived came from the mind and was first visually represented by way of the familiar image of dots on the number line. The post-spark representation of the (moving) image of waves is quite visually appealing and by virtue of this appeal it may reveal something about primes that the dots did not capture. This was a shift in perception from the dull and static to the alive and aesthetic. The new understanding of the old problem gained after the perceptual shift is the first and biggest step in mathematical problem solving.
See more about the teaching application of the prime waves in the next modules!
Go to the next module.
Observation – Imaging – Reinterpretation
Perception is the observer's transcription of sensory input into his own personal conception of the essence of the object he observes. We may perceive the same old object differently every day, all we need are new eyes to see with. This exercise in perception didn't start with a physical sensation, it started with numbers. Numbers are abstractions whose physical representations are everywhere; two birds, two beeps of the microwave oven, four thuds of four wheels over a speed bump, seven people, ten pages of a book… But what could smell, taste, look, sound, or feel, like the randomness of primes? In this particular case the idea is more abstract than simple numbers are, and readily available real life examples of it are very rare (though reproductive cycles of some plants and insects show prime patterns but we can’t readily perceive them because they are too spread out in time).
What are the stages of perception in this example? The object to be perceived came from the mind and was first visually represented by way of the familiar image of dots on the number line. The post-spark representation of the (moving) image of waves is quite visually appealing and by virtue of this appeal it may reveal something about primes that the dots did not capture. This was a shift in perception from the dull and static to the alive and aesthetic. The new understanding of the old problem gained after the perceptual shift is the first and biggest step in mathematical problem solving.
See more about the teaching application of the prime waves in the next modules!
Go to the next module.