Posted March 29, 2013 by Dr. Henri Montandon in art

Primal Chaos: Staring into the abyss

The scientific endeavor has  proven not to be a wild goose chase. One of the reasons is building blocks. Electrons, protons and neutrons are the building blocks of atoms. Atoms are the building blocks of molecules. Molecules are the building blocks of cells. Cells are the building blocks of tissue. Tissue is the building block of organs. Organs are the building blocks of creatures. And so on.

Each type of building block associates an order of magnitude and adumbrates one or more scientific disciplines. In the brain sciences, the puzzle of brain’s building blocks is one of the many lacking an agreed upon answer.

In mathematics, there has been and continues to be a great deal of interest in a strange kind of building block called a prime number. In the natural numbers from 1 to 10 we find 4 prime numbers. As we go further and further in the natural numbers, the primes thin out, but no one knows why and no one has found a pattern to the increasing infrequency of the primes. (The very first prime number is 2, and it gives rise to an old groaner. What is the oddest prime number? 2, because it is the only even prime number.)

In neuroscience, I have not been able to find research bearing on prime numbers per se. But prime numbers live in the number line, a mathematical model where lower numbers lie to the left, and higher numbers to the right. A number of studies have shown that people respond to lower numbers with faster left hand responses and to higher numbers with faster right hand responses ( the SNARC effect). It has been proposed that “that the attentional shifts induced by the mental number line are manifested at the earliest cortical stages of visual processing.”[1]

Despite the absence of direct research, prime numbers are of interest to neuroscience because some of the most interesting inventions/discoveries in science have to do with these exotic creatures.

Thus the website of Carlos Paris grips me in two ways. First, he tells the story of how he discovered a method for locating prime numbers, and second, the method uses a visualization algorithm of astonishing beauty:

This is an anecdote of how Prime Numbers revealed their elusive nature to me. I have recently learned that my concepts are not new. However, you will find that the images I’ve produced are quite shocking, and if anything, this website will help to popularize these concepts some more. Hopefully, you will rediscover these concepts along with me as I freshly explain how I innocently explored Prime Numbers, as well as some of their properties.

A section of the depiction of 3,232,000 circles used in the geometric sieve invented/discovered by Carlos Paris.

The trope of the “innocent explorer” pursuing his researches deep into the night after the kids have gone to bed must be a familiar one to many readers. One evening, excited by the news that there was a $1,000,000 prize for finding the largest prime, he began to think more deeply about them. Being a mechanical engineer who used AutoCAD daily in his work, he fired up his system. Seeking a visualization of the natural numbers, he drew 32,000 one unit circles on the horizontal, and below them 32,000 circles up to 101 units.

He wondered how all these circles would fit into each other, so he overlapped them on a common origin.

Seeing this pattern, he began to experience the untrammeled excitement of discovery: When I saw the pattern, it just exploded in my face. It is orderly chaos! That’s when I felt something from beyond the screen staring back at me. The patterns and shapes generated above never repeat, they always look different depending on your location to the right of the origin. And most importantly, now you can visually tell how all circles fit into each other! I call the image a “Map of Factors” because it tells you all the factors for each number. Wherever you have only 2 circles intersecting the horizontal axis at their right quadrants, you have a prime number! The green vertical lines show the locations of the prime numbers.

Mr. Paris’ explorations go far beyond what I have mentioned here. He found other patterns emerging from the graphics. He built a spreadsheet algorithm which automatically detects primes. He discovered advanced concepts regarding the twin prime conjecture, a mathematical puzzle unsolved for generations.

His work reminds me how far thought, inspiration and love of new ideas can take us. Viva Carlos Paris!

   When the circles are overlapped from a common point of origin, some of the unit circles touch with no intervening arcs. These places are marked with vertical green lines in the diagram. Each of these places is a prime number.

[1] Zaira Cattaneo   Juha Silvanto   Lorella Battelli   Alvaro Pascual-Leone  2009  The mental number line modulates visual cortical attention  NEUROSCIENCE LETTERS  253-256

Dr. Henri Montandon