Having recently discovered that the prevailing number spirals of our age are not nearly hexagonal enough for my needs, I invented new a circular spiral based on centered hexagonal numbers, which I have named the GRAHAM SPIRAL. The first rotation contains 6 dots, representing the first 6 natural numbers, and each subsequent rotation n contains n*6 dots, in the fashion of hexagonal tessellation. In the examples above, variable dot sizes represent the number of unique prime factors in a given number, while the uniform dots represent primes themselves, but of course any arithmetic attributes can be plotted in the same fashion. The resulting patterns clearly demonstrate the deeply hexagonal structure of the natural numbers, &c.
This article was transcribed by myself for Hexnet.org from a PDF file hosted on the Dozenal Society of America site, and originally published in the Atlantic Monthly, presumably some time ago.
From the Eskimo counting upon his fingers to the mathematical wizard producing split-second answers with a slide rule, we count by tens. In our critical age, such universality is phenomenal; it cannot be claimed for any religion, code of morals, form of government, economic system, principles of art, language, or even alphabet. Counting is one of the very few things which modern man takes for granted.
Note: Writing about dozenalism always presents some semantic complications. When discussing the natural numbers up to twelve, I have opted to spell out the numbers in English, since this is a clear and base-neutral way of representing them. After trying several different systems, I have settled on writing larger numbers in decimal. Unless otherwise specified, "10" means ten, not twelve, et cetera. When I use dozenal notation, and for clarification purposes elsewhere, I have prefixed the radix as an abbreviation before the number. Thus, "dec. 360" means decimal 360, and "doz. 260" means dozenal 260. "360" by itself, unless otherwise specified, refers to the former. Larger numbers spelled out, where it is stylistically appropriate to do so, will always be given in decimal. My first inclination, of course, was to put all numbers here in dozenal, but on further reflection I see no value in confusing people needlessly. (Confusing them for a good reason though is fine.)
Sexagesimal glyphs from the Babylonian numeral system of yore. Although perhaps only indirectly related to hexagons at first glance, this highly sophisticated counting system demonstrates the ancients's respect for the superior factoring abilities of multiples of 6 such as 6, 12, and 60 (doz. 6, 10, 50). Note well that the 60 numerals of the Babylonian system live on in our 60-minute hours, 60-second minutes, and 360-degree circles—the circle itself consisting of 6 "sides" of 60 units each.