Tagged: circles

Here we see another form of the Pythagorean decad, highlighting its hexagonal nature.

Geometries such as this have been used by people throughout history—Pythagoreans, Freemasons, &c.—to justify decimalism. This is a false understanding of the decad. In both this diagram, as well as in the Tetractys, it is clear that the fundamental organizational principle is hexagonal, not decadian. The concept of ten emanates from the underlying reality of the hexagon—not the other way around.

An astute reader recently brought to my attention the nascent movement afoot to replace π in common usage with the number now unfortunately known as 2π—viz., 6;349419 (dec. 6.283186):

• Pi Is Wrong! - By Bob Palais
• The Tau Manifesto - By Michael Hartl

(For a reasonably convincing argument on why the letter τ (tau) in particular should be adopted for this value, please read Mr. Hartl's manifesto.)

The fundamental point here is that, in trigonometry and all other manner of angle-measuring endeavors, what we care about is the radius of a circle, not its diameter. The one follows from the other to be sure, but at the end of the day the diameter is more usefully considered twice the radius than the radius is half the diameter. A circle is a circumference around a center—it is the measure of this distance between center and circumference that is elemental to the idea of a circle, not the rather incidental fact that its full width is twice that same distance.

This is a unit circle diagram using both dozenal notation (as with elsewhere on this site, using "A" and "B" for ten and eleven) and the newly proposed circle constant τ (tau), which is equal to 2π. The advantages of τ over π are numerous and obvious—instead of a full circle of arc being two of anything, it is just one τ. Put another way, τ is simply the number of radians in a circle.

NOTE: I have transcribed and edited this from various ancient translations of Euclid, augmented and tempered where necessary by at least the structure of more modern versions. I am pretty sure there are no errors in it. This is of course only one of many interesting Euclidean propositions involving hexagons, and for anyone reading this who does not in fact own a copy of Euclid I highly recommend purchasing one right now. Thank you.

Let ABCDEF be the given circle. It is required to inscribe an equilateral and equiangular hexagon in the circle ABCDEF.

Here we see the well-known "Flower of Life" pattern, consisting of nineteen interlocking hexagons in a cubic/hexagonal arrangement. While the particular term "Flower of Life" is, as far as I have ever been able to determine, of fairly recent and dubious origin, there is certainly no doubt that the pattern itself is of great antiquity, and can be found throughout the world among many different cultures.

The following video came to my attention recently. It presents, in my view, a perfect example of the sort of world-class hexagonal education we once provided our children in that bastion of cultural exceptionalism known as the 1980s, and which seems sadly lacking from today's undoubtedly clever yet somehow less challenging children's programming:

The following is a brief survey of some elemental properties of hexagons, and why they might be useful. It is not intended to be a comprehensive treatment of the subject. My specific concern here is with the mathematical properties of hexagons, and, to an extent, their role in the natural world. I have avoided discussing hexagons as they pertain to human culture, religion, history, and other "local" concerns, though there are many fascinating instances of hexagonality and sixness in these areas, and they will no doubt be treated more fully elsewhere at another time.