Tagged: geometry

A polyhex is a plane figure composed of n regular hexagons joined at their edges, in the manner of a regular hexagonal tessellation.

Polyhexes have perhaps achieved their greatest utility in organic chemistry, where they can be used to represent various configurations of aromatic hydrocarbons, but are also often employed in puzzles, logic games, and other recreational mathematical pursuits. A more speculative application of the first several orders of free polyhexes can be found in Patrick Mulcahy's article The Hexagonal Geometry of the Tree of Life, available as a PDF file in our Hexagonal Library.

We seem to have fallen a bit behind in terms of keeping this site updated with RELEVANT MATHEMATICAL INFORMATION about hexagons. This is a deficit we're looking to correct as soon as possible. First up, let us consider the permutohedra. We at HEXNET.ORG have been meaning to write something about permutohedra for a couple of years now, but have never really found a good opportunity to so. WE WILL NOT EXPEND GREAT EFFORT DOING SO NOW. It will suffice to merely describe the concept in conjunction with some helpful imagery, which will hopefully serve as a useful foundation for further geometrical observations and investigations in the near future.

As the harvest season draws nigh on to a close here in the northern climes, I thought it might be a good time to take a look at the year's HEXAGONAL CROP CIRCLES.

Now, I try not to get overly woo-woo here, so let me preface my remarks by assuring you, the gentle reader, that, pending the advent of more compelling evidence, I continue to maintain a strict agnosticism towards the phenomenon of crop formations. But I am comfortable making at least the following two assertions on the subject:

1) There are at least some crop formations from the past thirty years that simply were not made covertly, at night, by a small group of people with boards and crap. Not all of them, maybe not even most of them, but definitely some of them.

2) There is a spectrum of possible explanations for these formations between and beyond the false dichotomy of "pranksters running around fields at night with boards and wires" versus "zOMG ALIENS!" usually put forward by mainstream media and other consensus-reality-builders in our society.

Indeed, as an aside, I would like to point out that I find it highly unfortunate that both crop circles and the more well-documented UFO phenomenon have come to be associated, for little reason, with theories of extraterrestrial visitation. In the case of UFOs, the subjects are conflated so completely that you often hear people asking if one "believes in" UFOs, when they actually mean "Do you believe aliens are visiting Earth in nuts-and-bolts spacecraft from other star systems?" Which are obviously two completely different questions. (And how one could possibly "not believe" there are flying objects that are not, in fact, "identified," is itself utterly beyond my comprehension.) In the case of UFOs, at least there are actual apparent flying vehicles involved, so I can understand the conceptual leap. I have never understood at all how extraterrestrial speculation came into the crop circle issue, except by association with UFO culture, and a general lack of imagination as to the different ways intelligence may manifest itself in this universe. I mean, it could be extraterrestrials. It would be interesting if it were. But this is certainly not by default assumption, nor is it even remotely high on my list of plausible explanations.

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.

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:

In this image we see the symmetry group D₆ of a regular hexagon. The hexagon can be rotated six ways, and reflected six ways. Note that any combination of two or more of these operations will still result in one of these twelve configurations.

In this diagram we see that four of the five platonic solids can be projected in two dimensions as hexagonally-symmetric figures. The hexahedron, octahedron, and icosahedron can all be orthographically projected as regular hexagons, and the dodecahedron can be so projected as a somewhat lopsided yet equal-angled hexagon (not shown), or as a hexagonally-symmetrical dodecagon (shown here). The tetrahedron—or "freak" polyhedron if you will—can of course be projected as an equilateral triangle, which although not truly hexagonal is still of the same general angular family.

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.