Hexagonal Blogs

"Where is it writ large that talking monkeys should be able to model the cosmos?" – Terence McKenna

I remember reading a short but eloquent endorsement of hexagons by Alan Watts once, probably seven or eight years ago. Since I wasn't particularly interested in hexagons at the time, I quickly forgot about it, and where exactly I had read it. However, times have changed and my hexagonal thinking has evolved, and with the advent of Google Books and all, I recently decided to look into the matter further. After an exhaustive search, I found two mentions of hexagons in Watts' books, which I shall now share with you, and duly expound upon.

I am not entirely sure either of these is in fact the passage I remember reading, but if it were either of them it would be this first one, which can be found in the essay What on Earth Are We Doing?, from the collection Cloud-Hidden, Whereabouts Unknown:

"The whole secret of life and of creative energy consists in flowing with gravity. Even when he leaps and bounces our cat is going with it. This is the way the whole earth and everything in the universe beehives.*"

The footnote then explains:

"* Harrumph! Excuse the pun, but it is important, because bees live in hexagonal as distinct from quadrilateral structures, and this is the natural way in which all things, such as bubbles and pebbles, congregate, nestling into each other by gravity. It will follow, because 2 x 6 is 12, that—as Buckminster Fuller has pointed out—a number system to the base 12 (duodecimal) is closer to nature than one to the base 10 (decimal). For 12 is divisible by both 2 and 3, whereas 10 is not. After all, we use the base 12 for measuring circles and spheres and time, and so can "think in circles" around people who use only meters. The world is better duodecimalized than decimated."

This is, significantly, the first instance I have ever come across of anyone besides myself linking hexagons with dozenals. Watts was a keen observer of patterns in nature, and though he makes a rather abrupt arithmetic leap here from hexagonal packing to base twelve, I have no doubt that he understood—at least intuitively—the underlying hexagonal relationships that unite these two concepts.

But intriguingly, in Tao: The Watercourse Way, Watts writes the following:

"Even abstract and nonobjective paintings have the same forms that may be found in the molecules of metals or the markings on shells. As soon as this beauty is pointed out it is immediately recognized, though we cannot say just why it appeals to us. When aestheticians and art critics try to explain it by showing works of art with Euclidean diagrams superimposed on them—supposedly to demonstrate elegance of proportion or rhythm—they simply make fools of themselves. Bubbles do not interest one merely because they congregate in hexagons or have measurable surface tensions. Geometrization always reduces natural form to something less than itself, to an oversimplification and rigidity which screens out the dancing curvaceousness of nature."

This is a somewhat different, possibly even contradictory sentiment, and one that I believe is well worth addressing in the context of hexagonal awareness.

Anyone who is familiar with Watts' work will no doubt remember his general rap on geometry in nature—extolling the virtues of "wiggly" versus "straight" lines, et cetera. I don't disagree with his sentiments in this regard, but I do believe there may be a semantic difference between us here. Clearly, the wiggles and such of the "natural" world are but a geometry of a more subtle order than the more elemental forms we are familiar with in the human-conceived world (such as, for instance, certain simple convex polygons). In terms of the geometry of the living, growing, cellular world that dominates the surface of the planet we inhabit, this subtle geometry extends in several conceptual dimensions: on the one hand, there are the well-known geometrical and mathematical patterns that emerge in complex living organisms—the close-packing of seeds in a flower, the golden ratio in the curve of a nautilus shell or in the iterations of leaves, et cetera—and on the other, there is the inescapable fact that life itself is composed of biomolecules that behave, on the smallest scales, in a strictly geometrical and mathematical fashion.

Regardless of the particulars, it should be self-evident that even the wiggliest and most superficially "non-geometrical" natural systems can be understood as having geometrical order to them—geometry, after all, should be able to describe all forms and structures in space. Such geometrical order is, in turn, part of the larger continuum of mathematical order that governs the cosmos.

This is obviously the basic premise of modern science, in the broadest sense: that the universe can be understood mathematically. To the extent that some people seem to shy away from this fairly self-evident fact, I can't help but feel it is probably motivated more by an emotional distaste for "math" than by any real ontological disagreement. At any rate, it shouldn't be a very controversial statement to say that we live in a universe governed by mathematical principles—however subtle they may be—and thus that all things and systems within the universe can be understood mathematically. They can certainly be understood in other ways as well, but I do not feel it diminishes or subtracts from their grandeur or mystery to concede that, ultimately, they are mathematical processes. I would indeed go so far as to say the universe itself is mathematics, but that is perhaps an ontological question best left for another time (and which may in fact be meaningless).

Thus, I have never been entirely comfortable with the characterizations of those who would seem to limit mathematical interpretations of reality to only the crudest and most stilted expressions of constructive geometry. While I feel this is primarily a semantic disagreement, and that most who make these characterizations would not seriously challenge the mathematical underpinnings of reality (math is a pretty broad concept after all), I do feel that it points to the slightly different ways in which we relate to the universe. In passages such as this, Watts and others seem to suggest that there is a way to the flow of nature that is ultimately ineffable and undefinable, but which can be understood outside of the trappings of logic and mathematics. I would not deny that such a way clearly exists, but it is my strong suspicion that it need not remain beyond the realm of mathematical understanding. This is not to say that we should lose any sleep over defining the Tao mathemtically, or that it would even be feasible at this point in our understanding of the universe to do so, only that there must be some sort of mathematical order to it that is open to discovery and definition—probably involving hexagons.

This is all possibly rather beside the point though, since Watts clearly implied in the second quotation that the hexagonality of bubbles was interesting, only that they are not the limit of our interest. And I generally agree with that (though obviously, I find the hexagons particularly interesting). But I don't really believe we can divorce the hexagonal issue from our considerations of broader patterns in nature, or their aesthetic implications, however we choose to conceptualize them. Indeed, it is the very prominence and ubiquity of these hexagonal patterns—as Watts himself alluded to in the first quotation—that lead us to suspect they are indicative of a deeper order in nature. I will not go so far as to posit any sort of primal supremacy to the hexagon—the very concept is, after all, contingent on any number of mathematical axioms—but I will suggest that, of all possible realities, within all possible mathematical systems, we seem to find ourselves enmeshed in a uniquely hexagonal one, where hexagonal and parahexagonal principles play a conspicuous role in the spectrum of mathematical realities governing our existence.

To take this point a step further: if the set of all possible mathematics is greater than the set of mathematics that meaningfully describe our reality (and I assume that it is), the hexagon could be construed as something analogous to a musical key. As we say that a particular musical piece is played in the key of C, so we could say that our reality is a mathematical system played in the key of hexagons. That is, the hexagon could be some sort of ontological bridge between all possible realities and the broadest contours of our particular reality, or if you will perhaps something akin to a lense of sorts, focusing reality into a coherent, self-consistent mathematical universe. In this sense, there could be countless other realities—branching off at the most fundamental ontological level—that coalesce around mathematical objects that we, perhaps by definition, cannot even conceive of.

I don't actually understand what that would even mean, but I find it an interesting idea.

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.

To be sure, although 2π features prominently in all areas of mathematics dealing with angles and arc, regular π comes up a lot too. But dealing with τ in these situations would never be more of an issue than simply dividing τ by two or what have you. Again, the important point is that τ should properly be considered the more fundamental value, so even in cases were π leads to less verbose or more eloquent expressions, it would still perhaps be more informative and pertinent to use τ. Though perhaps not in all cases, I don't know. At any rate, there is no need for π to be "abolished" or anything, in the way that, say adopting dozenal notation would probably require the death—in practical usage—of the decimal radix. Both values can go on to have long and fulfilling lives. τ can be propagated in common usage by simply defining it and using it whenever π is required—there is no need for any sort of radical, society-wide transformation. Indeed, I can easily envision a situation where τ (or some other symbol, if τ ends up being impractical) slowly overtakes π as a the preeminent circle constant over the course of decades if not centuries.

The argument was presented so well by Mr. Palais and so exhaustively by Mr. Hartl that there scarcely much more for me to add on the matter—except perhaps to note that, of course, the prominence of six in the value of τ definitely highlights the hexagonal nature of the circle. Every time one considers the value of τ one is reminded that the circle differs from the hexagon by only 0;3494 radians of perimeter. Which is, in this blog's opinion, a worthwhile thing to remember.

Anyway, it is my intention going forward to use τ instead of π in all applicable formulae, expressions, equations, and what have you throughout Hexnet.org. I'm not sure how well I'll stick to that plan, but this certainly seems like an easier transition to make than the whole dozenal enterprise would be. One can always simply define τ on the fly for those who are unfamiliar with it, while it is very difficult to just go right out and use dozenal notation without at least explaining yourself first. But I consider both of these efforts to be part of the same grand movement to bring clarity and hexagonal sensibilities to modern mathematical semantics.

I am not yet sure what the ramifications of this development will be for Dozenal Pi Day (March 18;), since there is conspicuously no 34th day in June—in decimal or dozenal. Mr. Hartl apparently celebrates decimal "Tau Day" on June 28th (a perfect day in decimal), but my own preference would be to have a "universal" Tau Day on February 27., being 365./τ, though that is probably a bit too cerebral for most people, and it sacrifices the dozenal element. At any rate, I shall move forward with plans to celebrate Dozenal Pi Day in 11B7;—it is still τ/2 after all—and if nothing else I can use it as an opportunity to denounce both decimalism and π.

Anyone following the hexagonal news of late has no doubt noticed a flurry of stories related to the upcoming release of Civilization V, and in particular its new hexagonal tile system. In this blog's opinion, the adoption of hexagonal tiles by the Civ franchise is a long overdue development, and frankly one that should've been incorporated into at least the last two Civ releases. Indeed, many DOS-based strategy games have used hexagonal tiles going back to the late '80s, and one has to wonder why Civilization ever used square tiles. (I remember playing the original Civilization in the early '90s, well before the advent of my own hexagonal illumination, and thinking that, in fact, it would be better with a hexagonal grid. I can only imagine part of the media excitement here—aside from just the general awesomeness of hexagons—is due to the fact that many, many other people have had the exact same thought over the past twenty years.)

Even the most casual player of Civilization will immediately appreciate the advantages here of hexagons over squares. In the olden square-based system, a diagonal move between the corners of two squares was treated the same as a move across a square's von Neumann neighborhood to an edge-adjacent square. This had the effect of, among other things, making edge-adjacent movement fairly useless, except where necessary due to terrain, opposing units, et cetera. Whenever possibly, particularly when patrolling or exploring, it simply made sense to always move diagonally, and to move in a diagonal zigzag to traverse two or more edge-adjacent squares. Furthermore, the square system necessitated weirdly truncated city radii—the familiar 20-squre city radius being, essentially, a 5x5 square grid, minus 4 corner squares and 1 center square. As any Civ player knows, this leads to awkwardly-placed cities that either overlap or leave dead space between them, and to the preposterous fact that it takes three moves to leave a city radius edgewise while it takes only two moves to leave it cornerwise. A dangerous situation often arises where an enemy unit with two or more moves can enter from outside a city radius and attack the city in one turn, where if the same unit had been approaching edgewise it would have required an additional move. This is of course merely one of the idiosyncrasies one learns when playing Civilization—it's not a "flaw" per se, it's just a fact of life in the Civ universe—but it is certainly part of an overall tapestry of inconsistency and irregularity in gameplay necessitated by this bizarrely antiquated reliance on such an inefficient and inappropriate tiling system.

Conversely, in a hexagonal tiling, all moves are between six equally-spaced neighbors—there are no diagonal, corner-to-corner moves. This topologically simplifies and balances close-tile moving, as there is only one path between two adjacent tiles. Also, it allows for smooth-radius cities. I do not know how city radii will work in Civ V—rumor has it that there have been major overhauls to vast areas of game play (though the media seems mostly fixated on the hexagon issue)—but a two-tile radius on a hexagonal map would create eighteen-tile enclosures, which is fairly close to the traditional twenty. A hexagonal radius, in addition to "solving" the issue of inconsistent city radii, also allows cities to be tiled indefinitely on an ideal map. For the first time, city spacing will not be a compromise between spaatial efficiency and lebensraum. Cities will be able to be constructed back to back, in an indefinite tessellation, with no leftover space. (Again, assuming there hasn't been some underlying systemic change to the nature of city radii.)

None of these hexagonal principles are actually new to the world of civilizoidal gaming, of course, since Freeciv has provided a free, open source Civilization alternative with hexagonal tiles for years. But it is good to see the original franchise finally getting on board with our glorious hexagonal future.

Do you have money? Do you enjoy things? Why not order Civilization V today!

It looks like the hexagonal zeitgeist is finally floating to the surface of our collective media environment. Two stories in particular off of our hexagonal news feed caught my eye this morn:

Geometry lessons
". . . Just as the 1950s will always be remembered as the decade of atomic-inspired motifs, so may the early decades of the 21st century be remembered, by those of pop cultural bent, as belonging to the hexagon."

The Hexagon Harley 1.4301
"Six is the magic number on this handcrafted custom Harley Davidson by Horst Dzhangmen. Everything but the engine, a HD Shovelhead 1340 cm3 and the frame, has been cut and built into perfectly shaped hexagons and this exceptional work took its creator three years of hard work to achieve this perfection."

That's goddamn right.

One cannot, I believe, overstate the aesthetic value of hexagonal geometry. Its visually-appealing proportions, "nature"-evoking associations, combined with its outright utilitarian, mathematical functionality makes the hexagon a formidable motif for any commercial or cultural venture. It is remarkable how often hexagons are included in packaging, logos, et cetera, simply to convey the sense of balance and proportion they tend to evoke. For instance, yesterday I was walking through a bookstore, and I saw the side of a book on a table called The God of the Hive. Before I even saw the cover, I knew there would be a hexagonal pattern on it. I didn't even look at the cover thinking "Will there be hexagons on this book?" I looked at it thinking "I would like to see the hexagons on this book." And lo, there were hexagons. They weren't even like proper honeycombs per se—they were, as I expected, just geometrical figures in a stylized tessellation. That's all that is necessary to convey the hexagonal symbolism of the honeycomb. For some sort of deeply psychological and ontological reasons, we identify with hexagons as we identify with little else in geometry or mathematics, and this is why we shall, I believe, see a greater and greater hexagonal presence throughout society as we move into a more geometrically conscious age.

It is interesting to consider that perhaps the recent rise of hexagonal geometry directly follows a general return to visual imagery and spatial relationships in modern times, due both to broad social trends and the rise in certain enabling technologies (computers, television, et cetera), somewhat in the manner described by the late Leonard Shlain in his book The Alphabet Versus the Goddess. One could even, I believe, see the hexagon as the very embodiment of humanity's contrasting ways of conceptualizing the world—the hexagon is both an abstract, logical construct and a concrete, visual form in our sensory realm. This is one of many reasons I see the hexagon as the key to the very future of human consciousness itself—it is nothing less than the bridge between left and right brain, logic and intuition, math and art, et cetera, et cetera. And so on.

This is old but worth considering:

• Is the universe a dodecahedron?
• Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background

I have been completely unable to find any sort of followup to this since 2003. The whole idea may have been discarded by now, I'm not sure, but considering that Plato speculated—for no particular reason modern science is aware of—that the universe was shaped like a dodecahedron nigh doz. 1,500 years ago, this is obviously a significant conjecture to put forward.

While the other four platonic solids have traditionally been associated with the four classical elements, the dodecahedron has been identified with the universe, or the aether. Did the ancients have some insight into the cosmological structure of our universe that we are only rediscovering now? Maybe! Or maybe the whole idea is crap. Who knows! But the hexagonal implications are obvious—twelve being a multiple of six, the twelve faces of the dodecahedron forming six symmetrical pairs, et cetera.

I am hopeful that eventually more data from WMAP or its successors will shed further light on this hypothesis.