Nodes by hexnet

Good news everyone. The time has come, I believe, to announce the existence of this new and improved HEXNET site. The site has actually been up for several weeks now, and constitutes the third major iteration of Hexnet.org under the neohexagonal regime that began in 2010, replacing last year's tragically doomed Drupal 7 version.

First and foremost, as promised, I have finally moved OFF DRUPAL. Drupal is dead to me now, and shall forever remain so. Let us never discuss it again. Beyond that, the most obvious change, I think, is that the site now looks completely different. The main point I want to make about this is that the new layout is much closer than its predecessor to what I intended when I first started working on the Drupal 7 version last year. The ultimate result of that earlier redesign should be seen in retrospect as having been a tragic aberration. I became fixated with some very flawed ideas vis-a-vis how the UI should work, and pursued them long after it should have been clear that they weren't viable. Even after some much-needed fixes earlier this year, it remained, for me at least, practically unusable. It was a bad layout, I was wrong to make it, and I apologize for having done so. The new layout, conversely, should provide a pleasant and fully responsive experience on a wide variety of clients. I have tested it to my satisfaction on both iOS and Android browsers, and it is even marginally functional on IE 8 (for fuck's sake please stop using IE 8). There are definitely aspects of the design I consider sort of unfinished, and which I will continue working on in the coming weeks and months, but it meets my baseline requirement of presenting the site's content in a readable, navigable format that—hopefully—does not actively annoy the reader, and for this reason alone represents a vast improvement over its predecessor.

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.

The truncated octahedron is the fourth order permutohedron, and forms constituent cells in higher-order permutohedra. Its vertices represent every combination of the coordinates 1, 2, 3, and 4 in 4-space, in same way that a hexagon can be embedded in 3-space with vertices at every permution of 1, 2, and 3 (i.e., bisecting a cube spanning coordinates 1, 1, 1 to 3, 3, 3). It is also, along with the lowly cube, one of only two space-filling uniform polyhedra, and is one of only five regular-faced convex polyhedra able to do so—along with the aforeementioned cube, the triangular and hexagonal prisms, and the gyrobifastigium, whatever the crap that is.

Here we see the illustrious cuboctahedron, or vector equilibrium. Along with the truncated octahedron, it can be considered, in some sense, a three-dimensional analogue to the hexagon. Though it has no hexagonal faces, the cuboctahedron can be seen to consist of four hexagonal rings or planes arranged in the manner of tetrahedral symmetry. That is, if one took a tetrahedron, replaced its four faces with hexagons (as for instance with a truncated tetrahedron), and collapsed all four hexagonal sides so that they all shared a common center, the vertices of the hexagons would describe a cuboctahedron, with each vertex shared between two intersecting hexagons, collapsing the original 24 vertices of the four hexagons into the 12 vertices of the cuboctahedron. (Likewise, of course, the cuboctahedron can simply be seen as a sort of "expanded" tetrahedron, with four of its eight triangular faces representing the original four faces of an inner tetrahedron, the remaining four triangles representing its four vertices, and the square faces representing its edges.) Tetrahedral symmetry being the simplest type of polyhedral symmetry, and the only one suited to this sort of fitting together of hexagonal planes, the cuboctahedron represents a unique extension of and analogue to hexagonal symmetry in three dimensions.

Here we see the town of Grammichele, Sicily, built in the traditional hexagonal style. Located in the province of Catania, it was founded in 1693 after the nearby town of Occhialà was destroyed by an earthquake.

I want to take a moment to consider the conspicuous return recently of the so-called hexagon "trend." We first wrote about this phenomenon in a fairly forgettable post from 2010, but in the past year or so it seems to have resurfaced, more forcefully and more explicitly than ever, and we would be remiss in our obligations to the hexagonal community if we did not at least make note of it.

I don't really feel like carrying on with any sort of extended commentary on the matter this time—the trend speaks for itself, and its merits should be obvious to even the most casual adherent to hexagonal principles. It will suffice, therefore, to simply compile a list of recent articles on the subject, and let the reader draw what conclusions they will from them. Though it should be understood of course that we don't see this as a "trend" so much as a central—indeed, perhaps the central—turning point in the entire enterprise of human civilization. Our proverbial children and grandchildren, looking back on this era, will not be asking about the time hexagons were "trending," they will be asking about the time hexagons, as a concept, were finally rising above the clutter and minutiae of the collective human memescape to turn the very course of human civilization, putting us on our inevitable path towards a glorious and unimaginable hexagonal destiny.

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.

“We’re all going to be computers soon. Thank goodness.” – Hans Oberlander

Exciting news everyone,

Several weeks ago I was contacted regarding an opportunity for our involvement in promoting an upcoming show of hexagonal import being put together by V Squared Labs for the Dutch DJ Don Diablo. I had of course never heard of either of these outfits before, and was hesitant to wade into artistic and cultural waters so very foreign to my traditional areas of expertise, but nonetheless it struck me as exactly the sort of aesthetic articulation of hexagonal principles that I'd been quite interested in pursuing for some time now, but which I'd hitherto lacked an appropriate outlet to engage in, at least on any meaningful scale. The show, appropriately titled "The Hexagon," involves a hexagonal visual synthesizer—which I'm told carries with it a number of important technical advantages in addition to its obvious aesthetic import—and will be be performed live for the first time at Freshtival on 19 May, in Enschede, The Netherlands. SO MARK YOUR CALENDARS if you're going to be in the Low Countries next weekend for some reason.

Yes, I have returned from an AWKWARDLY LONG BREAK after my last blog post to once again announce HEXAGONAL AWARENESS MONTH. Since I do not, for the time being, want to be associated with this site at all, on any level, I have created a NEW SITE dedicated to this, the most glorious of all awareness months, mostly cobbled together from pieces of other sites I am currently developing. [Edit 2013-11: This is in reference to the previous, Drupal 7 version of this site, let us never speak of it again. The present incarnation is in fact mostly derived from the HAM site, which itself is no longer maintained and will be merged back in with the Hexnet site at some point. [Edit 2014-02: The HAM site has been slightly rehabilitated in anticipation of this year's festivities, and there are no immediate plans to eliminate it. Though it is still pretty lame.]]

It is fairly minimalistic, both like, in the good way, and in arguably the bad way. I am aware that the "forum" in particular, though keeping with the overall minimalism of the site, does not really come close to the baseline standards people expect from web fora in this day and age. Nonetheless it seemed preferable to confusing people with the "post index" terminology I have used here. I remain committed to my vision of content sites where there is no underlying difference between blog-style posts and forum-style posts, and look forward to more effectively implementing this scheme in the future. [Edit 2013-11: I no longer adhere to this paradigm at all, and have no intention of implementing it now or ever.]

I am informed by some interblags that March is Hexagonal Awareness Month. After some reflection, I have decided that yes, we will go with this.

HAPPY HEXAGONAL AWARENESS MONTH.

Please spread the word far and wide, that all may learn of hexagons and be aware of them. Also, feel free to use this tasteful logo I just made to celebrate this august occasion.

Elect Hexagonbot: For a prosperous and hexagonal future.

“The ancients who wished to illustrate illustrious virtue throughout the Kingdom, first ordered well their own states. Wishing to order well their states, they first regulated their families. Wishing to regulate their families, they first cultivated their persons. Wishing to cultivate their persons, they first rectified their hearts. Wishing to rectify their hearts, they first sought to be sincere in their thoughts. Wishing to be sincere in their thoughts, they first extended to the utmost their knowledge. Such extension of knowledge lay in the investigation of things.” – Confucius, The Great Learning

Today would be a good day I think to take a break from not-updating this site to remind everyone that this year's Hexagon Project at Interdependence Day Scranton is currently underway. I hadn't been planning on writing another post about it this year, but of late I received this lovely promotional poster, which I thought at least worth relaying to the wider hexagonal community. In commemoration of this year's festivities, I have also added to the Hexagonal Library a PDF of a presentation by Hexagon Day organizer Beth Burkhauser to the National Art Education Conference, "Global Interdependence and Art Education: Where Hexagons Make the Connection." More musings on the project can also be found in last year's blog post on the subject. In general of course HEXNET.ORG supports all efforts to relate the civic virtues of hexagons to their more familiar geometric ones, as both are clearly manifestations of the same unified underlying reality.

We at HEXNET.ORG have been experiencing a rash of weird, bot-like comments on various and sundry posts over the past several days. We're not sure what caused this evil to be unleashed on us, but it's possible Hexagonbot is at least partly responsible, as it seems to have attracted a lot of bot attention on Twitter. Anyway, our previously airtight CAPTCHA has failed us, so until we resolve this, anonymous comments will be moderated. We hope to return to normal as soon as possible, but we simply do not have time to deal with this crap at present.

Good news everyone.

After literally hours of research and development at our new GLOBAL HEXAGONAL HEADQUARTERS in Worcester, Massachusetts, Hexnet is pleased to announce the release of the world's first fully-automated hexagonal awareness retweeting service: Hexagonbot v.1.0.

Hexagonbot is a companion service to our Hexagonal Awareness Project reblogging blog on Tumblr, and is part of a broader effort by the global hexagonal community to bring hexagonal services to a variety of blogging and social media platforms.

Hexagonbot implements a proprietary algorithm through which it downloads a feed of hexagon-related tweets, and then retweets them. It's a very sophisticated process. Hexagonbot is not currently programmed to interact with other Twitter users in any way, though we anticipate this functionality may be added to future iterations of the platform.

I first became aware of the Hexagonismo y Arquitectura blog earlier this week, when I saw it on a list of traffic sources to this site. At first I didn't know what to make of it, being a bit too Iberian for my linguistic competencies, but upon further inspection it seemed to be affiliated with a "Hexagonismo" movement operating out of Ibiza, more of whose work, I think, can be found at Hexperience.org. [Edit 2013-11: This site unfortunately seems defunct at this point.]

This 2001 article from the Colonial Williamsburg Journal offers a fairly eloquent exposition of certain aspects of hexagonal symbolism that are worth noting. I found the hexagons themselves fairly interesting too—particularly in contrast with the rather hexagon-poor architecture of colonial New England that I'm more familiar with. I've never been a great fan of Anglicanism, or High Church aesthetics in general, but the Puritans clearly couldn't design an interesting building if their lives depended on it—indeed, they probably would've interpreted a hexagonal cupola as a sign of idol worship or something.

This intriguing hexagonal wind power project out of Japan came to my attention recently. Its precise details elude me, as the website is not particularly informative, but it seems to involve forcing air down some sort of central column and compressing it into a turbine on the ground, This evidently has advantages over conventional wind turbines. I'm not sure. But it involves HEXAGONS.

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.

Selected satellite images of SECRET LIBYAN HEXAGONS, near the city of Kufra in southeastern Libya. Locus: 24.13 N, 23.25 E.

We have launched a new reblogging blog on Tumblr for the purpose of raising hexagonal awareness and hexagon-consciousness among the Tumblr community. Going forward, I expect this blog to be an important aesthetic precursor to the eventual promulgation of HEXAGONALISM as a self-sufficient artistic tendency.

"Ten is doubtless a convenient number of fingers to have, though men have gotten along with less and a few people have been born with more. But as the purely arbitrary unit which determines the form of our numbers, it was a miserable choice." – F. Emerson Andrews

"Do not disturb my circles." – Archimedes

It's that time of year again—time to start thinking about DOZENAL PI DAY.

Overview

Regardless of its radical prejudices, Pi Day has become, in recent years, an annual cultural event of some magnitude, and the occasion for a certain nominal enthusiasm about "math" that people seem to find meaningful. At the very least, it seems to remind us of our own cleverness, which is something I have nothing against in principle.

Yet I am more convinced than ever that, unless we remain firmly mindful of what separates the arbitrary and symbolic in mathematics from the truly meaningful, Pi Day will ultimately remain a rather vacuous and misguided holiday. Thus, I once again propose the observation of DOZENAL PI DAY on 18; March (decimal 20th), to raise awareness of the superior dozenal radix, while also bringing attention to the underlying meaninglessness of our present decimal notation.

Here we see an arrangement of hexagonal clouds photographed by the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor aboard the Terra satellite on 6 June, 2002.

“Of course you forget, Peter, I was present at an undersea, unexplained mass sponge migration.”

I feel something must be said about this astrological "sign shift" business that has set the 'tubes abuzz in recent days. There being twelve astrological signs ("Ophiuchus" notwithstanding) arranged around a central point, the issue is clearly one of both dozenal and hexagonal import. Despite the fact that I'm not all that into astrology (though I've learned enough over the years to inform my general fluency in Western occult symbolism), I feel the hexagonal principles at stake must be defended.

First of all, this is of course old news. I don't really understand why it keeps popping up in the media every few years, but this is simply the latest incarnation of a story that we've all been well aware of since classical antiquity, and that certainly should not come as news to anyone, anywhere. "Ophiuchus" has always been there, the twelve signs have never been symmetrically arranged in exact 30-degree sections, axial precession has always been going on, we always knew about it, nobody has ever cared, and nobody cares now.

Here we see a selection of illustrations relating, I am told, to the so-called Chaldean Oracles of "Zoroaster"—which are, of course, neither Chaldean, nor Zoroastrian, nor oracles.

I've been unable to ascertain the origin or IP status of these images. But they are pretty amazing, so I'm posting them anyway, pending receipt of further information on the matter. They certainly look like olden-style woodcuts, or at least hand-drawn illustrations, but could just as easily be modern, and for all I know computer-generated. As always, if you have any information on the source of these images, please do not hesitate to contact us. Thank you.

Here we clearly see the hexagonal structure of the Discovery One spacecraft from 2001: A Space Odyssey. The engine housings as well as the connective structure between the engines and the habitation spheroid have an overtly hexagonal form to them.

Here we see the hexagonal structure of the RGB color system, as described on a flat hexagonal plane by the HSV coordinate space at V=1.

Human perception of color is tied to idiosyncracies in the hominid nervous system, and conceivably tetrachromats and pentachromats perceive far more complex spectra of colors that cannot be modeled on hexagonal principles at all. It may be, however, that the hexagonality of the human color spectrum is linked to some sort of innate neurological tendency towards hexagonal information processing. Indeed, it could be that the hexagonality of our color spectrum mirrors less-obvious hexagonal symmetries elsewhere in the subtle mechanisms by which our nervous systems structure reality.

Here we see several illustrations of John Leighton's proposal to divide Greater London into a series of hexagonal boroughs, as described in his book The Unification of London: The Need and the Remedy.

This just in, from SWEDEN:

Hexagons have won this year's Nobel Prize in Physics. More specifically, Andre Geim and Konstantin Novoselov won the prize for their work with hexagons.

In summary, for those who do not follow such things (and I have noticed that "graphene" is among the top keywords bringing people to this site, so it is quite possible you do follow such things): Graphene was "discovered," or if you will isolated, by Geim and Novoselov in 2004, by peeling off layers of graphite with scotch tape. It is essentially an indefinitely large aromatic molecule, and the flat, two-dimensional form of the buckyball or the buckytube. For a variety of reasons I won't get into here, it has numerous potential applications in electronics and nanotechnology, and is quite interesting all around.

It had never actually occurred to me prior to 2004 that graphene needed to be "discovered." I had always been taught that graphite, as a major allotrope of carbon, consisted of one-atom-thick sheets of carbon atoms arranged in a hexagonal lattice, and that this had been known for quite a while (since at least the advent of X-ray crystallography). Thus it really shouldn't have been too much of a conceptual leap to assume that, in fact, such sheets existed (though apparently the prevailing view was that the sheets would "roll up" when isolated). I still have a hard time wrapping my mind around the fact that it took until 2004 to actually discover this. Particularly with all the research that's gone on with fullerenes since the '80s, you'd think somebody, somewhere, would've taken the time to actually isolate a sheet of graphite. Apparently it is somewhat difficult, but crap I have scotch tape and pencils lying around, by all rights that Nobel could've been mine.

I have recently been made aware of a contest being held on Kotaku Australia regarding "hexagons that should not be." Apparently inspired by the hex tiling in Civilization V, they are giving away a free copy of the same to whoever most impressively photoshops a hexagonal shape onto an object that is not normally hexagonal. Unfortunately, the contest ends Monday night, so there's very little time to spread word of this to the larger hexagonal community. I wish I'd heard about it sooner. Anyway, samples of submitted work can be found on the following pages, and are most interesting:

• Some Of Your Civilization V Competition Entries
• More Civ V Competition Entries
• Your Six-Sided Take On The World

[Edit: The winners and runners up can be found here and here, respectively.]

I find it fascinating, obviously, both as a Civ fan (still trying to get 5 running under Wine!) and as a hexagonalist. Creative exercises of this sort, inducing people to think artistically about the roles and potential roles of hexagons in everyday life, are potentially very valuable tools in expanding hexagonal consciousness. It reminds me in some ways of last month's Interdependence Day Hexagon Project—in both cases, the hexagon is explicitly employed not for some incidental aesthetic reasons, but for its very hexagonality itself. This can only be, I think, a positive trend in our society, as we as a species come to terms with the implications of our glorious hexagonal future. I am hoping to see more and more of this sort of thing in the coming years, as hexagons rise to ever greater prominence in the collective human experience, perhaps leading at some point to the formation of an explicitly "hexagonalist" movement in art and design.

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.

Here we see, apparently, a cuneiform hymn to Iddin-Dagan, lugal of Larsa, from the 19th century BC, inscribed on a hexagonal artifact of some sort. It is my understanding that the Sumerians of yore made similar inscriptions on otherly-shaped blocks as well—square, pentagonal, etc.—but I would like to think there was some religious significance to the choice of a hexagon here.

Word on the street is that today's Google Doodle features an interactive truncated icosahedron in recognition of the 25th(.) anniversary of the discovery of the buckminsterfullerene (and, by extension, of fullerenes, nanotubes, &c. in general).

I normally don't actually pay attention to doodles—I find them rather played out at this point, and with the rise of integrated browser search fields, who even goes to Google's home page anymore. But it is worth noting that—according to sources who follow these things more closely than I do—this is ostensibly only Google's second "interactive" doodle, after this past spring's Pac-Man episode. THIS IS NOT AS INTERESTING AS PAC-MAN, but it's presumably indicative of the moral and intellectual gravity they assign to the discovery of fullerenes that they would consider the event worthy of such rarified and exotic treatment. The spinning of the fullerene here is not merely meant to amuse us for half a second, but rather to instill in us an understanding that the event being commemorated is, on balance, probably more important than, you know, the Australian federal election, or Chinese Valentine's Day, or what have you.

The following is a transcript of Chapter VI of The Pythagorean Triangle: Or, the Science of Numbers by the Rev. Dr. George Oliver, a noted 19th century English Freemason. I have compiled this from several sources, including the Internet Archive, Google Books, and my own copy of the work. I slightly reworked the layout where necessary, including sort of rejiggering the chapter opening here. I've made several minor typographic modifications as well, such as removing the spaces before semicolons (a form of punctuation the good Doctor seems to have had a particular fondness for).

The rest of the book consists of similar musings on the other natural numbers, up to ten. I do not necessarily agree with Dr. Oliver's views, either on the hexad in particular, or on Christian numerology in general. I certainly don't agree with his cryptodecimalist ontology. But it is nonetheless an interesting treatment of the subject, and I offer it here as a noteworthy historical artifact of antiquarian hexagonal thought.

A PDF version of the full book can be downloaded from our Hexagonal Library.