The permutohedron

Posted by hexnet ::

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.

The Hexagon

Posted by hexnet ::

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

Don Diablo presents The Hexagon 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.

RYGCBM hexagon

Posted by hexnet ::

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.


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.

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

Saturn-style hexagon "recreated" in lab

Posted by hexnet ::

Interesting article from Science's popular science outfit about the Saturnine hexagon. While it is certainly an intriguing development, and a step forward from the well-known "spinning bucket" experiments, they should get back to us when they've made a hexagon 15,000 miles wide that lasts for 30 years. (Which, for practical purposes, one wouldn't expect any time soon.) Fluid dynamics can be vastly different at different scales. I would also like to know what the viscosity is of this "water" they speak of relative to the atmosphere of Saturn.