3 dimensions and their 6 directions
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From: nuny on 3 Apr 2010 17:01 On Apr 2, 4:56 am, James Dow Allen <jdallen2...(a)yahoo.com> wrote: > On Apr 2, 11:43 am, Danny73 <fasttrac...(a)att.net> wrote: > > > But here on the three dimensional earth grid it > > is 6 directions --- > > North,South,East,West,Skyward,Earthward. ;-) > > Let me try to inject a serious question I have into > this thread. ;-) > > In a hexagonal grid, each point has six immediate neighbors; > what should their names be? (I asked this question before, > with the only answer being the ugly "solution I was > already using: West, Northwest, Northeast, East, SE, SW.) That assumes you orient it east-west. You could just as easily orient it north-south, or ignore "conventional" map coordinate systems completely. > Hexagonal grids have big advantages over square grid > but are seldom used. It sounds silly, but perhaps > lack of the msot basic nomenclature is one reason! NSEW nomenclature is tied to the rotation of the Earth. What do you want to tie hex map nomenclature to? What is *available* to tie it to? Note the origin of the word "orient". Mark L. Fergerson
From: BURT on 3 Apr 2010 17:44 On Apr 1, 4:45 pm, BURT <macromi...(a)yahoo.com> wrote: > Directions are: > > Up down > Right left > Front back > > When we move through space we are moving in a 6 directional space grid > in only 3 of these directions. > > Mitch Raemsch For the 4th dimension surface there is two more directions in the universe for circling the hypersphere. Hypersphere geometry gives 8 directions in its surface. Mitch Raemsch
From: Tim Golden BandTech.com on 4 Apr 2010 08:15 On Apr 2, 7:56 am, James Dow Allen <jdallen2...(a)yahoo.com> wrote: > On Apr 2, 11:43 am, Danny73 <fasttrac...(a)att.net> wrote: > > > But here on the three dimensional earth grid it > > is 6 directions --- > > North,South,East,West,Skyward,Earthward. ;-) > > Let me try to inject a serious question I have into > this thread. ;-) > > In a hexagonal grid, each point has six immediate neighbors; > what should their names be? (I asked this question before, > with the only answer being the ugly "solution I was > already using: West, Northwest, Northeast, East, SE, SW.) > > Hexagonal grids have big advantages over square grid > but are seldom used. It sounds silly, but perhaps > lack of the msot basic nomenclature is one reason! > > James Dow Allen Yes, and in fact only three directions are needed on the 2D hexagonal grid. Likewise just four directions are actually necessary to address 3D space. These are the simplex coordinate systems, whereby we can return to our original location by stepping once in each of those directions: http://bandtechnology.com/PolySigned/Lattice/Lattice.html The ray is more fundamental than the line. The simplex coordinate system is already encoded into the real number due to the behavior - 1 + 1 = 0 which generalizes via polysign to both an algebra and a geometry. This geometry in lattice form is different than ordinary geometry, for it still carries its directional origins along with it http://bandtechnology.com/PolySigned/Lattice/P3Signon.gif - Tim
From: spudnik on 5 Apr 2010 19:37 NB, quaternions are not "quadrays" (for an amateur attempt at homogenous co-ordination), but you can "do" special rel. with them (according to Lanczos .-) thus: The "cap & trade" omnibus bill -- what Waxman-Markey should be known as, being so fundamental to the Stupid, economy -- is at least as old as Waxman's '91 bill to ameliorate acid rain. One must really stop and consider, just who really opposes this "last hurrah" for Wall Street (like- wise, the healthcare bill, also under Waxman's House committee, and which, after all, is geared toward funding a smaller aspect of the S-- the economy, already tremendously leveraged by the "voluntary" cap & trade, which the bill would essentially mandate, a la the much-larger, market- making EU scheme). Not so long ago, there was a guest-editorial in the WSJ, which mentioned that an actual carbon tax would achieve the same thing, more or less, as the total "free" market apporach of cap & trade; oh, but, there're certain, so- called Republicans, who refer to the bill as "cap & tax!" Well, before any "reform" of the financial system, why would one put all of one's eggs into such a casino -- especially considering that the oil companies have not bothered to release the carbon-dating "fingerprints" that they use, to determine whether two wells are connected, underground; so, guys & gals, how old is the stuff, on average, anyway? Surely, the green-niks who lobby for "renewable" energy, do not think that oil comes only from dinosaurs, and their associated flora -- all, from before the asteroid supposedly offed them (I refer them to the recent issue of Nature -- several articles that may be related!) Finally, note that, in a sense, the whole world is going a) nuclear, and b) into space, while we are essentially frozen into '50s and '60s techniques in these crucial frontiers. (While some folks dither about Iran's nuke-weapons policy, they are rapidly achieving a full-scale nuke-e and process-heat capbility for industry & infrastructure.) --yr humble servant, the Voting Rights Act o'65 (deadletter since March 27, 2000, when Supreme Court refuzed appeal in LaRouche v. Fowler ('96))
From: Michael Moroney on 7 Apr 2010 17:45
James Dow Allen <jdallen2000(a)yahoo.com> writes: >On Apr 2, 11:43=A0am, Danny73 <fasttrac...(a)att.net> wrote: >> But here on the three dimensional earth grid it >> is 6 directions --- >> North,South,East,West,Skyward,Earthward. ;-) >Let me try to inject a serious question I have into >this thread. ;-) >In a hexagonal grid, each point has six immediate neighbors; >what should their names be? (I asked this question before, >with the only answer being the ugly "solution I was >already using: West, Northwest, Northeast, East, SE, SW.) A hex grid has 3 coordinates. Using your alignment, they'd be North-South, NE/SW, NW/SE. However, they are not independent, if you know any two, the third is defined. Also, nothing special about those directions, turn the grid 30 degrees and you get a different alignment. Also the NE/SW and NW/SE directions are approximate. >Hexagonal grids have big advantages over square grid >but are seldom used. It sounds silly, but perhaps >lack of the msot basic nomenclature is one reason! One disadvantage is that a basic hexagon isn't subdividable into smaller hexagons or easily combined into larger ones. In rectangular coordinates, the map gets divided into small squares. Each square is easily divisible into n^2 smaller squares by dividing each side into n parts. You can't divide a large hexagon into smaller ones. If you want to have fun, extend the hexagonal mapping into three dimensions. There are two ways - the first is to add a Z axis to a hex map, kind of like making a 2D polar coordinate graph into 3D cylindrical coordinates, like stacking honeycombs. The other way is more interesting - add an axis at 60 degrees to the plane of the graph. You now have 4 coordinates for each volume in 3D space. Like the 2D case, you need to know any 3 of them to define a volume region. Once you know 3 the 4th is defined, it's not independent. All of space is divided into 12 sided 3d solids. I don't remember what the shape is called. It is _not_ the platonic dodecahedron with pentagonal faces, but instead, each face is a rhombus. In this shape, all faces and all edges are identical, but all vertices are not identical. |