On Apr 19, 2:51 am, Ostap Bender <ostap_bender_1...@hotmail.com> wrote: > On Apr 18, 1:16 pm, BURT <macromi...@yahoo.com> wrote: > > > > > On Apr 8, 5:13 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com> > > wrote: > > > > On Apr 7, 5:45 pm, moro...@world.std.spaamtrap.com (Michael Moroney) > > > wrote: > > > > > James Dow Allen <jdallen2...@yahoo.com> writes: > > > > > >On Apr 2, 11:43=A0am, Danny73 <fasttrac...@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. > > > > It's the rhombic dodecahedron: > > > http://bandtechnology.com/PolySigned/Lattice/Lattice.html > > > I agree with what you say above. The shape, which I call a signon, > > > does pack (though I don't have a formal proof) and is general > > > dimensional. Most importantly when you take this shape down to one > > > dimension then you are left with the usual real line segment as a > > > bidirectional entity. There is then one more beneath that level whose > > > dimension is nill and whose solitary direction matches the behavior of > > > time, in which we observe no freedom of movement yet witness its > > > unidirectional character coupled with space. > > > > But rising up in dimension the geometry of the signon maintains its > > > unidirectional qualities, so that we can argue that your square > > > implementation has four directions whereas the simplex system has only > > > three. This is because each line of the cartesian construction is > > > bidirectional. The cells have a flow form about them, and I have seen > > > this shape characterized as 'nucleated'. When the lines connecting the > > > interior of the shape are filled in, and the hairs put on the lines, > > > then the signon and the simplex coordinate system become more > > > apparent. > > > > Getting away from the lattice the usual vector characteristics do > > > apply to these coordinate systems and while there is an additional > > > coordinate there is likewise a cancellation so that on the 2D > > > (hexagonal) version: > > > (1,1,1) = 0 > > > Note that the real number (1D) version has the behavior > > > (1,1) = 0 > > > which is just to say that > > > - 1 + 1 = 0 > > > and so this is a way to bear the polysign numbers, for in the 2D > > > version we can write > > > - 1 + 1 * 1 = 0 > > > where * is a new sign and minus and plus symbols take on different > > > meaning than in the two-signed real numbers. Arithmetic products are > > > easily formed from there. > > > > It can be shown that there is a savings of information in high > > > dimensional representations by using the polysign or simplex > > > coordinate system. Because the coordinates of the > > > (a,b,c,d,...) > > > representation do not carry any sign and one of them can be zeroed we > > > can communicate a 1 of n value and then a series of magnitudes. For > > > large dimension this method saves roughly n bits of information. So > > > for instance a 1024 dimensional data point would save roughly 1014 > > > bits of information by using the simplex geometry. This is because we > > > saved all of those sign bits, and needed just 10 bits to communicate > > > the zero component. This is an esoteric savings because the size of > > > each magnitude will likely be a larger cost. Still, the savings is > > > real. > > > > I believe that there will be a more natural form a Maxwell's equations > > > on the progressive structure > > > P1 P2 P3 ... > > > which will bear productive physics. The rotational qualities of > > > Maxwell's equations are somewhat built into this structure, as is > > > time. Study more closely and many details are in alignment with > > > existing theory, both relativity and string/brane theory. Should the > > > electron's spin be inherent rather than tacked onto a raw charge? In > > > some ways this is the ultimate in existing Maxwellian thought. A > > > stronger unification lays in structured spacetime. Relativity theory > > > is a first instance of structured spacetime, not a tensor spacetime. > > > > - Tim- Hide quoted text - > > > > - Show quoted text - > > > Aether field of dimension. 8 directions for 4D space aether > > No that you have figured out that 4 times 2 is 8, here is a new puzzle > for you: what is 5 times 2? Take your time.
No. There is no need for five times two. It's just five direction for a 4D space. They balance so that (1,1,1,1,1) = 0. This is the simplex geometry. The components do not require any sign and instead the construction is the generalization of sign, just as the one dimensional form is (1,1) = 0 which is to say that - 1 + 1 = 0 . Five signed numbers do have inverses but each individual sign does not carry a direct inverse as they do in the two-signed numbers.