
Re: 3 dimensions and their 6 directions
Posted:
Apr 21, 2010 9:25 AM


On Apr 20, 2:56 pm, Thomas Heger <ttt_...@web.de> wrote: > Tim Golden BandTech.com schrieb: > > > > > On Apr 19, 10:04 pm, Thomas Heger <ttt_...@web.de> wrote: > >> Tim Golden BandTech.com schrieb: > > >>> 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 > >>>>>>> NorthSouth, 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 twosigned 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 twosigned numbers. > >> Hi Tim > > >> long time no see.. > > >> Don't want to disturb, but you should have a look at my latest version. > >> The doubletetrahedron is generating such a hexagonal pattern. This is a > >> symbol for complex fourvectors or biquaternions. That two are > >> tetrahedrons acting in opposite directions.http://docs.google.com/Presentation?id=dd8jz2tx_3gfzvqgd6 > >> (it is now more or less finished, but I have still not many reactions) > > >> Greetings > > >> Thomas > > > Hi Thomas. > > If you can point me to one section you'd like me to review that would > > be great. > > I think everything boils down to something in between octonions and > biquaternions. But the physicists are sooo stupid. It would be > essential to search for the solution 'somewhere near' these constructs. > Once you do this, the final problems could be circled in. It is > important to do something like the text, that I have written and think > about, what to search for and why and then search. > Otherwise, everything gets soo pointless. > Even your polysigned numbers don't address some kind or real problem > (even though they are a good approximation). It is simply an assumption, > that nature could be modeled by an appropriate algebra. I think, that > could be the case, but by one, you could never use, because the 'real > thing' is smooth, continuous and infinite. > > Greetings > > Thomas
Well, the problems are open. I'm pretty sure that the mainstream respects Hamilton's quaternion and its EM representation, but also accepts that the Heaviside expressions are the cleanest expression of EM. These are the modern form of Maxwell's equations. At least this is the way that I look at it. I haven't yet read Heaviside so that portion is hazy to me.
Regardless of these differences I feel pretty sure that the quaternion struck Hamilton as having spacetime correspondence. Still, why then is time unidirectional in nature? Even relativity is a structured spacetime and is forced into a linguistic interpretation of sign as timelike or spacelike and really just abuses the tensor representation in order to get its result, all the while preaching the tensor and isotropic behavior.
The quaternion does not stand alone. It stands in a chain real>complex>quaternioin>octonion>... The 4d relativistic representation does not stand alone. It stands in a chain 1D>2D>3D>4D>5D>... But on the chain of polysign a natural breakpoint exists P1 P2 P3  P4 P5 ... and unidirectional time exists consistent with unified spacetime. The electromagnetic qualities are already present there in the geometry with a rotational disk P3 and an (informationally) orthogonal P2 axis. Still, I have yet to express this as it is done on the other theories.
Whether spacetime should be arbitrarily built out of general dimensional phenomena or whether it should be derived... This is a question that polysign answers affirmatively. The existence of polysign numbers already expose the flaw in an arbitrarily constructed spacetime from a theoretical perspective. The purpose of theory is not just to fit a curve or match a dimension mathematically; it is to provide some reasoning why such a limitation might exist. In that theory explains observation then the existence of 3D space already is the observation.
A spacetime theory does not exist, except for polysign. We are free to pull things out of a hat, but the best things to pull out of the hat are the fewest things necessary. A Euclidean geometry with twenty axioms will gain little attention if ten axioms will do the job. The direction of progression here is pertinent to modern theory. The question Why SpaceTime? is valid and relativity makes no theoretical statement on spacetime itself from the quantity of dimensions. Now we have string/brane theories which have truly expressed this problem. Some have caught onto this paradigm, but most just leave it feeling vague, at least I think this must be so. Even Newton and Kant did not grapple with this. It was beyond them. It lays still beyond us and is a critical step in the advancement of pure theory. So it goes undiscussed as do many conflicts.
 Tim

