Nick
Posts:
1,720
Registered:
1/29/05
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Re: 3 dimensions and their 6 directions
Posted:
Apr 19, 2010 5:53 PM
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On Apr 19, 7:49 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
wrote:
> 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.
>
> - Tim- Hide quoted text -
>
> - Show quoted text -
In the hypersphere surface is the first three dimensions round in
gravity curve.
Mitch Raemsch
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