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Re: 3 dimensions and their 6 directions
Posted:
Apr 19, 2010 10:04 PM
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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
>>>>> 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.
Hi Tim
long time no see..
Don't want to disturb, but you should have a look at my latest version.
The double-tetrahedron is generating such a hexagonal pattern. This is a
symbol for complex four-vectors or bi-quaternions. 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
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