3 dimensions and their 6 directions
in [Physics]
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From: BURT on 7 Apr 2010 17:56 On Apr 7, 2:45 pm, moro...(a)world.std.spaamtrap.com (Michael Moroney) wrote: > James Dow Allen <jdallen2...(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. There is always a direction in the 4th dimension. Mitch Raemach
From: Tim Golden BandTech.com on 8 Apr 2010 08:13 On Apr 7, 5:45 pm, moro...(a)world.std.spaamtrap.com (Michael Moroney) wrote: > James Dow Allen <jdallen2...(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. 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
From: bert on 8 Apr 2010 11:56 On Apr 8, 8:13 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Apr 7, 5:45 pm, moro...(a)world.std.spaamtrap.com (Michael Moroney) > wrote: > > > > > > > James Dow Allen <jdallen2...(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. > > 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 - Macro has 3 dimentions+ spacetime Thanks to Witten micro string theory on space dimentions is down to only 6 (that is a lot better than 11) O ya TreBert
From: BURT on 8 Apr 2010 17:11 On Apr 8, 8:56 am, bert <herbertglazie...(a)msn.com> wrote: > On Apr 8, 8:13 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > > > > > On Apr 7, 5:45 pm, moro...(a)world.std.spaamtrap.com (Michael Moroney) > > wrote: > > > > James Dow Allen <jdallen2...(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. > > > 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 - > > Macro has 3 dimentions+ spacetime Thanks to Witten micro string > theory on space dimentions is down to only 6 (that is a lot better > than 11) O ya TreBert- Hide quoted text - > > - Show quoted text - There is only one higher spatial dimension and it is hypersphere geometry. Mitch Raemsch
From: Ostap S. B. M. Bender Jr. on 8 Apr 2010 20:04
On Apr 3, 2:44 pm, BURT <macromi...(a)yahoo.com> wrote: > 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 So: In 1 dimensions there are 2 directions. In 2 dimensions there are 4 directions. In 3 dimensions there are 6 directions. In 4 dimensions there are 8 directions. Can you find the amazing pattern here? Can you predict how many directions there are in 5 dimensions? If you do so - you will outdo yourself, disproving the widely held misconception that people with severe mental handicaps can't think abstractly. |