Defining a point in space with two parameters.

[John Cuthber]

OK, how do I know if 111222333 is 11,1222,333 or 111,222,333 or 11122,23,33?

You can do it if you specify the number of digits in advance and pad the numbers with leading zeros but, if you say in advance how many digits you will use then you can't have arbitrary accuracy.

[michel123456]

OK, how do I know if 111222333 is 11,1222,333 or 111,222,333 or 11122,23,33?

You can do it if you specify the number of digits in advance and pad the numbers with leading zeros but, if you say in advance how many digits you will use then you can't have arbitrary accuracy.

Well I can tell the number 111,222,333 IS a number (and not a coordinate) like 111 kilometers 222 meters 333 millimeters is a distance (a point number upon the number line). It is just a matter of convention.

i can as well associate a single number to any single point of space. As I said above it is only a matter of knowing if one infinity is bigger than the other one.

 

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Intuitively the infinity of points of spaces are a thousand times bigger than the infinity of numbers from the line of numbers, but I doubt that intuition gives the correct answer when talking about infinities.

Edited June 3, 2013 by michel123456

[John Cuthber]

Since we are talking about counting spheres or blocks to get there, I'm pretty sure the infinities are all aleph null.

 

 

The point remains that your "convention" limits resolution.

 

"Intuitively the infinity of points of spaces are a thousand times bigger than the infinity of numbers from the line of numbers, but I doubt that intuition gives the correct answer when talking about infinities."

I'm fairly sure it doesn't.

You can imagine the infinite number of coaches, guests and rooms here

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel#Infinitely_many_coaches_with_infinitely_many_guests_each

 

as the x,y and z coordinates.

there are just as many of them as there are numbers on a line

[tar]

John Cuthber and Iggy,

 

OK, I thought all day, without coming up with the impossible plan. But I am not ready to throw in the towel yet.

 

As michel123456 suggests, it may be a matter of convention and agreement.

 

Let's say for instance we were to agree only on the fact that no more than twelve same size balls can fit, in the space around a ball.(touching the center ball)

 

And they can be so arranged as in the fashion described in the pictures and diagrams in this thread. That this arrangement is real, exact, reproducable by anyone, anywhere, and would be true, and the same figure with the same relationships, anywhere it was constructed, on any scale it would be contructed and would stay the same figure under any and all rotations, with the infinite amount of rays emminating outward only, from the center of the center ball, passing through every concievable point in space.

 

Then we agree that one of these infinite number of rays passes through the center of ball A and continues outward in that direction infinitely, passing through every point that lies exactly in that direction. We call that direction A. We do the same for ball B, and call that ray, direction B. We then consider the infinite amount of rays in the fan of rays that lie between A and B, and give them each a name by a regular and descriptive convention, let's say for instance 59A1B would be that unique ray that went outward one degree away from A and 59 degrees away from B. This would give us a name for that ray, with no reference to any inclination, plane, altitude. There is only one percise and distinct direction defined by 59A1B. Under this convention we might also call direction A, by another name 60A0B

 

Thusly proceeding we could name every ray in the fan between A and B, the fan between B and C and the fan between C and A. And we have those directions named. Then someone more mathematically clever than I comes up with the extension of the convention to include the rays inside the triangle funnel. What would the ray in the middle of the three, be named? Does 30A30B30C work out? If so, then we could name every possible ray within the triangular funnel passing thru that face of the dudecahedron.

 

Then the task is to figure the convention for the square funnel bounded by the 60 degree fans AB, BL, LK and KA, a little more challenging but probably mathematically possible. The BK fan is 90 degrees worth of rays, as is the AL fan.

 

If the conventions were agreed on, we could drop the lattice and balls and just consider that any point in space must in the singular direction of one and only one of the named rays.

 

Regards, TAR2

 

Infinite precision can be achieved by decimal degrees, or minutes and seconds or whatever.

P.S. But alas you could probably do the same thing with an octahedron. Consider each equilateral triangle, dropped into xyz funnel, normal to the origin, and name your rays through it.

 

I don't know if that means you can name a direction and a distance and define a point, or whether it just means you need at least three and no more than three dimensions to define a point.

 

Well yes I do know. It's the latter.

 

In the words of John Cuthber, "Its not a debate, its a fact."

 

I yield.

Edited June 4, 2013 by tar

[Iggy]

If the conventions were agreed on, we could drop the lattice and balls and just consider that any point in space must in the singular direction of one and only one of the named rays.

 

Infinite precision can be achieved by decimal degrees, or minutes and seconds or whatever.

 

That is called the spherical coordinate system. You need two angles and a distance to be sure to be able to find any point in 3d space.

 

If you could be sure that any point was between two named balls then you'd only need one angle, but if it doesn't you need a second

[tar]

Iggy,

 

Agreed.

 

Regards, TAR

[John Cuthber]

Thusly proceeding we could name every ray in the fan between A and B, the fan between B and C and the fan between C and A. And we have those directions named. Then someone more mathematically clever than I comes up with the extension of the convention to include the rays inside the triangle funnel.

Yes, it's easy.

You add another parameter.

You can then use something like this

http://www.chemix-chemistry-software.com/school/ternary-plot.html

 

Incidentally, one of the numbers is redundant in your scheme

59a 1b

Since the a and b have to add up to 60, the 1 (or, if you prefer, the 59) is not needed- what other value could it have?

[tar]

John,

 

Well redundant in that you know they have to add to 60, but not redundant in naming b as the other axis involved, and in giving you the visual that you are 59 degrees from B and one degree from A, in positioning your ray. The triangular funnels seemed doable in this fashion, as in the ternary scheme you linked. But the square funnel seemed more unfriendly to triangulation, and seemed to me to add a flexibility that did not "force" a ray to be in an exact position. Sort of why a triangle is a strong structural shape, where a square can fold up, with two corners going out and two corners coming in. Nothing to insure the corners remained at 90 degrees.

 

But I've dropped that scheme, as being sufficient to name a point with two parameters.

 

However, I had a thought about 45minutes ago, that I would like to explore with you guys and gals that just might still work.

 

One of the aspects of considering the matrix of hexagonal and square planes one can imaginarily build off the figure I pictured, is that each ping pong ball, is a possible "origin" that has the same exact twelve directions as any other pingpong ball in the matrix.

 

Now, let's say our convention was that your first parameter was one of the twelve directions, combined with a number.

 

Say F5.985,940.84757. This first part of the convention would mean travel out in the F direction 5,985,940.84757 units.

 

This position you would be at at this point then becomes the center of a ping ball in a new matrix, that is shifted .84757 units out along the F ray, but in all other regards is oriented in the same directional sense as the first matrix.

 

We then describe our second parameter FROM THERE. Say B598,822.3.

 

F5.985,940.84757 meters , B598,822.3 meters, describes a particular point in space. Any resolution you desire can be used.

 

Only thing left is to mathematically prove that given only the twelve singular directional rays sweeping through space along one ray will completely blanket space.

 

For visual reference, consider a ball with six lethal rods thru it in the agreed upon pattern going off in the twelve directions, as far as the eye can see, floating 100 yards from you. One of the rods is a track that the ball will take in one direction or the other, and the other 5 (10) rods are fixed to the ball. You do not know which of the 6 axis is the track and which of the other 10 rods are fixed, but you know the ball is going to move along the track rod in 1minute in one of the twelve directions at 40000 miles an hour.

 

Is there a position you can take where you know you will not be killed?

 

Regards, TAR2

In motion, in a singular direction, an axis becomes a plane.

 

I am thinking that not only might you be able to name every point in space with this new scheme, but you might be able to name each point in space several different names.

Additionally, back to the ternary scheme, and the octahedron. Name the six points of the octahedron A, B, C, D, E, F (as in X, Y, Z, -X. -Y, -Z) and you have the eight triangles XYZ, XY-Z, -XY-Z, -XYZ, X-YZ, X-Y-Z, -X-Y-Z, and -X-YZ. Or naming Y as A, Z as B, X as C, -Y as D, -Z as E, -X as F, you have the more positively named octahedral faces, CAB, CAE, FAE, FAB, CDB, CDE, FDE and FDB, naming the eight general directions, that describe every possible direction, in that every possible ray from the origin, or the center of the octahedron, will either pass thru one of the octahedral faces, or pass thru an edge between two faces or pass thru a point where three faces meet. And thusly be namable as in C20A44B36, C64A36, and C.

 

Thusly naming each direction by the percentage in the ternary schemes 100% total, we have a name for every direction possible and we don't need to travel up a certain amount and over a certian amount and forward a certain amount. We can just make one trip in one direction, and get to any point in the universe, near or far.

 

Only thing left to do is decide what distant quasar is in the A direction.

Edited June 6, 2013 by tar

[John Cuthber]

The fact that you are using 4 parameters to indicate a point strongly suggests that you can find more than 1 name for each point in space.

But it's also consistent with the observation that you can't just use two parameters.

[tar]

John Cuthber,

 

Well I am tending to agree with you. The octahedral plan uses three axis to locate the place the ray will intersect which triangular face, so I suppose its sort of after the fact, to call that ray a singular direction. Well it is, but you have to use three others to define it, so I've failed there.

 

The movment along an axes however, and then taking one of the 12 directions from there, I have not been able to rule completely out yet. Question. What rules that out?

 

Regards, TAR2

[tar]

Major apology in order.

 

Every time I mentioned the word dudecahedron I was thinking about the The Cube Octahedron.

 

I was never speaking about the dudecahedron.

 

I have an explanation, in that I also have some dudecahedrons lying about, and it is a figure that I "discovered" in drawing on a sphere and trying to determine the "equal" spaces that surround a point, and I simply called the one thing the other, but I have no excuse. It was just a gross, misdirecting, incorrect, mistake. Please accept my apology and replace any instance of "dudecahedron" with "the cube octahedron", in the above thread.

 

Regards, TAR2

[John Cuthber]

Not that it matters, I think you mean dOdecahedron.

[tar]

Ok wrong thrice.