steveupson

Additional Question About Surfaces in Higher Dimensions

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46 minutes ago, steveupson said:

“we can establish a one-to-one mapping between any Rm and any Rn.”

That is mapping any value in Rm to Rn.

You appear to be trying to map a function. That doesn't work, in general, because you will lose information going from, say, 3D to 2D. You can't reconstruct a 3D image from a single 2D photograph.

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17 hours ago, Strange said:

That is mapping any value in Rm to Rn.

You appear to be trying to map a function. That doesn't work, in general, because you will lose information going from, say, 3D to 2D. You can't reconstruct a 3D image from a single 2D photograph.

Only if you require that the map be continuous. There are (highly discontinuous) bijections between R^m and R^n for any positive m and n.

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I think Strange was referring to the structure which may well not be preserved by the mapping.

I think this whole thread has arisen because I failed to make this distinction when I first talked about the one-to-one correspondence.

Cantor provides a simple example within one of his proofs.

You can put the even (or odd) positive integers into one-to-one correspondence with all the positive integers.

The structure of continually increasing magnitude is preserved, but the odd/even characteristic is not.

 

 

 

 

 

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20 hours ago, Strange said:

That is mapping any value in Rm to Rn.

You appear to be trying to map a function. That doesn't work, in general, because you will lose information going from, say, 3D to 2D. You can't reconstruct a 3D image from a single 2D photograph.

This is directly on point.  Thank you Strange.  It’s sort of jarring to hear how staggeringly ignorant and naïve I must sound to everyone, but this is truly the crux of the matter.  I am going to keep quoting you throughout this post in order to try and keep the focus on this precise issue. I hope you don’t take offence or take it the wrong way.  I just want to emphasize that what you cannot believe is actually true.

 “you will lose information going from, say, 3D to 2D”

This is well known, and if we describe it such that 3D is a bucket of information that we can pour into a 2D bucket, we find that the 2D bucket is much smaller and cannot contain the contents of the original bucket.  For the most part, this information that won’t fit will be lost, and there really isn’t any clean way to identify precisely what this lost information is.

My claim is that we’ve stumbled across a mathematical method that exactly identifies this “missing” information, and also gives a great deal of insight into why and what this information is.

“you will lose information going from, say, 3D to 2D”

The “tridentity” exists in 3D and not in 2D.  Mathematically and conceptually, this formula contains precisely the information that is currently lost when we map 3D to 2D.  The lost information doesn’t fail to exist; it just doesn’t appear in the new mapping. 

When we look at how topologies work, when we create polygons and perform operations on them, we can see that they are related to one another in a manner that is called “locally Euclidean.”  What this means in 3D is that a surface, such as a sphere, has a tangent plane located at every point on the surface.  Each of these relates to the adjacent planes in a very specific manner, and this is the information that is captured when we map to 2D.

If we construct a bunch of surface normals to each of these tangent planes, then we can create a new symmetrically complimentary “map” that shows how all non-adjacent points are related to one another (perhaps in a non-locally Euclidean manner.)  They will be related to one another by a quantity of direction that does not lie in the same plane as the tangent plane.

“you will lose information going from, say, 3D to 2D”

The direction information that doesn’t lie in the tangent plane is what is captured and quantified by the tridentity. The fact that it is quantified is what makes the picture of Euclidean 3-space come into focus, together with the lost information that doesn't occur in 2-space. This lost information is not of any concern in Galilean spacetime, but it is a real issue when we try to look at relativistic spacetime or curved spacetime that is affected by something like a gravity field. 

We could also say, to express it another way (although this is just an idea and isn’t being put forward as a argument) that 3D space contains 27 degrees of freedom and that only two degrees of freedom exist in 2D.  The tridentity function deals with how the other 25 degrees of freedom are organized.

I know that I only have a precious few more chances to try and reach you guys. If anyone is offended by the tone or content of this post, trust that it isn’t intentional.  We desperately need your help.  Without it the project will be at a standstill for an inordinate amount of time.

“you will lose information going from, say, 3D to 2D”

It is imperative that we get help in order to continue.  I actually do understand how ridiculous this all sounds, and why everyone believes it to be a waste of time.  But think of how much time has already been spent on it.  We need skeptical reviewers to take a look at our work.

Edited by steveupson

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40 minutes ago, steveupson said:

Mathematically and conceptually, this formula contains precisely the information that is currently lost when we map 3D to 2D.  The lost information doesn’t fail to exist; it just doesn’t appear in the new mapping.

We already know exactly what information is lost (and how to recover it if, for example, we have two 2D images). 

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Once again, that's precisely the point.  The formula that is presented, the tridentity if you will, contains information about spatial relationships that don't exist in 2D2.

I understand that what you believe to be true makes a lot of sense, it just happens that the function proves it to be false.

Look at the math and try and understand how and why it is the way that it is.  

edited to add>>>

 

Mathematically, the relationships in the function cannot exist with simply the trait of perpendicularity.  They require orthogonality in order to exist.

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41 minutes ago, steveupson said:

it just happens that the function proves it to be false

You mean you believe it proves it false. You have not yet demonstrated any such proof, just claimed it.

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In the OP there is a graph where different angles are plotted using some different colors.  There should be some reason for the different colors being unique curves.

Does anyone know the mathematical explanation for the curves being different?  There must be some explanation.   What is it?  If our explanation is wrong, what's the correct explanation?

Some experts have argued that this is all just highschool level math, simple rotations, and that I'd understand it better if I'd learn more about vector calculus.   I'm quite sure that my math skills have nothing at all to do with the reason why the curves are different.  Those exact curves in that particular graph were plotted by another member just before they asked what does it all mean.  No one seems to know the answer, which is fine, but isn't anyone even a little bit curious?  

Isn't it even worth some discussion?  Why are the curves different?  What is actually being plotted? 

Edited by steveupson

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1 hour ago, steveupson said:

Does anyone know the mathematical explanation for the curves being different?  There must be some explanation.   What is it?  If our explanation is wrong, what's the correct explanation?

It is a plot of alpha (y-axis) vs lambda (x-axis) for values of upsilon = pi/2.2, pi/3, pi/4, pi/6

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