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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.

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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Sorry, bad choice of words.  It indicates that it is somewhat different than what you say.

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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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Alpha and lambda relate to one another geometrically.

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Steve, this is a suggestion for you.

Applied Mathematics acknowledges something called a continuum which roughly means XY or XYZ space (or whatever).

There is a whole area of ApMath called continuum mechanics which studies movement within a continuum.

When rigid bodies and non rigid bodies move, they change their xyz coordinates.

These changes can be split into linear deformations and angular deformations.

The angular deformations can be composed into a single Vector (2D) or Tensor (3D) known as the Spin Vector or Spin Tensor.

So imagine a point and a direction arrow emanating from that point  on a block of rubber that is being deformed.

The point will move from X1Y1Z1 toX2Y2Z2 and the arrow will rotate to point in a new direction during the deformation.

These values will change as the deformation proceeds.

Alternatively the distance between two points may well change as will the direction of the arrow linking them.

I wonder if you are trying to describe the spin vector/tensor part of this since it is very similar to the idea you are trying to put over.

A word of warning if you try to look this up. This is not the spin vector of quantum mechanics and the theoretical Physics approach using Noethers theorem is not directly relevant either.

The mathematics of this requires advanced calculus and will be found in applications in diverse fields as fluid mechanics, earth science, stress analysis, electrodynamics etc.

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I appreciate that you think this must be something that someone somewhere has seen before; it cannot actually be a new way to look at Euclidean 3-space.  The tridentity has never been published.

I really want this thread to focus on specific mathematical realities, what I call facts or proofs.

We seem to need a landmark in order to begin communicating rather than talking past one another.  The best place to start is with Chapter XVII (Chapter XIV) – Arcs Drawn to Fixed Points on the Surface of a Sphere, which can be found in SPHERICAL TRIGONOMETRY by Todhunter.

We have the identity:

$\cos TA^2 + \cos TB^2 +\cos TC^2 = 1$

We seem to be at an impasse as to whether or not this is a valid identity.  Sure, it is in three dimensions, but it isn’t in two dimensions.

Let’s agree to call this type of equality a tridentity.

The thesis is complicated, but I assure you one more time that no one has ever published anything along this line of mathematical investigation.  I assure you one more time that, unlike the calculus that you are conflating with this question, the mathematical effect of considering these newly found relationships is non-existent in Galilean spacetime and is only of concern in relativistic spacetime (or, at least that is how it seems to be at this time.)

I’ve been searching for jargon to facilitate understanding of this new approach and there just isn’t any that I’ve been able to find.  Let’s call the circle functions and conic sections “static” functions as a way to distinguish what’s happening mathematically from what happens with a sine curve which I’ll call a “dynamic” function.  If there is any existing terminology to draw this distinction I am not aware of it.

The dynamic sine curve is to the static circle function as the new dynamic tridentity is to the old static spherical tridentity from a couple hundred years ago.

Edited by steveupson

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36 minutes ago, steveupson said:

I appreciate that you think this must be something that someone somewhere has seen before; it cannot actually be a new way to look at Euclidean 3-space.  The tridentity has never been published.

I really want this thread to focus on specific mathematical realities, what I call facts or proofs.

We seem to need a landmark in order to begin communicating rather than talking past one another.  The best place to start is with Chapter XVII – Arcs Drawn to Fixed Points on the Surface of a Sphere, which can be found in SPHERICAL TRIGONOMETRY by Todhunter.

We have the identity:

cosTA2+cosTB2+cosTC2=1

We seem to be at an impasse as to whether or not this is a valid identity.  Sure, it is in three dimensions, but it isn’t in two dimensions.

Let’s agree to call this type of equality a tridentity.

The thesis is complicated, but I assure you one more time that no one has ever published anything along this line of mathematical investigation.  I assure you one more time that, unlike the calculus that you are conflating with this question, the mathematical effect of considering these newly found relationships is non-existent in Galilean spacetime and is only of concern in relativistic spacetime (or, at least that is how it seems to be at this time.)

I’ve been searching for jargon to facilitate understanding of this new approach and there just isn’t any that I’ve been able to find.  Let’s call the circle functions and conic sections “static” functions as a way to distinguish what’s happening mathematically from what happens with a sine curve which I’ll call a “dynamic” function.  If there is any existing terminology to draw this distinction I am not aware of it.

The dynamic sine curve is to the static circle function as the new dynamic tridentity is to the old static spherical tridentity from a couple hundred years ago.

Thank you for replying, a pity you are not listening.

It is common practice in spherical geometry to mean only great circle arcs when using the term arc

Quote

Todhunter p 19

There are no great circles in 2D.

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Yes, exactly, that's exactly it.

Finally we seem to be on the right track.

What makes the stuff in the OP new is that it uses small circles, not great circles.  We've had this discussion before, a long time ago, but this time I think you might be beginning to look at things a little differently, I hope.

When we move from great circles to small circles an entire new set of relationships becomes visible.  These relationships are not visible by other means.

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27 minutes ago, steveupson said:

Yes, exactly, that's exactly it.

Finally we seem to be on the right track.

What makes the stuff in the OP new is that it uses small circles, not great circles.  We've had this discussion before, a long time ago, but this time I think you might be beginning to look at things a little differently, I hope.

When we move from great circles to small circles an entire new set of relationships becomes visible.  These relationships are not visible by other means.

So why mention great circles or introduce references to them?

1 hour ago, steveupson said:

Let’s agree to call this type of equality a tridentity.

If you must call it a tridentity, then OK.

At least you have found a suitable nice new word that sort of echos what you want to say and does not redefine an existing term.

This is really good as it avoids confusion well done.

1 hour ago, steveupson said:

I appreciate that you think this must be something that someone somewhere has seen before; it cannot actually be a new way to look at Euclidean 3-space.  The tridentity has never been published.

The technique I was referring to will serve for Euclidian and non Euclidian space.

The term refers to how you calculate distance, not angle.

The big stumbling point as I see it is most people look at what you are proposing and say

So what?
What does it offer me or tell me?
When would I want to use it?

Show me an examaple of its use (however trivial; trivial is good for an initial example as complexity hides the underlying message)

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I feel like I must respond to the comment about not listening.  I agree with everything you said.  What more can I say?  Yes, you're correct, you're right, that is exactly what they teach about this stuff.  I understand and do not dispute that everyone believes that direction and distance and time relate to one another in a certain way, and that you are correctly presenting the current state of the art.

Because the relationships between distance and direction are well understood, we think that we know how to create a non-Euclidean space that perfectly reflects our natural reality when time is combined with these two things.  The insight that can be had from understanding the tridentity is that direction is affected by relativistic effects along with distance, and that the interplay between direction and distance in spacetime is different than what we think it is.

There are no trivial applications of relativity to the interplay between distance and direction.

There are some well known phenomena that are currently unexplained which are resolved by understanding this interplay between time, direction, and distance.  In a manner similar to what happens to distance under relativistic transformations, direction undergoes a process that causes it to be different than what it is normally.

The three most obvious cases where phenomena are explained using this approach are quantum entanglement, dark energy, and dark matter.  All three of these apparently unrelated observed physical anomalies are explained as being caused by the math that we are using to model spacetime.  The relationships that are identified in the OP are just as relevant to nature and physics as pi or Pythagoras.  They must be considered, which is not being done.

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2 hours ago, steveupson said:

The three most obvious cases where phenomena are explained using this approach are quantum entanglement, dark energy, and dark matter.  All three of these apparently unrelated observed physical anomalies are explained as being caused by the math that we are using to model spacetime.  The relationships that are identified in the OP are just as relevant to nature and physics as pi or Pythagoras.  They must be considered, which is not being done.

Gosh your thought empire has expanded greatly.

But you still haven't offered anything for others in your new proposed property you call direction.

Here is an example.

I propose a new property called squiffyness.

This is what you can do with it.

When you are making a jelly (Jell-O), measuring the squiff allows you to tell if the jelly will stand up as a rabbit or slump to a sloppy mess, when turned out of its mould.

You measure the squiff by taking a sample in the bulb of a squiff meter, filling to the set-one line in the measuring tube.

Squeezing the bulb causes the contents to rise up the tube, the distance up the tube being graduated from 1 to 10 in squiff.

A reading of 3 squiff of less means that the turned out jelly will stand and wobble.

Greater than 3 squiff will result in a slump.

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Part of the theory is that direction dominates distance in the quantum scale.  This is because the distances become nearly insignificant while the magnitude of the direction doesn’t change as significantly. This may be helpful in determining squiff size.  Once we recognize direction as a base quantity then it should be possible to express it in Planck units.

For particles (or waves) to exist, their future must lie in a direction that is coming forward in our universe.  Mathematically their direction is toward or away, and in either case the quantity is between zero and infinity, so that the sign is always the same. This is important at both the macro and quantum scale.  On the macro scale, particles/waves whose future/past lies perpendicular to our own future/past are not moving forward (their direction is not toward or away) and so they do not exist.  This is independent of the observer.  Also, if this premise is correct, then when the math is corrected for this peculiarity then the existence of the 2π relationship with dark energy will be explained.  Anything that has a future which is perpendicular to our future isn’t there.

The tridentity provides a mathematical understanding of the rate at which this disappearance from our universe occurs.  The angles in orthogonal space are all related to one another, not only by distances (which change with regard to time), but also by other angles.  Every angle relates to every other angle in the same way as every distance relates to every other distance.

Because these angles can now be quantified, they can also be used to create a new coordinate system where the metric is direction.  There should be a way to pass back and forth between these coordinate systems by marrying the Planck values and the orthogonality axiom to each system.

The Planck direction has an impact on physics for both the quantum and the macro scale. As previously stated, the perpendicularity of futures makes them nonexistent to one another. This perpendicularity is determined or bounded by the size of the Planck direction.

At the quantum level, the Planck direction establishes or bounds the conditions for entanglement.  This may be somewhat easier to explain than the previous example for dark energy.  Because of the way the math works, the futures of entangled particles lie in the same direction.  The entangled particles each have futures that are coming toward us, due to their size.  This is definitely weird, sure, but the math indicates that this is how spacetime is able to wrap back on itself.  It must wrap back on itself in some fashion in order to satisfy the condition where c is a constant.

The alternative expression of Euclidean 3-space (using direction in lieu of distance) changes things in a way that affects the mathematical treatment of spacetime. In flat or Newtonian spacetime the effect is indistinguishable from the normal perception. It's only when we look at this model in relativistic spacetime that the ramifications appear. This is probably due the difference in ranges for the quantities that are being combined (length, direction, and time).

When distance ($l$ength) and $t$ime are combined in spacetime the result is bounded by $-c$ and $+c\::$

$-c\leq lt \leq +c$

When direction ($\tau$urns) and $t$ime are combined in spacetime the result is bounded by zero and infinity:

$0\leq \tau t\leq \infty$

Note that when structured this way, the sign is actually associated with length rather than direction. This has an effect when spacetime is curved. A subtle anomaly arises and has to be accommodated when this geometric relationship is considered alongside the equivalency principle.