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April 9th, 2018, 12:21 PM   #21
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Quote:
 Originally Posted by studiot None of this has anything to do with what I said. One small correction. Most vectors have a length, the zero vector has zero length, but I thought you were not using vectors?
I'm not following what you are getting at then. If we're not using vectors then why would you want to talk about the additive identity of the additive group of vectors? I can't imagine what you are trying to say, or why you might think that this would be useful.

April 9th, 2018, 12:23 PM   #22
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Quote:
 Originally Posted by Maschke See, now if I called you an idiot, that would be a personal attack and an ad hominem argument. So I won't do that. But what you wrote is idiotic. You clearly haven't the slightest notion of why the Banach-Tarski theorem works. All you've done is embarrass yourself.
I can, on the other hand, understand perfectly why you insist on trying to make this discussion about me.

You don't seem to understand what you just said, at all.

April 9th, 2018, 12:26 PM   #23
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Quote:
 Originally Posted by steveupson I understand that this criticism isn’t helping to explain anything. I think that you may better get the idea of what we’re talking about by looking at the Banach-Tarski paradox. They break the sphere into an infinite number of directions in 3D, then they use a reference that is not part of the surface (sphere center) in order to re-position these directions in space, and then they reassemble the surface using this infinite number of directions, only in another orientation.
Awesome! As you are undoubtedly aware, the Banach-Tarski paradox works basically because of a paradoxical decomposition of the free group on two generators. At least this is the standard way of proving the Banach-Tarski paradox. So how would you phrase this comment of yours on the free group of two generators? What are the infinite number of directions there?

 April 9th, 2018, 12:40 PM #24 Senior Member   Joined: Oct 2009 Posts: 770 Thanks: 276 Waaaait, so you just solved a basic spherical trigonometry question, right? The type of things you'd find in spherical trig textbooks. That's all you did, no? So where is the amazing breakthrough you claimed? Am I missing something???
April 9th, 2018, 12:58 PM   #25
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Quote:
 Originally Posted by steveupson I'm not following what you are getting at then. If we're not using vectors then why would you want to talk about the additive identity of the additive group of vectors? I can't imagine what you are trying to say, or why you might think that this would be useful.
Part of what (I think) you are trying to say is already part of conventional mathematics, and that part is done with vectors in a way entirely consistent with the rest of conventional mathematics.

The rest has the difficulties I outlined in my sketch, which by the way is 2 dimensional (as are all planes) not 3 dimensional as you seem to think.

April 9th, 2018, 02:29 PM   #26
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Quote:
 Originally Posted by studiot Part of what (I think) you are trying to say is already part of conventional mathematics, and that part is done with vectors in a way entirely consistent with the rest of conventional mathematics. The rest has the difficulties I outlined in my sketch, which by the way is 2 dimensional (as are all planes) not 3 dimensional as you seem to think.
Having reread this a few more times, I do think that I understand what you are asking. You want to know how any of this works without a reference position. I think that I can answer this question, if that is what you are asking.

In the block diagram in the OP it shows that position is a derived quantity, or in other words, that it is the composition made up of two of our base quantities. Without both, we cannot express position. In this system, we cannot know position, which is in keeping with observational data.

Instead, we can know something completely different, based solely on quantities derived from the two base quantities, direction and time. Conceptually, we now have a coordinate system where we can express the difference in orientation between two intersecting lines in space, without position or distance. The two intersecting lines are the cardinal and ordinal axes, and either line can be the cardinal or ordinal, it doesn’t matter at all. What we are expressing is the quantity that defines the difference between their orientations. This quantity has a magnitude.

One of the features of this coordinate system (in addition to not having positions) is that we know the difference between the two directions for the cardinal and ordinal axis, but we don’t the orientation of both at the same time. In other words, if we are referencing from the cardinal axis, the ordinal axis will lie somewhere upon the surface of a cone having the proper aperture.

(I understand how to create a turns ratio using vectors. If we use dimensional analysis on a formula that has this in it, how does this pan out dimensionally? Can you point me to any info on that? It would be a great help.)

April 9th, 2018, 02:43 PM   #27
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Quote:
 Originally Posted by Micrm@ss Waaaait, so you just solved a basic spherical trigonometry question, right? The type of things you'd find in spherical trig textbooks. That's all you did, no? So where is the amazing breakthrough you claimed? Am I missing something???
Yes, you are completely missing something. The geometric identity cannot be found in textbooks. You should have noticed that this has been mentioned several time in this thread, and is explained in great detail in the introduction to the proof of the equation.

The reason that this particular geometric identity is important is because it should have been discovered a century ago, when science naturally needed it. Instead, as is also outlined in the introduction to the proof, they developed vector analysis, derived all the equations for spherical trigonometry, and then used that alternate method to solve this type of problem.

And sure, all that there was to it was to solve the isosceles spherical triangle, once another member, at another forum, during another collaboration, noticed that the intersection and the pole contained the same angle.

Then there's also the timespace model that was developed during another online collaboration with some other generous folks. Hopefully, we'll be able to continue in this spirit. So, yes, of course, you missed the whole point of the thread.

April 9th, 2018, 02:47 PM   #28
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Quote:
 Originally Posted by Micrm@ss Awesome! As you are undoubtedly aware, the Banach-Tarski paradox works basically because of a paradoxical decomposition of the free group on two generators. At least this is the standard way of proving the Banach-Tarski paradox. So how would you phrase this comment of yours on the free group of two generators? What are the infinite number of directions there?
No, no, you're awesome.

And once again, my understanding has nothing at all to do with whether or not this is correct. If there is an error, then please, contribute something.

April 11th, 2018, 05:10 PM   #29
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Math Focus: non-euclidean geometry
We've revised the timespace block diagram following more discussion at another forum. The project still needs to find someone who can make a graph of $\displaystyle \upsilon$. The changes to the diagram are based on comments by danshawen, "Funny. I just noticed, we really have reversed the numerical order of Newton’s calculus in order to describe timespace."

I'm pretty sure that this new version is correct. What we are really talking about is Newton without the clock in that lower block. I'm still trying to rearrange things into a more coherent form. Force is an instantaneous change in direction acting over some distance. Without the distance there is no acceleration. Without the instantaneous change in direction there is no mass (or inertia?)
Attached Images
 block diagram.JPG (31.7 KB, 12 views)

April 30th, 2018, 08:16 AM   #30
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Math Focus: non-euclidean geometry
These animations are intended to facilitate the discussion of the tridentity:

$\displaystyle (\cos\frac{\phi}{2}\,\sin\frac{\lambda}{2})^2 +(\cot\frac{\phi}{2}\,\cos\alpha)^2=1$

This first one is an example of a member of the family of functions.

I'll have to create some links, the gif won't post. It may take a while.
Attached Images
 FamilyMember.jpg (5.4 KB, 1 views)

Last edited by steveupson; April 30th, 2018 at 08:37 AM.

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