steveupson

That's the reason why I have been arguing that length doesn't commute in spacetime. There will be only one case (condition) where the length from the first end to the other is the same as the length from the other end to the first. This would be analogous to the hairy ball thing. If spacetime is isotropic then there cannot be multiple instances of a null field (in any particular reference frame), i.e. where a length measures the same in both directions, or where it commutes. This sounds like an argument that direction does commute in spacetime, which sounds correct to me. Distance appears to be the property that is affected by time, not direction.

Direction. I think that I understand perfectly what your question is about. Historically, this type of information medium was once attributed to the aether, and it’s probably best described (not any more imprecisely than you described it in your question) as Mach’s principle. Many modern theories discount the existence of an aether or ether and yet these theories have no explanation of how the physical property (attribute, quality, quantity, whatever) that we call direction manifests itself. From wiki: “Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).” and: “In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight-line distances along each axis from O to P…" When you examine the mathematical representation of position, it is simply a length and a direction (referenced from another position), although the meaning of the thing we call direction is never quantified. We begin with the axiom that xyz are orthogonal to one another and leave it at that. Hence, every direction in two dimensions is expressed as either the ratio between two perpendicular lengths (Pythagoras) or as the ratio between the lengths of the diameter and circumference of a circle (π). In either case, the mathematical representation relies on length, which implies that a metric has already been applied to the quantity (mathematical representation of direction) before its even evaluated. No one seems to be sure about this, and for some reason (probably because no one knows exactly how this can be) it’s very controversial to talk about this. Since we all seem to believe that velocity is a derivative of position, and since the position of a point is a relationship involving distance and direction from some other reference point, then all points must be related to one another by direction. All particles that are in motion have a future that lies in a specific direction. Since all these directions are relative to one another, no particle can change direction without affecting the direction of all other particles. Although this doesn't mean that there's an absolute direction, it does seem to imply that there is a universal direction, or system of directions. In spacetime it might be time that produces universal direction. There’s a paper that I keep posting as a reference to some of this which no one has yet to understand why I keep posting it. It concerns the mechanism by which force manifests itself. Although the paper talks about the analysis of a static system, the exact same principles are also present in dynamic systems. 2015 The Physical Origin of Torque and of the Rotational Second Law Daniel J. Cross I wish someone would think this through wrt kinetics. My take on it is that it is fundamental. And although their analysis seems correct, their conclusions are speculative. It would be equally correct (mathematically) to simply change the direction rather than the position of the force, thus a non-rigid body is not necessarily a requirement. (welcome Rasher Null, and since you're new here, let me warn you that everyone everywhere calls me a crank (without actually doing the math) so take everything that I say with a grain of salt)

The Mathematica model, NewSphericalTrigFunction, Nr 2, v9.cdf, Filesize: 0.057 MB, can be downloaded here: http://s000.tinyupload.com/index.php?file_id=81679108911074343506

I think I can do that, and I'm fascinated by it.

Yes, let's discuss the [math]2\pi[/math] question. Do you have any suggestions as to why that is? Do you think the answer could be geometry, or do you rule that out as a possibility?

I think there must be some misunderstanding about what constitutes an animation and what constitutes a computer model. Kepler's conjecture was recently proven with a computer model. Why do you think that was? Was it because the authors were afraid of math or because they had no one to "reverse engineer" the math for them?

I've moved on, as instructed by the moderation staff here. Why are you reintroducing a topic that I was explicitly told not to reintroduce? Why not respond to what I actually said?

There's a possible explanation for that geometric relationship. If our understanding of direction in spacetime is superficial, if there is a deeper way to view it, then that would account for that particular relationship rather neatly. I think that this can be proven mathematically, but I need help with the algebra. No one seems to take any stock in mathematical models prepared with mathematical modeling software. A pencil and paper have been recommended as the proper way to go about proving this out.

It probably is time to close the thread. Everyone stopped listening long ago. We're rehashing stuff that is already redundant in this discussion. Thanks everyone. I'll probably resume these discussion at in few days. Anyone interested stop by and join in. If anyone out there gets the gist of what I've been saying, please join in. Or if you just have questions, join in. If anyone figures out the formula for the function, please post it. Thanks again for hosting this discussion.

Now we're getting somewhere. The example uses a rotating body in a fixed reference frame. The same exercise can be done with a fixed post and a rotating reference frame. The rules are different because there we are dealing with a non-Euclidean frame. What I'm talking about is a geometric truth that can be applied to either case. We only believe that length is constant and direction is changing because that's how we've always done the math. The reason we've always done the math that way is because the way we express direction doesn't commute, while length does. What I'm talking about is a technique for expressing direction in a manner that commutes mathematically in the same way that length does in our conventional methods.

The input and output of a function have been provided. Why is that not math?

The tangent plane contains the tangent of the small circle at the intersection point and the center of the sphere (which isn't part of a sphere, by the way).

I apologize for getting frustrated. Yes, it is either complicated or simple. One or the other. The model was produce from these three animations posted to youtube. Mathematica does the rest, internally. https://www.youtube.com/watch?v=ho8XCHIT-Oo https://www.youtube.com/watch?v=xwjIeHC3Nb0 https://www.youtube.com/watch?v=6OnSZki9yp0

Yes, of course. I believe that this is one of those instances that you spoke of earlier in the thread where pi/4 is a special case with special behavior.

We are both on the same page in this regard We can't ignore blue. Blue and green are at an angle to one another. This angle is part of the function. It is the tangent angle. The tangent plane is the blue plane.

The blue plane and the green plane are a pair. Their relationship to one another is established by the surface normals of those two planes. The yellow plane and the beige plane are another pair. Their relationship to one another is also established by their surface normals. The model returns the function of: tangent angle (blue/green) is a function of the elevation angle (yellow/beige) Those others show spherical trigonometry using Napier.

What trig function? This is not what you think it is. The graph shows the elevation angle plotted against the tangent angle. There is no other data or perspective or projection that can be used. It is simply the plot of how the tangent angle changes as the elevation angle is varied. The animation is is a gif of the model, yes. The model is the cdf that was used to create the gif. The author of the model assured me that that isn't the case. The original problem was to try and formulate (compose) the function from the various trig operations. Hans Milton found a way to implement the function in Mathematica without it being necessary to compose a function. I believe that Mathematica has a spline function that creates the input/output angles for the animation. If, however, you open the cdf file you can move the slider manually and for every elevation angle it will give you a corresponding tangent angle. How it does this is internal to the modeling software. See post #17 here: http://www.thephysicsforum.com/trash-can/9252-defining-new-function-merged.html

[math] |\pi|[/math] I know that this doesn't look right. It has a particular meaning. I've tried to explain it many times. What I was asking in the previous post was for Mordred to tell me what he thinks I've been trying to say. I would like to know what he thinks the function does. That way I can correct his misconceptions about what I am trying to say and correct his misconceptions about what the function actually does. It has nothing at all to do with wobbles or dihedrals. I hope you can allow the tread to continue at least until someone tells me what they think I've been saying. Everyone says it's just incoherent ramblings, but that isn't the case. A dozen pages of Q&A and for what? I could easily show what I mean if I could use Mathematica, or if someone who does know how to use it would produce two more graphs of two more functions. Without someone else, besides me, doing the math, I cannot explain what the math means. [math] |\pi/4|[/math] Is the worked example that has been produced for this thread. Mathematically the model has a meaning. It produces a function that has been graphed. What more, specifically do you require?

The function is unlike any other that has ever existed. Try and figure out what it does.

It would help a lot if someone would tell me what they think the function represented by the blue line in the graph does. Does anyone have an educated guess as to what the function does?

y=x

My understanding is that it is done that way by convention. If there is a mathematical significance, I'm not aware of it. The same things could be shown if the diagram were composed of a 180 degree angle and one 90 degree angle rather than a 90 degree angle and a 45. on edit> there's no particular significance for why the model uses 45 degrees, other than it was easier to describe it having that form. The thing to remember is that the model produces a function, and that function will change depending on what the angle is between the ordinal and cardinal directions. second edit> after further thought my first response isn't correct. I spoke too soon. The 45 is used because it places time and space on symmetrical axes.

That's the only significance of the model.

Yes, but the new quantity is invariant under Lorentz transformation. Relativistic invariance.

Yes, and I want to show you that. We agree. What is special about this function is that the angle between the two directions (45 degrees) is specified in the space surrounding the plane that the two directions (45 degrees) lie in. In other words, we don't have to know how they relate to one another in the plane that they are both in. If we know that they have this relationship to one anther in space then we know they are 45 to one another in the plane they both occupy. The angle between the two is specified in the space surrounding them. There is nothing, no metric, used in the plane that they are in. Tell me what you think that means.