Formalizing length as an antivector - algebra help requested (once again)

[steveupson]

Help is needed, once more, in expressing these equations using proper algebra and proper nomenclature. The math seems to work ok, it just needs to be corrected for grammar and puctuation, so to speak.

 

To try and avoid some confusion as to exactly what this topic is about, some new terminology is introduced in order to distinguish this technique from other, more standard, methods. The main addition is the term antivector. This is no different (mathematically) than a standard vector where the roles of length and direction have been reversed. Basically, this new terminology has been invented in order to help assure other members that this author understands what a standard vector is, how it works, and more importantly, why it works the way it does, which seems to be a somewhat controversial subject. I'd like to avoid any controversy as much as possible, and I'm open to any reasonable suggestions on how the terminology should be approached if this nomenclature is unacceptable.

 

In order to create a system that uses antivectors, it is first necessary to quantify direction in a manner that is more similar to the way that length is currently quantified for use in standard vectors. The ability to define this quantification mathematically allows direction to be expressed as a scalar value which becomes the magnitude of the antivector, with length becoming the antivector quantity.

 

The method used for quantifying direction is shown in the attached file below.

 

The method returns a [latex]2D[/latex] graph. The area [latex]A[/latex] below the curve in the graph is the scalar value of angle [latex]\upsilon[/latex]. To create the antivector we will use this value (area [latex]A[/latex]) to locate a point [latex]D[/latex] in space such that the distance [latex]d[/latex] from the origin will be expressed as magnitude of direction. Begin this part with a blank sheet, which is not to be confused with the other coordinates used in the preliminary draft paper referenced above.

 

First, name a non-zero point [latex]P_x[/latex] somewhere along the positive [latex]x[/latex] axis. Construct a cone with the apex at [latex]P_x[/latex] and the base at the [latex]yz[/latex] plane. The axis of this cone is coincident with the [latex]x[/latex] axis and the aperture will be 90º.

 

(More specifically, the aperture of the cone does not have to be specified in degrees or radians, but instead the entire surface is parameterized by the area [latex]A[/latex] beneath the curve for the function of angle [latex]\upsilon[/latex], where [latex]\upsilon[/latex] is 45º. In other words, if I’m saying this correctly, [latex]A=(f)\upsilon[/latex] where both [latex]A[/latex] and [latex]\upsilon[/latex] have unique geometric interpretations.)

 

Next, where the cone intersects the positive [latex]y[/latex] axis and positive [latex]z[/latex] axis (call these points [latex]P_y[/latex] and [latex]P_z[/latex] ), construct two more congruent cones with bases in the [latex]xy[/latex] and [latex]xz[/latex] planes.

 

Make point [latex]D[/latex] a point somewhere within the volume of space occupied by the three cones. Make the [latex]x[/latex] coordinate of [latex]D[/latex] ([latex]x[/latex] coordinate of the antivector) equal to [latex]A_x[/latex] where [latex]A_x=(f)\upsilon_x[/latex] and [latex]\upsilon_x[/latex] is the angle between the [latex]x[/latex] axis and point [latex]D[/latex] with vertex at [latex]P_x[/latex]_. The [latex]y[/latex] and [latex]z[/latex] antivector coordinates are expressed similarly.

 

We should have three areas, [latex]A_x, A_y,[/latex] and[latex] A_z[/latex], which are the direction coordinates of point [latex]D[/latex] in the antivector space. We should also have a condition where [latex]\frac{A_x+A_y+A_z}{3}[/latex]

is a parameterization of a sphere of radius [latex]d[/latex].

 

 

on edit > the draft document has been reloaded with a more current version

Abridged Draft (1) (1).pdf

Edited December 13, 2016 by steveupson

[studiot]

Hello Steve,

 

You need to beef up Fig 5 by labelling the axis and then providing some explanation.

 

You state that the blue trace is [math]\left| {\frac{\pi }{4}} \right|[/math] but the modulus of pi divided by four is a constant.

 

So how does this work?

 

Otherwise you have some very pretty pictures that have come a long way from your original thesis.

 

As regards you terminology, I suggest you draw a distinction between what are known as pseudo-vectors or axial vectors to avoid confusing readers.

 

Edited December 13, 2016 by studiot

[steveupson]

The figure should have been fixed several iterations ago. Sorry about that. The axis and the curves were both wrong.

 

As far as how it works, I can't really explain it without using the math, which is probably a lot murkier than what I think it is. It must be impenetrable or else this would have all been done before.

 

There is a relationship between angles that occurs in 3D that does not occur in 2D. There is at least one of these relationships, and I would guess that there are more, probably.

 

The best that I can do to explain it is to solve a spherical right triangle and show these two simultaneous equations:

 

 

[latex]\cot\alpha = \cos\upsilon\tan\frac{\phi}{2 }[/latex]

[latex]\sin\frac{\lambda}{2 } = \sin\frac{\phi}{2 }\sin{\upsilon}[/latex]

 

 

I know that it must be possible to compose a function for [latex]\upsilon[/latex] from these equations that will define a curve similar to the one shown in Fig 5.

 

 

.

[AbstractDreamer]

If i understood you.

If A_x is an area, its the area of the base of a right cone.

[math]

A_x=

\pi(x tan(\nu_x))^2

[/math]

 

If its a coordinate the its just the radius of the circle of the base of right cone.

[math]

A_x = x(tan\nu_x)

[/math]

 

pi is the relationship between the circumference and diameter of a circle on a plane.

your function is a relationship between what and what? Between alpha and lamda?

 

I can't get my head around these planes.

Edited December 14, 2016 by AbstractDreamer

[steveupson]

The natural reaction is to try and specify a length somewhere in order to make this comprehensible with regards to other stuff that we already know about. I need to explain the concept of an antivector a little more.

 

With a standard direction vector in 3D, the length is a constant and the coordinates combine mathematically in order to specify the direction that lies along a subtended line though a specified point that is a distance of one from the origin. The direction doesn't have a magnitude or value that is independent of the ratio of lengths of the straight line distances specified by the values of the xyz coordinates.

 

With the antivector in 3D the opposite occurs. The direction coordinates, when taken together, sum to a particular magnitude and this magnitude represents a particular length or distance that the specified point is displaced from the origin. In other words, the magnitude of the three direction coordinates, once mathematically combined, will specify a sphere. The radius of the sphere doesn't have a magnitude or value (length) that is independent of the quantity of the direction values of the xyz coordinates.

[AbstractDreamer]

Ok so what is the difference between (the three coordinate antivector) and (the scalar value of the radius of the sphere)?

 

So summing the coords of the antivector gives you the radius?

 

So say you have the three-3D-direction coords perpendicular to each other

 

[math]

L_1 (x_1, y_1, z_1)

[/math]

 

[math]

L_2 (x_2, y_2, z_2)

[/math]

 

[math]

L_3 (x_3, y_3, z_3)

[/math]

 

Then the antivectors

[math]

A_x = (x_1 + x_2 + x_3),

A_y = (y_1 + y_2 + y_3),

A_z = (z_1 + z_2 + z_3),

[/math]

 

But then there would be three solutions to the radius? What am i missing?

 

What use does the three-coord-antivector give us?

Edited December 14, 2016 by AbstractDreamer

[steveupson]

Ok so what is the difference between (the three coordinate antivector) and (the scalar value of the radius of the sphere)?

 

So summing the coords of the antivector gives you the radius?

 

So say you have the three-3D-direction coords perpendicular to each other

 

[math]

L_1 (x_1, y_1, z_1)

[/math]

 

[math]

L_2 (x_2, y_2, z_2)

[/math]

 

[math]

L_3 (x_3, y_3, z_3)

[/math]

 

Then the antivectors

[math]

A_x = (x_1 + x_2 + x_3),

A_y = (y_1 + y_2 + y_3),

A_z = (z_1 + z_2 + z_3),

[/math]

 

But then there would be three solutions to the radius? What am i missing?

 

What use does the three-coord-antivector give us?

 

 

The direction coordinates are different than what you think they are. They are each the area under a curve which is found by using the function described above. Each of these coordinates is simply a number with a range of 0 to 1 (I think that's the range, but I need some help to make sure about it) that represents the 3D space enclosed by a cone which has point D on its surface.

 

When we combine these three numbers we get a result that is proportional to the distance between D and the origin, or in other words a radius of sphere of a certain size. What makes this representation different is that length is expressed as a formula or an equation that doesn't use length as a parameter.

 

As for its usefulness, there's probably something wrong with the math that makes it impossible for it to work properly. It's very likely that I have developed a bias about how this all works and that this bias is leading me to some wrong conclusions. My result are very inconsistent trying to do this on my own and I tend to accept expected results, no matter how sloppily obtained, while also thinking I must have made mistakes when there are unexpected results, no matter how careful I've been. Help is needed with the algebra in order to be able to work a few examples in a convincing manner.

 

Do you think you'd be able to help compose a function? Even some help running more graphs with different angles would be appreciated.

[AbstractDreamer]

Happy to get involved i need to learn stuff, but trying to visualise what going on first

 

Area under a curve is found by integration. I'm not sure how you can normalise the result to between 0 and 1. The result of integration can only be a constant if the integrand f(x) is such that x is only raised to the power of 1. This means you only get a non-x-value result (or a constant) if f(x) is a line not a curve. (I think). There are some odd techniques but i don't see how a curved function can have an area (within any significant limits) that doesn't have a variable that changes with respect to x.

 

I don't have mathematica. Also my experience in maths is around 2-3 weeks lol

 

So take your x-axis cone of unit length, where

[math]

\nu = \frac{\pi}{4}

[/math]

(i just drew two lines with limits, figuring out how to plot a real right cone with varying \nu)

 

 

 

 

 

So point D is somewhere on the two lines? The area enlosed between the x-axis, and f(x), between x=1 and x=D_x Would be

 

[math]

\int_{D_x}^1 (-x+1)dx

[/math]

Edited December 14, 2016 by AbstractDreamer

[steveupson]

I'm not explaining this very well. There isn't really a unit cone. There really isn't a unit anything because there are no lengths, but If you have to think of it as having a unit anything, it would have to be the sphere that is defined by the way the three cones intersect one another. The unit is a unit of direction in this scheme, not a unit of length. It becomes a length because of how these three individual directions are combined mathematically, and this length is expressed as three separate directions, analogous to the way a normal vector expresses direction as three separate lengths.

 

Again, we must start at the beginning because the math is important to understand. The two equations must be used in order to assign a value to each cone, and this value is not anything that has something analogous to it that I've been able to find in any of my research. The function expressed in the equations behaves differently than other functions so it is critical that the function be understood, or at least rewritten to show [latex]\alpha[/latex] and [latex]\lambda[/latex] are going to produce a different curve for each value of [latex]\upsilon[/latex]:

 

[latex]\cot\alpha = \cos\upsilon\tan\frac{\phi}{2 }[/latex]

 

[latex]\sin\frac{\lambda}{2 } = \sin\frac{\phi}{2 }\sin{\upsilon}[/latex]

 

When [latex]\upsilon[/latex] is near zero the function approaches a sine curve and the area beneath the curve will be 0.707 and when [latex]\upsilon[/latex] is near [latex]\frac{\pi}{4}[/latex] the function approaches a step function and the area beneath the curve approaches 0.0 (or 1, because the function wraps around somehow, depending on which side of the step function you look at.)

 

Each cone will have a value between 0 to 1 assigned to it, depending on the aperture, and this number is determined using the function. I suppose that the cone with an assignment of 1 could be referred to as a unit cone but this would be very misleading since the 1 simply specifies that the aperture is [latex]\frac{\pi}{2}[/latex] and doesn't really specify any unit length.

 

So, each of the three cones has point D on the surface and is represented by a value between 0 and 1. This value is the area under the curve produced by the function. The radius of the sphere is defined by the direction that results when the three values are combined.

Edited December 14, 2016 by steveupson

[swansont]

 

In order to create a system that uses antivectors, it is first necessary to quantify direction in a manner that is more similar to the way that length is currently quantified for use in standard vectors. The ability to define this quantification mathematically allows direction to be expressed as a scalar value which becomes the magnitude of the antivector, with length becoming the antivector quantity.

 

 

How do you deal with orthogonality? How is one length orthogonal to another?

[steveupson]

How do you deal with orthogonality? How is one length orthogonal to another?

I'm not sure that these answers will make any sense if the math behind the function isn't clear.

 

There is no specification of any length, other than what would be necessary to define where the surface of a sphere is positioned in space. This particular attribute (psuedo-length?) has no defined direction. It's a sphere located in space. The dimension (pseudo-length) is defined solely by the three direction coordinates, which don't relate to any particular length.

 

This scheme is separate (completely separate) from the standard system. There's no trivial method that I know of that would allow the ability to move back and forth between this system and the normal system.

 

As for the orthogonal question, orthogonality is an axiom, but I don't think that's what your question is asking. I can't seem to put the question in any context for which I have an answer.

[wtf]

Help is needed, once more, in expressing these equations using proper algebra and proper nomenclature. The math seems to work ok, it just needs to be corrected for grammar and puctuation, so to speak.

...

For the sake of those of us with limited attention spans, can you explain in simple words what is the purpose of all this? Are you inventing a new coordinate system for three-space? Or what, exactly. I skimmed your paper and the diagrams are beautiful but I have no idea what you're trying to do.

 

The metaphor is the "elevator pitch." In the field of tech startups, your elevator pitch is what you say to a venture capitalist if you happen to run into them in an elevator and hope that they will fund your company. You haven't got five hours or even five minutes. You have abut 15 seconds. Uber: "Airbnb for taxicabs." Boom. Three words, everyone gets the concept. If you saw the movie The Player, Hollywood does the same thing. You have a pitch meeting in which you have a couple of minutes to get the favorable attention of some studio executive who hears a dozen stories a day.

 

Can you just explain to me what you're doing? New coordinate system? Better in some way? Like stereographic projection perhaps? Or the roll/pitch/yaw system used by airplane pilots, in which the origin of the coordinate system moves with the airplane. Clever idea, right? "Coordinate system that moves with the observer." Boom. I got that in less than a second.

 

I need your elevator pitch please. I'm fairly sure I'm not the only one. To know if you've succeeded, it should be at most a sentence or two but ideally just a few words. "Improved coordinate system," or "Pretty pictures."

 

By the way, "reverse vectors" isn't helpful. Not to me anyway. I have no idea what a reverse vector is.

 

Can't help mentioning that we have recently seen a striking example in public life. "Make American great again" defeated "I'm with her." You can never overestimate the importance of your elevator pitch.

Edited December 17, 2016 by wtf

[studiot]

@wtf

 

So why wasn't your post at elevator length?

 

But I really like it anyway +1

[steveupson]

Im trying to toss a small pebble across the chasm of ignorance.

 

The chasm is documented extremely well in this video: www.youtube.com/watch?v=ZihywtixUYo

 

A lot of movers and shakers in the scientific community are convinced that the next big breakthrough will be in mathematics. Physics relies on a very specific geometry for almost all of the mathematics. What if there were another geometry that was possible, one in which the mathematical roles of length and direction are reversed?

 

What would a transverse Doppler effect of a transverse wave look like if it were modeled in a geometry based on direction instead of length?

 

What would quantum entanglement look like if it were modeled in a geometry based on direction instead of length?

 

This is the elevator pitch, I guess, but I have been discouraged many times from using it, though, because next Ill be pitching the method for extracting free energy from the N-layer using Teslas secret method.

 

The online math experts all claim that this is simply trivial math, simple rotations and stuff that is learned in a first-year course. Only none of them have been able to express the function. Why is that? Its been months (years) now.

 

I know that a function should be able to be composed from these two parallel equations. I know that there should be two arguments, [math]\upsilon[/math] and [math]\lambda[/math], and that someone should be able to express [math]\alpha[/math] as a function of [math]\lambda[/math] while [math]\upsilon[/math] is held constant.

 

If this isnt possible to do, then does anyone have any clue as to why not?

Edited December 18, 2016 by steveupson

[steveupson]

"Develop a new geometry."

[wtf]

"Develop a new geometry."

All I saw was pretty pictures in Euclidean 3-space. I think Euclidean 3-space is a solved problem, as they say. I am not understanding what you think is wrong with it.

 

The other thing is that you are talking about applying this to physics. But physics is all about infinite-dimensional function spaces in which the "length" of a vector is some norm given by an integral; and "angles" are replaced by the inner product.

 

If you are doing anything at all in Euclidean 3-space this is at best an exercise in second-year calculus. Physicists use manifolds and advanced abstract differential geometry.

 

I think in your exposition you are failing to put your ideas in proper context. Are you proposing a new way of looking at Euclidean 3-space? In terms of physics, are you trying to reform classical mechanics perhaps? Because certainly your work does not appear to be challenging the established orthodoxy in differential geometry.

 

I'm just tossing out ideas. I think what happens is that when people have pet theories, they have everything very clear in their own mind and they don't realize that they are failing to communicate a lot of context to their readers.

Edited December 19, 2016 by wtf

[steveupson]

Are you proposing a new way of looking at Euclidean 3-space?

 

 

 

Yes.

[swansont]

What would quantum entanglement look like if it were modeled in a geometry based on direction instead of length?

 

Entanglement isn't based on geometry, so probably looks no different.

 

 

But if you're implying that some phenomenon would be easier to understand or solve with your alternate geometry, how about solving a problem using your geometry.

[steveupson]

Entanglement isn't based on geometry, so probably looks no different.

 

 

But if you're implying that some phenomenon would be easier to understand or solve with your alternate geometry, how about solving a problem using your geometry.

 

I'd love to work through a few examples. That's the whole reason for this discussion. And, I sort of understand everyone's unwillingness to help me.

 

Is there some reason why these two equations can't be expressed as a function? Do you know the answer to that question?

 

I can understand that you wish to know why I would ask this question, and I've done my darnedest to explain. If you can't help then that's fine. There's no reason to challenge me in order to get me to explain. I've been trying to explain. Like much of physics, it's not really possible to explain certain things without using math.

 

One problem is that I don't look at things the same way as I used to, so it's difficult to recall the way I used to see things. An example: It's impossible for me to understand polarization without geometry. For me, it's based on geometry. Direction is not what I used to think it was.

[Strange]

And, I sort of understand everyone's unwillingness to help me.

 

 

That is a little unfair. You have had masses of help. The original animation created by someone else. The Mathematica model was created by someone else. The mathematics and most of the description in your paper was developed by someone else. People have tried to understand and help in this thread. (In fact, I might be tempted to ask what you have contributed? )

 

You now have a simple analytical description of the relationship between the angles (which I always said was possible, but you claimed was not). As to whether that can be described as a function or not: that is way beyond my math skills. But a function maps between an input value and an output value. It is not clear (to me) which of [latex]\alpha , \upsilon , \phi , \lambda[/latex] is the input and which is the output. Perhaps if you can answer that (if you have already, I apologise) then maybe someone can solve for those (or say why it isn't possible).

Edited December 19, 2016 by Strange

[steveupson]

 

 

That is a little unfair. You have had masses of help.

 

 

Of course you’re right. It’s not only unfair what I said, it’s also unreasonable for me to expect that everyone will automatically understand what I’m thinking. I’m not really frustrated with the help I’m getting, it’s my inability to communicate that’s making me nuts. My feelings of inadequacy are helped by knowing that you don’t know how to do this either. Thank you for your honesty about this. I know people only want to help.

 

The function is supposed to map [math]\alpha[/math] as a function of [math]\lambda[/math]. The angle [math]\upsilon[/math] is the angle of interest, and it will be an acute angle. For each value of [math]\upsilon[/math] there will be a different map, which I think means that there will be a different function for each of these values. This is because a function maps a set of inputs to outputs, and the set of outputs will vary depending on the value of [math]\upsilon[/math].

 

It’s been explained to me that since [math]\alpha[/math] has been expressed as a function of [math]\phi[/math], and [math]\lambda[/math] has been expressed as a function of [math]\phi[/math], that this is a parameterization of [math]\alpha = f(\lambda)[/math]. I know (I think) that in order to get the equations into the form that I’m after it will be necessary to cancel out [math]\phi[/math] in order to make one relationship. Also, I think what we're trying to do is to hold [math]\upsilon[/math] constant for each instance of [math]\alpha = f(\lambda)[/math]. I don't know how to write this algebraically.

 

Please understand that I'm not trying to change any existing methods for expressing mathematical representations of physical phenomena. This adventure would be in addition to existing methods which are not lacking in any way, except perhaps around the fringes when we approach the chasm.

 

 

 

 

 

I had a new idea of how to try and explain the overall strategy. The previous draft paper was an explanation of how [math]\alpha = f(\lambda)[/math] is derived. The following explanation is supposed to explain how the function is to be utilized. In order to try this approach, I’ve modified some artwork that’s been taken from another publication that I wish to credit:

 

https://www.researchgate.net/publication/282979400_Computing_Nash_Equilibria_and_Evolutionarily_Stable_States_of_Evolutionary_Games

 

Each of the two figures have points [math]xyz[/math]. We also have two points [math]a_1[/math] and [math]a_2[/math] shown on a spherical surface. There is an angle a_1(0,0,0)z [math]a_1z(0,0,0)[/math] that we’ll call the output of the function for [math]\upsilon_{a_1z}[/math]. There is an angle a_1(0,0,0)y [math]a_1y(0,0,0)[/math] that we’ll call the output of the function for [math]\upsilon_{a_1y}[/math], and there is an angle a_1(0,0,0)x [math]a_1x(0,0,0)[/math] we’ll call [math]\upsilon_{a_1x}[/math].

 

<on edit: xyz are the vertices>

 

These three angles are each expressed as a separate 2D value based on the function that we’re trying to compose.

 

We also have three more angles that relate to point [math]a_2[/math] on the surface of the sphere which are [math]\upsilon_{a_2x}[/math], [math]\upsilon_{a_2y}[/math], and [math]\upsilon_{a_2z}[/math]. Each of these is also expressed as a 2D value.

 

The theory is that, when done properly, [math]\upsilon_{a_1x}+\upsilon_{a_1y}+\upsilon_{a_1z}=\upsilon_{a_2x}+\upsilon_{a_2y}+\upsilon_{a_2z}[/math]. All the points on the surface yield the same sum.

 

The entire explanation can be repeated for points [math]b_1[/math] and [math]b_2[/math] in the second illustration. All the points on that surface will yield the same sum, which will be different than the one in the first example.

 

In both cases, the magnitude of the sum is the "reverse vector" or "antivector" of the length represented by the radius of the surface. It’s exactly a mirror image (mathematically) of the normal vector. The length doesn’t have any specified direction and it’s completely specified by the three direction coordinates, or angles.

 

Or, I'm imagining things, in which case our work will be wasted but you'll still have my undying appreciation.

Edited December 19, 2016 by steveupson

[steveupson]

The function [latex]\alpha=f(\lambda)[/latex] turns out to be:

[latex]\alpha={\cot}^{-1 }(\cos\upsilon\tan{\sin}^{-1}(\frac{\sin\frac{\lambda}{ 2}}{ \sin\upsilon}))[/latex]

It is a smooth function that approaches a sine curve when [latex]\upsilon\to0[/latex], and that approaches a hyperbola when [latex]\upsilon\to\frac{\pi}{2}[/latex].

[steveupson]

The legends for the Bizarro vector are not correct in the above post. In any event, the basic method is that there are angles [latex]\infty\,\,{x_0}\,\,{a_1}[/latex], [latex]\infty\,\,{y_0}\,\,{a_1}[/latex], and [latex]\infty\,\,{z_0}\,\,{a_1}[/latex]. When these are all expressed as the area [latex] A=(f)\upsilon [/latex] then the sum of these angles is equal to the sum of the similar angles for point [latex]a_2[/latex].