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  1. Additional Question About Surfaces in Higher Dimensions

    Alpha and lambda relate to one another geometrically.
  2. Additional Question About Surfaces in Higher Dimensions

    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?
  3. Additional Question About Surfaces in Higher Dimensions

    Sorry, bad choice of words. It indicates that it is somewhat different than what you say.
  4. Additional Question About Surfaces in Higher Dimensions

    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.
  5. Additional Question About Surfaces in Higher Dimensions

    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.
  6. Additional Question About Surfaces in Higher Dimensions

    The concept that is being examined here in this thread is the one where it was stated that “we can establish a one-to-one mapping between any Rm and any Rn.” The model for [latex]\alpha=f(\lambda)[/latex] is the example of [latex]R^3[/latex] that cannot be mapped to [latex]R^2[/latex], or at least that's how it appears. The point [latex]S_O[/latex] at the origin, and unit sphere S, are all that exists in a form that can be mapped, and these are given, not derived. There is nothing in that accessible paper that will help with this particular mapping. Please, if it isn't too much to ask, can you tell me what you think I am saying? We are not talking to one another. What is it that you hear me saying? What is it that the math is showing you? As is the case with the circle becoming a point, the entire map vanishes in [latex]R^2[/latex].
  7. Additional Question About Surfaces in Higher Dimensions

    This a verbal description of what you get when you set r to zero. The circle doesn't exist (or does it still exist as is being hinted at, because if it still exists then this answer will have to be rethought.) When the circle does't exist then there is no arc length [latex]\phi[/latex] because this arc length is part of a nonexistent circle. [latex]\lambda, \upsilon, \alpha[/latex] will also fail to exist because they have either had one or two endpoints merged into the single point at the North poles SN. Also, since z is the distance of the center of the unit sphere from the center of the coordinate system, all we seem to have remaining is this unit sphere, without anything else to relate to anything else. I am assuming that at this time we are left with [latex]0=f(0)[/latex]. One more time, the relationship either exists in 3D or it doesn't exist at all. If [latex]\upsilon[/latex] is one infinitesimal from zero the function is one infinitesimal from the sin function and when [latex]\upsilon[/latex] is one infinitesimal from [latex]\frac{\pi}{2}[/latex] then the function is one infinitesimal from the hyperbolic function. I don't know if there are any more ways to express what is going on here. on edit>>>> probably should be: [latex]\varnothing=F(\varnothing)[/latex]
  8. Additional Question About Surfaces in Higher Dimensions

    You're not arguing that a point is a circle are you? Because that was the question I was answering.
  9. Additional Question About Surfaces in Higher Dimensions

    As for what happens when a geometric object becomes a point, I think it fails to exist. According to Euclid: "A point is that which has no part."
  10. Additional Question About Surfaces in Higher Dimensions

    The divide-by-zero condition only exists because the model of the object cannot be completely represented in only two dimensions. Whenever the model is completely parameterized it must have three dimensions in order to describe it, and the divide by zero error does not occur. This does explain why the tridentity is only an identity in three dimensions and cannot be represented in only two. I see how the math works, but is that the reason why the function cannot be applied to surfaces? Even surfaces of many dimensions? I think this may be the case. This would be the reason why tensors have not been able to accurately describe fields that alter 3-dimensional space. They are unable to express this particular aspect of it.
  11. Additional Question About Surfaces in Higher Dimensions

    Why should that matter? We still have proper expressions for [latex]sin\upsilon[/latex] and [latex]cos\upsilon[/latex]. Why doesn't the identity hold true? How does the identity know not to be in effect whenever [latex]\upsilon,\lambda,\phi[/latex] are in the same plane?
  12. Additional Question About Surfaces in Higher Dimensions

    No, I hadn't seen it, but it isn't important to this problem. In this problem the tangent plane is tangent to cone N at point P. I think that it would be useful to apply some other method such as vector analysis or quaternions or something besides spherical trigonometry to validate the solution obtained through spherical trigonometry. If there are discrepancies between methods then knowing what these discrepancies are would be useful. We have already uncovered a major anomaly concerning the trigonometric identity. We should probably work toward a better understanding of why the identity exists in 3D and not in 2D, and what the implications are concerning the spacetime question. Don't misunderstand me. This information deserves great skepticism. That's a good thing. Since everyone probably doesn't have a spherical trigonometry textbook:
  13. Additional Question About Surfaces in Higher Dimensions

    Right. Thanks. I do tend to remove my genius spectacles from time to time.
  14. Additional Question About Surfaces in Higher Dimensions

    I apologize. I edited my previous post once I realized that I had not furnished an answer to your question. I don't see anything new in your post at all. Once more, what I'm showing you is new. And I'm sort of hurt that you think it is arrogance. It's total and complete frustration. I keep answering the same questions over and over again, using different approaches, and everyone seems to just ignore what I say and assume that I don't know what I am talking about. What does the "identity" mean to you at this point? Does it mean anything to you? The Coordinate System In the model as illustrated in this particular case, small circle C has a circumference that is 45º of latitude. We will be using the conventional terminology where a circle on the surface of a sphere that is made by the intersection of the sphere with a plane passing through the sphere center is called a great circle. All other circles, where the intersecting plane does not pass through the sphere center SO, are called small circles. If we view the model such that the equator of sphere S is the intersection of the surface with the tan-colored plane, and where the main axis of the sphere is along the intersection of the green-colored and magenta-colored planes, then circle C can be said to be tilted 45º from the natural orientation of a circle of latitude. In this case, circle C intersects the north pole SN and is tangent to the equator. The center CO of circle C, together with point SO and point SN, form angle ∠COSOSN. There is a main axis passing through SN and forming one ray of this angle which we'll call the cardinal axis, and there's another axis that passes through CO which we'll call the ordinal axis. The relationship between these two axes is that they are 45º to one another in the instance shown, but a similar model can be constructed using a small circle that is other than a 45º latitude. As an example, there could be a small circle with a circumference of 10º that is tilted to an angle of 10º from the horizontal, and this circle will also intersect the north pole. In this case the angle between the cardinal and ordinal axes will be 10º. This angle between the cardinal and ordinal axes is the upsilon angle, [latex]\upsilon[/latex]. A "casual observer" would think that the initial position of the colored rectangles in the animation represents the planes defined by the coordinate axes (the xy-, xz-, and yz-planes). However, that’s not the correct interpretation. The point where the rectangles meet (Fig. 2) is the center SO of unit sphere S, having position (0,0,sz), and is not, in general, the origin of the coordinate system (0,0,0). Circle C is, in general, a small circle, and is the intersection of the xy-plane (which isn't a plane indicated by any of the animated rectangular planes) and sphere S. Its orientation can be regarded as a circle of latitude of a specific size (in the example, 45º that has been tilted by the same angle (45º) such that it intersects north pole SN of sphere S. Circle C can be described as the base of cone N (not shown), having sphere center SO as the apex, having the ordinal axis as the cone axis, and having the cardinal axis as a generatrix. Line COSO lies along the ordinal axis and line SNSO lies along the cardinal axis. In the model as illustrated, the angle between the axis of cone N and the axis of sphere S is 45º, giving an aperture of 90º (maximum angle between two generatrix lines) but this angle is variable. The axis of cone N is the z-axis. The circle center CO is the origin of the coordinate system (0,0,0). North pole SN is facing toward the top in the animation, and lies along the x-axis, which is oriented such that the positive x-axis passes through SN. The radius of circle C is r. Since sphere S is a unit sphere, 0 < r < 1 The y-axis is oriented such that the positive y-axis is toward the viewer in the illustration. Figure 2. Coordinate System The Planes The two planes that remain stationary in the animation each contain SO and a line tangent to circle C. For tan-colored plane [latex]\mathbb{T}[/latex] this tangent is at (-r,0,0) and for magenta-colored plane [latex]\mathbb{M}[/latex] it is at SN which is also (r,0,0). Because these two planes contain the center of the sphere, which need not be in the xy-plane, these stationary planes are not parallel to, or perpendicular to, the xy-plane except for when r is at either limit r = 0, or r = 1. The point on C where 3 moving planes intersect is P. Blue-colored plane [latex]\mathbb{B}[/latex] contains SO and a line tangent to the circle at point P. The animation starts with P at (r,0,0) and then moves P along a 180º arc of circle C. Green-colored plane [latex]\mathbb{G}[/latex] contains the point P and pivots about an axis as P moves, but this axis is not one of the coordinate axes. It is an axis defined by a line containing the line segment from SO to SN, which is the main or cardinal axis of sphere S. Yellow-colored plane [latex]\mathbb{Y}[/latex] (or light green) contains points SO and P, and is perpendicular to [latex]\mathbb{G}[/latex]. The Angles Plane [latex]\mathbb{G}[/latex] can be considered the longitude plane because the great circle made by its intersection with sphere S is always a line of longitude. Plane [latex]\mathbb{B}[/latex] can be considered the tangent plane since it always contains a line tangent to circle C at point P. The dihedral angle between planes [latex]\mathbb{G}[/latex] and [latex]\mathbb{B}[/latex] is angle [latex]\alpha[/latex], the angle of interest. Plane [latex]\mathbb{Y}[/latex] can be considered the elevation plane and it stays perpendicular to plane [latex]\mathbb{G}[/latex] and passes through sphere center SO and point P. The smallest angle between the cardinal axis of the sphere and plane [latex]\mathbb{G}[/latex] is angle [latex]\lambda[/latex] , or ∠SNSOP. Point P can be defined as an arc length along circle C equal to angle The Problem The objective is to define a family of functions which are based on different values of [latex]\upsilon[/latex], and which express α as a function of [latex]\lambda[/latex] . The approach will be to find [latex]\alpha[/latex] and [latex]\lambda[/latex], each as a function of [latex]\phi[/latex]. One of the methods used here will be to solve a spherical isosceles triangle using spherical trigonometry. At the top right corner of the animation there is a set of numbers representing [latex]\lambda[/latex] that change between 0° and 90°, while below them, in the rectangular box, there is another set of numbers representing [latex]\alpha[/latex] that also change between 0° and 90°. The mathematical model used for the animation defines a specific member of a family of functions. The z-coordinate for the center of sphere S defines which member of the family we are analyzing. The model shows [latex]\alpha[/latex] as a function of [latex]\lambda[/latex] with angle [latex]\upsilon[/latex] equal to 45°, or z = [latex]\frac{1}{\sqrt{2}}[/latex]. Also, because we are using a unit sphere, r = sin [latex]\upsilon[/latex] = [latex]\sqrt{1-{s_z}^2}[/latex] (Unit sphere center SO can move up or down along the z-axis and the north pole will remain the north pole and circle C will still pass though the pole SN because angle [latex]\upsilon[/latex] changes accordingly in order to keep this true.) It should be mentioned that, although a sphere is used for construction of the model that is presented here, there actually is no sphere involved in the function itself. In other words, there is no two-dimensional surface involving spherical excess (parallel transport) or anything like that. The sphere is simply used as an aid in visualizing how the object is constructed and spherical trigonometry used in solving for some of the unknowns which can be found by analysis of the sphere’s gross occupation of Euclidean 3-space. Lambda as a Function of Phi Point SC is the intersection of the z-axis with sphere S such that ∠SNSOCO = [latex]\upsilon[/latex] = ∠SNSOSC. There is a spherical triangle ∆abc that, when solved, will express the relationship between [latex]\alpha[/latex], [latex]\phi[/latex], [latex]\lambda[/latex], and [latex]\upsilon[/latex] such that: Side a is great circle arc of length [latex]\lambda[/latex] Side b is great circle arc of length [latex]\upsilon[/latex], or ∠SCSOSN Side c is great circle arc of length [latex]\upsilon[/latex], or ∠SCSOP [latex]\alpha[/latex] = ∠[latex]\mathbb{BG}[/latex] [latex]\phi[/latex] = ∠SNCOP = dihedral ∠SN-SOCO-P [latex]\lambda[/latex] = ∠PSOSN [latex]\upsilon[/latex] = ∠SNSOCO Since both point P and point SN are on the unit sphere with center SO, the triangle that these 3 points form is an isosceles triangle with a pair of sides, PSO and SNSO, each having length = 1. The angle between the two equal sides is [latex]\lambda[/latex]. If we take [latex]\lambda[/latex] as given, we can find the length h of the base of this isosceles triangle, which is the distance between SN and P. Then, using length h we can find [latex]\phi[/latex] by regarding the solution as a 2-dimensional problem that is set in the xy-plane where circle C lies. For the first step, h is the chord of the great circle arc [latex]\lambda[/latex] of sphere S, or h = [latex]{2}\sin\frac{\lambda}{2}[/latex]. For the second step, with a circle of radius r and center (0,0), h will be the chord length of the arc segment from (r,0) to P, or h = [latex]{2}{r}\sin\frac{\phi}{2}[/latex]. These equations give a parametric representation of [latex]\lambda[/latex] as a function of [latex]\phi[/latex]. Figure 3. Isosceles Spherical Triangle ∆abc Alpha as a Function of Phi Referring to Fig. 4, since b and c both have endpoints on circle C, Δabc is an isosceles triangle with dihedral angle [latex]\phi[/latex] between the two equal sides. If we construct a great circle arc from point SC to midpoint D of side a, we will bisect Δabc into two congruent right spherical triangles. The bisected dihedral angle at SC[latex]=\frac{\phi}{2}[/latex], and side c = b = [latex]\upsilon[/latex]. We can solve for the remaining dihedral angle α at P, or at SN, since these are equal. Since side c is perpendicular (Fig. 4) to the great circle arc made by plane B, this dihedral angle at P will be α. [latex]\cot\alpha=\cos\upsilon\:\tan\frac{\phi}{2}[/latex] Then, solving for side a (which was bisected) we get: [latex]\sin\frac{\lambda}{2}=\sin\frac{\phi}{2}\:\sin\upsilon[/latex] Also, because we are using a unit sphere, [latex]{r}=\sin\upsilon=\sqrt{{1}-{s_z}^2}[/latex] (Unit sphere center SO can move up or down along the z-axis and the north pole will remain the north pole and circle C will still pass though the pole SN because [latex]\upsilon[/latex] angle changes accordingly in order to keep this true. Figure 4. Dihedral Angle α Variables C = circle made by intersection of the xy-plane and unit sphere S CO = (0,0,0) = origin = center of circle C r = [latex]\sqrt{1-{s_z}^2}[/latex] = radius of circle C (r,0,0) = intersection of positive x-axis with unit sphere S = SN (north pole) (-r,0,0) = intersection of negative x-axis with unit sphere S (0,r,0) = intersection of positive y-axis with unit sphere S (0,-r,0) = intersection of negative y-axis with unit sphere S z = length from origin to center of unit sphere S = unit sphere SO = (0,0, sz ) = center of unit sphere S, where -1 < sz < 1 SN = (r,0,0) = north pole of unit sphere S SC = (0,0,1-z) = projection of line segment between SO and CO to the surface of sphere S N = cone having circle C as its base and sphere center SO as its apex P = (r cos[latex]\phi[/latex], r sin[latex]\phi[/latex],0) = tangent point on circle C where planes [latex]\mathbb{B}[/latex], [latex]\mathbb{G}[/latex], and [latex]\mathbb{Y}[/latex] intersect h = arc length of circle C from P to SN h = [latex]{2r}\sin\frac{\phi}{2}[/latex] (from analyzing a chord in circle C) h = [latex]{2}\sin\frac{\lambda}{2}[/latex] (from analyzing isosceles triangle ∆SNSOP, which has two sides of length 1) [latex]\mathbb{T}[/latex] = plane containing point SO and also the equator of sphere S [latex]\mathbb{M}[/latex] = plane containing point SO and also the tangent (in the xy-plane) to circle C at point SN [latex]\mathbb{B}[/latex] = plane containing point SO and also the tangent (in the xy-plane) to circle C at point P [latex]\mathbb{G}[/latex] = plane containing point SO and point P and point SN [latex]\mathbb{Y}[/latex] = plane containing point SO and point P and plane [latex]\mathbb{Y}[/latex] is perpendicular to plane [latex]\mathbb{G}[/latex] a = great circle arc λ b = great circle arc υ c = great circle arc ∠SCSOP D = midpoint of side a [latex]\alpha[/latex] = ∠[latex]\mathbb{BG}[/latex] [latex]\phi[/latex] = arc length in degrees from (r,0,0) to point P = ∠SNCOP = dihedral ∠SN-SOCO-P [latex]\lambda[/latex] = ∠PSOSN [latex]\upsilon[/latex] = angle between the axis of cone N and the cardinal axis of sphere S = ∠SNSOCO.
  15. Additional Question About Surfaces in Higher Dimensions

    Of course I read your post. You are talking about an angle. It has no bearing at all on the math that I have presented. I think that all of us know about what you are showing in your post. It's available in any book on rotations or basic kinematics. Trust me that I know what you said. You are the person who won't listen to me. What I have shown you is not in any books. I read every word you say. We agree on almost everything. Try and focus on the math that we disagree about. on edit>>> sorry, I forgot to answer your question. The black loop is a small circle on the imaginary spherical surface that allows us to do spherical trigonometry on a 3D object (for instance a ball, or any other object, including empty space.) The size of the small circle is related to the size of the sphere through the value or magnitude of [latex]\upsilon[/latex]. It is shown in the animation in order to help illustrate how the intersection of the three moving planes can be tracked along the spherical surface. The slope of the tangent to this circle is [latex]\alpha[/latex]. This is the first of two angles, oriented in space, that relate to one another geometrically. The other angle, [latex]\lambda[/latex] is the angle PSOSN. Its used to solve this particular geometry by another method that does involve spherical trigonometry. The circle is used in order to create an isosceles spherical triangle that is solved in order to find [latex]\alpha[/latex] and [latex]\lambda[/latex] in relation to [latex]\phi[/latex], which is the arc length from SN to P along the black loop curve, which is otherwise called the small circle.