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Ray vs. Line: Equivalent Freedoms?

  1. Sep 15, 2012 #1
    I’m curious, but no mathematician. Euclidean geometry is the only type for which I've had training (45 years ago). Apologies in advance for misused terms.

    A line, as I recall, extends indefinitely in opposite directions. It has no definite “origin” or "center". If we view “translational freedom” as the potential for unrestricted displacement, a line can be said to offer both forward and reverse translational freedoms.

    A ray, by contrast, has a locatable origin (a closed end) and extends indefinitely in one direction. So, a ray has only forward translational freedom. Structurally then, a ray appears to be half of a line. In fact, two rays joined back-to-back (180° angle), create a line.

    Still, I am of the impression that a ray and a line offer equivalent total freedoms. By this, I include rotational freedoms for which I find a symmetry.

    A ray, having a single reference direction “away” from its origin, would seem to offer both an absolute forward (say clockwise) and reverse (say counterclockwise) rotations, with respect to that direction.

    A line however, having no single reference direction, can offer only forward (non-negative) rotation. For example, observers on either side of an object rotating about a line would disagree about its direction being clockwise or counterclockwise. It rotates or it doesn't, but there's no absolute direction.

    Is it reasonable to assert that a ray offers twice the rotational freedoms of a line?
     
    Last edited: Sep 15, 2012
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  3. Sep 15, 2012 #2

    chiro

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    Hey FaraDave.

    What exactly do you mean by rotational symmetry with regards to a line? (I've heard about symmetries involving two or higher dimensional objects but never 1).

    I'm going to assume you mean that if you put a line in some plane, you can rotate it twice 180 degrees in the plane and it won't change, but for a ray you can't rotate it all to preserve the nature of the object (Since the ray is oriented).

    In symmetry, orientation plays a big role in how the number of symmetries are determined: typically if you invert the chirality of an object you don't get anything in the normal space of realizations (i.e. if you are considering a group with a 3D frame that is right handed, then a left-handed version is not going to be included in general).

    However because you only have one object (i.e. a one dimensional ray), you don't really have any notion of chirality since this requires two objects and typically, rotation needs to be considered in the context of the chirality of the basis involved, and rotations are usually defined to keep the chirality of the space (through having a determinant of +1 as opposed to -1 which changes things).

    Maybe you could outline what you mean by a rotational freedom.
     
  4. Sep 16, 2012 #3
    Thanks chiro. I should have avoided the word “symmetry”, as I use it as a layperson. Here’s what I meant.

    A line offers two directions of unrestricted translational displacement, but I see just one kind of rotation. 2 + 1 = 3 freedoms (of movement).

    A ray offers one direction of unrestricted translational displacement (away from the origin) but has two distinct rotational directions (clockwise and counterclockwise) with respect to the ray’s arrow (by which all observers can agree). 1 + 2 = 3 freedoms.

    Even though a ray is structurally half of a line, it seems to compensate for its decrease in translational freedoms with an increase in rotational freedoms.

    For the sake of familiarity, I view the line and the ray embedded in a 3-D space (manifold?). By “rotational freedom” I mean the freedom to move in a distinct, unlimited angular sense, agreed by all observers.

    Suppose you have a circular space station in deep space. Everyone can agree it's rotating by the presence of artificial gravity (centrifugal force). But is it rotating clockwise or counterclockwise? That question is meaningless unless a ray (a reference direction) is imposed on the axis of rotation. Having a line as the axis of rotation does nothing to answer the question.
     

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  5. Sep 16, 2012 #4

    chiro

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    So basically you're saying that if a ray rotates, it's change in orientation and position can be agreed on all observers (if they are all in the same space relative to the reference frame the ray is in) to have specific properties in their own reference frame?

    The idea of orientation in any dimensional space requires two objects that have both a length and a direction, and the general formulation to find said orientation properties of one vector with respect to another is known as the inner product.

    The outer product allows you to find another relative quantity where you get a resultant vector by checking the order of one vector relative to another, but in a specific order.

    But both require quantities that have magnitude and length: you can't do either of them without the two and the absolute basic requirement is that you are dealing with a two-dimensional space at the minimum.

    So your thinking about having a ray as a reference point is partially correct, because all notions of relativity between objects consider not only the direction but the length.

    You can not really do an entangled relative analysis between two "infinite" vectors in the case where you are finding the relativity using the common geometric algebra techniques (i.e. inner and outer products).

    You could transform them to unit vectors and gauge the relativity that way, and this is exactly what we do when we only want to consider the direction component but not magnitude, and this process is known as normalization.

    In your space example, you are right in that you need a directed quantity to gauge a sense of relativity between things (i.e. vectors), but again be careful about introducing rays as vectors with an infinite length: if you only want to look at direction without magnitude then just consider normalized vectors which do this for you.
     
  6. Sep 16, 2012 #5
    To be clear, by “rotational freedom” about a ray or line, I mean that the length of the object is the axis of rotation. There is no change in orientation of the ray or line. The axis of rotation of the circular space station for example, can be infinite in length, and does not undergo a change in position or orientation when the space station begins to rotate.
     
    Last edited: Sep 16, 2012
  7. Sep 16, 2012 #6
    My fault - Begin Again:

    Considered as an axis of rotation, is it reasonable to assert that a ray offers twice the rotational directions as a line?
     
    Last edited: Sep 16, 2012
  8. Sep 16, 2012 #7

    chiro

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    You can't really rotate around a line since you need an orientation to rotate around.

    A rotation always requires an orientation no matter how you put it and a line doesn't have one.

    Everything is relative and relativity requires some kind of way of relating two things by direction and magnitude (hence why we have vectors, and why they are useful).

    Even if you don't want to think about an axis having a direction, the effect of the rotation certainly has an orientation of its own.

    The line doesn't have any orientation whatsoever and trying to think about it in a relative way by considering rotations on that object is not going to be consistent.

    The main point is that somewhere along the line you will need to relate something to another, and when that happens you will need direction and length and that means arrows (vectors).
     
  9. Sep 16, 2012 #8

    So can you better define "translational freedom"? Say we take the plane the line is in to be the x,y plane, and lets say the line is x=1; A vertical line going through '1,0', So what I think your saying is that we can add or subtract Δy to all the points and it leaves the same line right? I would think that in the case of the ray, lets say it starts at (1,0) and extends infinitely vertically, if you add anything to either the x's or y's you change the graph, it doesn't restrict it to one direction, it restricts it to none.

    If that is what you mean then also with rotation, you can rotate the line by any amount other than 180° and change the graph(And agree that it was either x° to the left or 360-x° to the right, If you are using the line/ray as the axis of rotation(2d in the case of the space station you see a ring and a point in the center that is the line/ray), then it is the same as the ray in that neither change. Since the rotating point obviously doesn't change, and the ray (in the same orientation) is either a point or not a point, either way rotation leaves it unchanged. But with any other axis of rotation you can tell equally well that the graph has changed in any instance that the line doesn't flip exactly 180°... Does that make sense?
     
  10. Sep 16, 2012 #9
    I parse your reply for clarity. I do so with respect and appreciation for your patience. I've thought about this a long time and hope it will be food for thought to others. But I also suspect that the subject must have been dealt with long ago.

    An object in Flatland can turn around 360° (or any angle). When it does so, it lacks even a dimension for an axis, let alone specifying the axis as a line vs. a ray. The axis of rotation is imaginary.

    Even so, any outside observer can say, "It rotates clockwise." or "It rotates counterclockwise." Here, orientation is a function of the necessarily, self-centered observer. That is, the observer imposes a reference direction (a ray) on the system being observed.

    Correct, a line does not have AN orientation (forward or reverse). I suggest that a line has TWO (opposing) orientations. I consider a line to be a compound linear object, comprised by two rays, each providing an orientation.

    Rotation about a line seems to me, rotation in two directions at once, simultaneous rotations about each of the two rays comprising the line. I see rotation about a line axis as essentially "bipolar" rotation (about two opposite-pointing polar vectors)..

    Feynman (Six Not-So-Easy Pieces, p.35) identifies vectors of two types, "polar vectors" (ray-like) which reflect oppositely (arrow reversed) and "axial vectors" (line-like) which reflect unchanged (having essentially, arrows on both ends).

    It might be a good idea for me to restate my question in the General Physics thread but since mathematics has contributed so much to physics, I thought there might be a "pure" answer here.

    Well, yes and no (IMHO). Some phenomena, such as centrifugal force (e.g. artificial gravity), don't seem to care if rotation is in one direction or another (or both simultaneously). Others, such as magnetism, do.

    Consider a spinning electron. An electron is an electric monopole, purely negative charge. But when it spins (about a spatial axis, a line), it becomes a magnetic dipole! That is, it appears that the electron invokes both a right hand rule (for the North magnetic pole) AND a left hand rule (for the South magnetic pole), at the same time.

    The magnetic poles diverge maximally from the poles (opposite ends of the rotational axis), yet at the equator, where the two rotations meet, no magnetic field lines emerge, as if the rotations cancel each other.
     
    Last edited: Sep 16, 2012
  11. Sep 16, 2012 #10
    By “translational freedom” I mean the ability to displace without restriction. Consider a bead on a string. If the string is line-like, it extends infinitely in two directions. The bead has freedom to displace linearly, without restriction in two directions. (I call them “forward” and “reverse”, without specifying which is which.)

    If the string is ray-like, having an origin and extending infinitely from there, the bead has freedom to displace unrestricted in just one direction (“away” from the origin). It might be able to displace toward the origin but that would be limited by the end of the string, thus not qualifying as a freedom, by my definition. The bead is imprisoned on the closed end.

    By “rotation” I mean the ability to roll the object (line or ray) as if rolling a pencil between your fingers (or the bead on the string). By “rotational freedom” I mean the number of directions this can be done for which all observers could agree. Because a ray has a direction which can be defined for all observers (i.e. away from the origin), it provides two recognizable directions for rotation (clockwise and counterclockwise). This is easily seen by rolling a pencil with an eraser on one end.

    I assert that a line, because it lacks a single reference direction, provides only that it is rolling or not, but observers will not uniformly recognize the direction to the roll. Think of rolling a pencil with points on both ends. I suggest the roll is bidirectional.
     
    Last edited: Sep 16, 2012
  12. Sep 16, 2012 #11
    I don't know anything about mono/di -poles, but since the north and south poles are opposite, and the field lines I believe are said to originate at one and terminate at the other, I would think that there is only one of those rules, causing loops to "move" through the center in an oriented line, from either south to north or north to south depending on naming convention. Just how it seems to me...perhaps I should stay out of this one.
     
  13. Sep 16, 2012 #12
    Also, if you consider an empty universe with a line going through it and it's just me and that line, then yeah, you are correct, it just rotates I guess, but if you have some arbitrary point of reference then you can know you are either + or - that point, and come up with a convention, just as we have for the rotational vector in rotational kinematics, a clock can't really be said to turn clockwise or counterclockwise if you remove the face and backing and made it where you could only see hands turning from either side. But luckily in all geometry I have had to worry about there -is- a point of reference even for a line, if I am considering a point p on the line, and there is a convention to determine which way it is rotating.

    again...I think that is right...
     
  14. Sep 17, 2012 #13
    I could have done a better job describing the Right Hand Rule as it applies to spinning electrically charged particles. By convention, if a positively charged particle rotates about an axis indicated by the thumb of your right hand, a north magnetic pole is found in the direction of that thumb when the spin is indicated by the direction of your curled right fingers. (In the picture, the current in the wire is considered to be positive charges and can, by reducing the radius of the coils to zero be treated as a single spinning positron.)

    You’re correct that an opposing south pole is always found as well. But that’s one of the reasons for my question. Every spatial axis is a line, extending infinitely in two directions and thus, equivalent to two opposing rays. Science hasn’t found any magnetic monopoles in nature but we don’t appear to have access to any ray-like axes either. A version of my question could be, “What would result by spinning an electron about a dimensional ray instead of a line?” I’ll save that for a thread in the physics section.
     

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  15. Sep 17, 2012 #14
    To summarize my take up to this point.

    A ray, by virtue of its singular direction ([itex]\rightarrow[/itex]), provides two distinct spin modes, labeled by convention, Right-handed and Left-handed, with respect to that direction.

    A line, because it extends in two opposite directions ([itex]\leftrightarrow[/itex]), offers a single spin mode combining Right- and Left-handedness.

    If I join my hands at the small digits and use my thumbs to indicate line extensions ([itex]\leftrightarrow[/itex]), there is just one spin mode, indicated by my curled fingers. Can you tell if the photo below was reversed vertically or horizontally?
     

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    Last edited: Sep 17, 2012
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