Flatland being of course the famous fictional 2D universe. I’m having some difficulty wrapping my brain around how rotation works in Flatland. Of course objects in 2D can change their orientation; for example, a triangle can turn so that a vertice facing north now faces east, and other 2D entities can see this with no problem. However, the axis of such a rotation is at right angles to Flatland, extending into 3D space, and is not perceptible to Flatlanders and can only be thought of by them in terms of mathematical abstraction. IOW, I wonder if the concept of rotation can’t truly be defined in less than three dimensions, since a plane of rotation seems to inherently imply an axis. Except of course 2D objects can change their orientation in a plane; so am I wrong there or do the Flatlanders consider a change of orientation something other than what we would call “rotation”?
The real difficulty came when I tried to generalize the concept from two to three dimensions. A 2D object can only rotate in one specific way without altering it’s shape and size in the plane. And for a 3D object you not only have multiple possible planes of rotation, but the axises (sp?) of those rotations are also in 3D space. I tried to imagine what 3D movement would be the analogy of a 2D rotation- the object changes orientation without altering it’s shape or size, but the axis of rotation is in a higher dimension- and couldn’t. Any ideas there?
IANA mathematician, but it seems to me if you’re looking for an analogous situation, you don’t just need another imaginary two-dimensional axis. You need an imaginary three-dimensional structure which is at simultaneous right angles to two planes of movement in a three-dimensional orbit.
The word “rotation” can mean many different things. For example, it can mean any continuous motion which never changes size or shape. It’s only in three dimensions that these can be specified as by some angle around some axis; in higher dimensions, such a rotation can fix more than one axis simultaneously, and more than one such movement can cause a swing by the same angle around the same axis.
I guess I don’t understand why you consider your axis to need a third dimension.
It seems to me that the residents of Flatland would be perfectly happy to identify their axis as a 2-D point, and would scratch their heads at your question.
Thinking about it a bit, in a rotation, you’re rotating about every axis perpendicular to the plane of rotation. You could as easily say you’re rotating about the time axis, or (assuming you’re rotating in the X-Y plane), the T+Z axis.
You just naturally pick the Z-axis because it’s the one you know about that’s left.
The residents of Flatland would be perfectly happy to consider rotation a concept that has nothing to do with an axis, as is a natural thing to do.
But, as I said, there are different accounts of what “rotation” is. It could mean circular movement within a 2d plane. It could mean arbitrary rigid continuous linear transformation (probably the way mathematicians use the term most often). It could mean rigid linear transformation fixing a single line. These concepts all happen to coincide in 3d, but they all come apart in higher dimensions, just as they aren’t quite the same in 2d.
Sorry, I should’ve said “continuously parametrized linear transformation” here, or something like that. Equivalently, a motion with determinant 1. Various such things.
As other posters have alluded to above, it’s the notion of a single “axis of rotation” that doesn’t generalize to spaces with more or less than three dimensions. In any dimension, you can think of a rigid rotation as a transformation “within a plane”. In two dimensions, with coordinates X & Y, you have one rotation that rotates the XY-plane. In four dimensions, with coordinates W, X, Y, & Z, you have six possible independent rotations: in the WX-plane, WY-plane, WZ-plane, XY-plane, XZ-plane, and YZ-plane. Only in three dimensions can you draw a one-to-one correspondence between the number of independent rotation planes you have (XY, XZ, and YZ) and the number of axes you have (Z, Y, and X, respectively.)
I blame cross products. Kids are taught in geometry classes that there’s a kind of vector multiplication that lets you define the product of two vectors as another vector. Well, strictly speaking, that’s true, but what they neglect to mention is that the resulting vector is not part of the same vector space as the original two. In three dimensions, it’s easy to neglect this, since the vector space of the resulting vector happens to have the same dimensionality as the original space, but that’s just because n*(n-1)/2 = n for n = 3.
Right, and even in 3d, there are numerous reasons why one would still want to refrain from identifying the codomain of the cross product with the domain of its two arguments (avoiding an arbitrary selection of unit magnitude and the pseudoproblem of pseudovectors/“handedness”). Life would be so much easier for students if we bothered to entrust them with abstract vector spaces (and the rest of abstract linear algebra) where useful instead of uglily pretending everything has to be done in R^n. But old pedagogical habits apparently die hard.
Other than “doing nothing” and “turning to face the other direction”, there is little one might want to say purely about Lineland which I would be inclined to bring under the familial umbrella of the term “rotation”. But there are still those two; 0 and 180 degree rotation, in some sense, though there be nothing in any smaller increments. (This is a discrete account of rotation, as opposed to the potential accounts I gave above in which rotation would have to be a continuous rigid movement from start to finish)
Note: This 180 degree rotation defines precisely the negation operation (the “real” number -1), and should a Linelander be so cosmopolitan as to ponder higher dimensions, it is the generalization of this to rotation in smaller increments (through a two-dimensional plane) which defines most naturally and immediately the idea of the “complex” numbers. Those who have considered the latter somehow mysterious as opposed to the former are simply exhibiting a kind of Linelander provinciality, unable to comprehend how one could have such an “imaginary” entity as half of a 180 degree rotation, though every Flatlander child is familiar with the mundane idea of a 90 degree rotation.
As spark240 said, the plural of “axis” is “axes”. Also, the singular of “vertices” is “vertex” not “vertice”. People know what you mean anyway and I wouldn’t normally harp on spelling but it sounded like you were sincerely interested in knowing these kinds of things.
Sorry I don’t have interesting things to say about your actual question like the other posters. It’s a good one.
While I can easily visualize rotating objects in 2D space (you rotate around a point) how would a line segment pull it off? Neither 2D nor 3D objects can face a different direction without going through all the other directions in between. Furthermore, a flatlander cannot rotate himself about a line that exists in his own space, any more than we Spacelanders can rotate about a plane. (By analogy, if you consider the fourth dimension to be time, that would mean rotating oneself to be going backward in time, facing and thus seeing the the things in the future, rather than those in the past.)
That’s what I meant by calling it a discrete account of rotation; there’s a certain kind of mapping from 1-dimensional vectors to 1-dimensional vectors which one might want to consider a “rotation”, on some account of what one might want to use “rotation” to mean, even though there’s no continuous way to transform from the identity mapping into this one [and thus it’s not a “rotation” on accounts which do require this homotopy property].
(Mind you, in the same sense, “Flip around the Y axis (i.e., negate the X coordinate while keeping the Y coordinate fixed)” might be considered a rotation by Flatlanders, even though, again, there’s no way to pull it off as a rigid continuous transformation entirely within the plane from start to finish. But we Spacelanders are perfectly happy to call this a 180 degree rotation around the Y axis, and the properties which might lead us to call such a transformation a “rotation” are properties which a Flatlander may be able to appreciate as well)
Think of a large cylinder that you see from the side, spinning like a record. It doesn’t change apparent shape or size either. And it’s “axis” is invisible to you from the side, the only way to actually see it would be to peer into the fourth dimension. But yet, you can still imagine the axis point, the same way you can imagine the insides of things for which you have seen cutaway diagrams. The flatlanders would be able to see the rotation, or spinning the same way we do for a cylinder, by the movement of the surface grain. If you are talking about ‘flipping’ they would understand this the same way we do, by looking in a mirror.
I thought you were going to ask a different question - how do they physiclaly do it? If I decide to spin in a circle I’m using the friction between me and the floor to give myself angular velocity, but of course in Flatland they don’t have a floor, so what do they do?
Um. That doesn’t seem to be the question you’re interested in though. NM, carry on…
Check out my illustration of Flatland at 茄子视频懂你更多-茄子视频懂你-茄子视频懂你的更多app下载安装-茄子视频懂你更多app Click on the intro pages to start. The side view has “height” to make it visible, but one can argue that to flatlanders it is infinitely high.
