Length, width, depth, time...I always figured a dimension was simply a linear measurement; this many inches, that many seconds, etc.
The higher dimensions of supergravity/sypersymmetry/superstring/M theory are said to be "curled up" on the order of the Planck Scale...the Sub-subatomic size.
I figure this tiny curling up is part of particle spin; if so, then is spin and curled up dimensions non-linear, but angular?
Can an angular dimension even be mathematically/geometrically described?
angular dimensions?
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Enola Straight
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angular dimensions?
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Kuroneko
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Didn't you already ask this question before?
I'm unclear of what you mean by angular dimension because such a property would be something said of coordinates. For example, if you have a cylinder, you can introduce a coordinate θ as an angle from its axis in a particular embedding, or you can work in completely Cartesian coordinates (e.g., x to measure distance along the axial direction, and y for distance along the azimuthal direction perpendicular to it). Doing the latter would show that the cylinder is intrinsically flat, having the standard Euclidean metric ds² = dx²+dy², with the only curious property for the y direction is that going along it far enough ends one back to the starting point.
If by angular dimensions, you mean something of the latter sort, then of course such things are mathematically and geometrically describable: the standard sphere is a trivial surface of two dimensions that are both "angular" in that manner, and the above-mentioned cylinder is another. But I suspect that you mean something else entirely.
If your question is about spin in physics, then spin is intrinsic angular momentum. We can have spin even in classical physical--for example, for an electromagnetic field, the spin density is proportional to E×A, where A is the electromagnetic vector potential (because of gauge symmetry, not all components are physically observable, but some are). The main difference with quantum mechanics is that in QM, even point-particles can have spin, rather than only fields.
As to what compactified dimensions have to do with spin... well, nothing, in themselves. Manifolds need extra structure defined on them, creatively called 'spin structure', for that to be applicable. This is not always possible, although compact manifolds of a small number of dimensions always have a definable spin structure (and often multiple incompatible ones). The primary reason that the extra dimensions of string theory and the like "need" to be of very small size is that if they weren't, we'd already be aware of them.
I'm unclear of what you mean by angular dimension because such a property would be something said of coordinates. For example, if you have a cylinder, you can introduce a coordinate θ as an angle from its axis in a particular embedding, or you can work in completely Cartesian coordinates (e.g., x to measure distance along the axial direction, and y for distance along the azimuthal direction perpendicular to it). Doing the latter would show that the cylinder is intrinsically flat, having the standard Euclidean metric ds² = dx²+dy², with the only curious property for the y direction is that going along it far enough ends one back to the starting point.
If by angular dimensions, you mean something of the latter sort, then of course such things are mathematically and geometrically describable: the standard sphere is a trivial surface of two dimensions that are both "angular" in that manner, and the above-mentioned cylinder is another. But I suspect that you mean something else entirely.
If your question is about spin in physics, then spin is intrinsic angular momentum. We can have spin even in classical physical--for example, for an electromagnetic field, the spin density is proportional to E×A, where A is the electromagnetic vector potential (because of gauge symmetry, not all components are physically observable, but some are). The main difference with quantum mechanics is that in QM, even point-particles can have spin, rather than only fields.
As to what compactified dimensions have to do with spin... well, nothing, in themselves. Manifolds need extra structure defined on them, creatively called 'spin structure', for that to be applicable. This is not always possible, although compact manifolds of a small number of dimensions always have a definable spin structure (and often multiple incompatible ones). The primary reason that the extra dimensions of string theory and the like "need" to be of very small size is that if they weren't, we'd already be aware of them.
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Enola Straight
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I don't think you know what a dimension is. Put shortly, it's just one of the measurements needed to fix the location of a point in a system.
It doesn't matter what you measure in as long as you take a measurement. We use length and angle frequently, but there's nothing stopping you from describing the location of a point on a plane by using the dimensions J and K - the first establishing a square of side J centered at the origin and facing an arbitrary "up"; the second a decimal K determining what fraction of the way around the square from the top-right corner an ant would have to walk to get to your target point. It would be mathematically clumsy compared to the more elegant coordinate systems we use, but there's nothing fundamentally wrong with it.
We have angular dimensions all the time. You use them. "Five miles southeast of Memphis" is a point in a 2-dimensional coordinate system with its origin on Memphis, one distance dimension R to establish a circle of radius R around Memphis on which the target point lies, and one angular dimension (with the compass directions replacing degrees) to determine where on that circle is the target point.
In 3-space (3d objects) you can also use angular dimensions (up to two, because distance from the origin comes in units and angles are unitless, so we need something to carry the units). This is presumably how the guns on a battleship are aimed: two angular dimensions at right angles to each other, one from horizontal, the other from forward.
Similarly in 4, 5, 6, and n-space, angular dimensions are perfectly valid, often useful, and also fundamentally unnecessary.
It doesn't matter what you measure in as long as you take a measurement. We use length and angle frequently, but there's nothing stopping you from describing the location of a point on a plane by using the dimensions J and K - the first establishing a square of side J centered at the origin and facing an arbitrary "up"; the second a decimal K determining what fraction of the way around the square from the top-right corner an ant would have to walk to get to your target point. It would be mathematically clumsy compared to the more elegant coordinate systems we use, but there's nothing fundamentally wrong with it.
We have angular dimensions all the time. You use them. "Five miles southeast of Memphis" is a point in a 2-dimensional coordinate system with its origin on Memphis, one distance dimension R to establish a circle of radius R around Memphis on which the target point lies, and one angular dimension (with the compass directions replacing degrees) to determine where on that circle is the target point.
In 3-space (3d objects) you can also use angular dimensions (up to two, because distance from the origin comes in units and angles are unitless, so we need something to carry the units). This is presumably how the guns on a battleship are aimed: two angular dimensions at right angles to each other, one from horizontal, the other from forward.
Similarly in 4, 5, 6, and n-space, angular dimensions are perfectly valid, often useful, and also fundamentally unnecessary.