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In the penultimate paragraph, they say, "As a result, the authors were able to prove that when you begin by fixing the angle in advance, the maximum number of e

by zellyn 10y ago

In the penultimate paragraph, they say, "As a result, the authors were able to prove that when you begin by fixing the angle in advance, the maximum number of equiangular lines is 2d – 2 for one particular angle (approximately 70.7 degrees), and no more than 1.93d for any other angle." Earlier, they show that in 3 dimensions, you can arrange 6 lines. 2 > 1.93, so I guess either I or they misunderstood something. [Edit] "for…sufficiently large n" https://arxiv.org/abs/1606.06620 https://arxiv.org/abs/1606.06620 “in this paper we prove that for every fixed angle θ and sufficiently large n there are at most 2n−2 lines in ℝn with common angle θ”

jerf 10y ago
One of the things that makes understanding higher dimensionality so challenging for us is that all the cases we can visualize (0, 1, 2, and 3, to a limited extent 4 if you use time as a dimension, which is suitable for raw visualization but makes rotation hard) are very frequently special cases when you ask about some property in n dimensions. I suspect (obviously without proof) that while a 100-dimensional being might have a hard time directly visualizing a 101-dimensional space since all their specialized neural-equivalent-hardware might be set up for 100 dimensions that they would at least have less mathematical trouble with going up one dimension than we do. By that point the dimensions are becoming more regular in a lot of ways, whereas for us we're still stepping from something that's still often a special case (3) to something else that is also quite often a special case (4), with only the ability to visualize things that are super-special cases below us to guide us. Too much special case going on with too many properties for us to get a general understanding of dimensionality very easily. To give one example of a special case we live in where even going to 4 doesn't help much in escaping it, we are used to the unit sphere taking up most of the volume of the enclosing cube. This turns out to be a special case for low dimensionality; in the general case it takes up a vanishing fraction of the volume. This makes sense and is perhaps even obvious if you think about it completely algebraically; if I'm picking variables from -1.0 to 1.0 for each dimension and doing SQRT(a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2...), the more I add inside that expression the less likely the resulting value is going to be less than 1.0. But even as I know this quite easily algebraically, the geometry portion of my brain is screaming "No, it cannot be! The sphere is large!", because it's too trained on special cases and generalizing those special cases into the general case incorrectly. (Yes, the SQRT doesn't do much here since we're doing the unit circle, but it's part of the form.)
- hashhar 10y ago
  Your last paragraph had me intrigued. I'm now wondering what other observations we have that are simply special cases due to low dimensionality.
  - rmidthun 10y ago
    Rotations can be composed in up to 3 dimensions only. That means that for any sequence of rotations, you can come up with a single rotation that is equal to that rotation. In 2d, this is just adding scalars and rotation has no meaning in 1 or 0 dimensions. In 4 dimensions, this is not true. A rotation in XY cannot compose with one in ZW so at least two rotations are required.
    - pandaman 10y ago
      Could you explain more about 4D rotation? In my understanding, a rotation is a transform by an orthogonal matrix with determinant equal to 1. For the 4D rotations to be not composable the product of two 4x4 orthogonal matrices should not be an orthogonal matrix. Is this really true?
      - abecedarius 10y ago
        By 'rotation' is meant rotating a plane that goes through the origin, carrying along the other points that are off the plane according to their projection onto it. In 4-d if you rotate around the y-z plane, then around the w-x plane, there's no plane that can express a single composed rotation with the same effect. But as you say, it's composable as a unitary matrix.
        pandaman 10y ago
        Hmmm, I am still confused. Rotating "around plane" makes sense only in 3D, where a 2D plane is a hyperplane i.e. has a single normal and you can rotate around that normal (which you implied by mentioning projections along that normal). In 4D a hyperplane is a 3D subspace, orthogonal to some normal. So, while it may be true that you cannot find a 2D plane describing a composite rotation in 4D it does not seem to have any significance since "rotation around y-z" plane does not describe one either :) because y-z plane has an infinite number of normals in 4D and any rotation around any of them will be technically a rotation "around y-z plane" while a different rotation in 4D. I am sure you can find a 3D hyperplane in 4D for any composite rotation. Am I wrong?

by zellyn 10y ago

jerf 10y ago

One of the things that makes understanding higher dimensionality so challenging for us is that all the cases we can visualize (0, 1, 2, and 3, to a limited extent 4 if you use time as a dimension, which is suitable for raw visualization but makes rotation hard) are very frequently special cases when you ask about some property in n dimensions. I suspect (obviously without proof) that while a 100-dimensional being might have a hard time directly visualizing a 101-dimensional space since all their specialized neural-equivalent-hardware might be set up for 100 dimensions that they would at least have less mathematical trouble with going up one dimension than we do. By that point the dimensions are becoming more regular in a lot of ways, whereas for us we're still stepping from something that's still often a special case (3) to something else that is also quite often a special case (4), with only the ability to visualize things that are super-special cases below us to guide us. Too much special case going on with too many properties for us to get a general understanding of dimensionality very easily. To give one example of a special case we live in where even going to 4 doesn't help much in escaping it, we are used to the unit sphere taking up most of the volume of the enclosing cube. This turns out to be a special case for low dimensionality; in the general case it takes up a vanishing fraction of the volume. This makes sense and is perhaps even obvious if you think about it completely algebraically; if I'm picking variables from -1.0 to 1.0 for each dimension and doing SQRT(a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2...), the more I add inside that expression the less likely the resulting value is going to be less than 1.0. But even as I know this quite easily algebraically, the geometry portion of my brain is screaming "No, it cannot be! The sphere is large!", because it's too trained on special cases and generalizing those special cases into the general case incorrectly. (Yes, the SQRT doesn't do much here since we're doing the unit circle, but it's part of the form.)

hashhar 10y ago
Your last paragraph had me intrigued. I'm now wondering what other observations we have that are simply special cases due to low dimensionality.
- rmidthun 10y ago
  Rotations can be composed in up to 3 dimensions only. That means that for any sequence of rotations, you can come up with a single rotation that is equal to that rotation. In 2d, this is just adding scalars and rotation has no meaning in 1 or 0 dimensions. In 4 dimensions, this is not true. A rotation in XY cannot compose with one in ZW so at least two rotations are required.
  - pandaman 10y ago
    Could you explain more about 4D rotation? In my understanding, a rotation is a transform by an orthogonal matrix with determinant equal to 1. For the 4D rotations to be not composable the product of two 4x4 orthogonal matrices should not be an orthogonal matrix. Is this really true?
    - abecedarius 10y ago
      By 'rotation' is meant rotating a plane that goes through the origin, carrying along the other points that are off the plane according to their projection onto it. In 4-d if you rotate around the y-z plane, then around the w-x plane, there's no plane that can express a single composed rotation with the same effect. But as you say, it's composable as a unitary matrix.
      - pandaman 10y ago
        Hmmm, I am still confused. Rotating "around plane" makes sense only in 3D, where a 2D plane is a hyperplane i.e. has a single normal and you can rotate around that normal (which you implied by mentioning projections along that normal). In 4D a hyperplane is a 3D subspace, orthogonal to some normal. So, while it may be true that you cannot find a 2D plane describing a composite rotation in 4D it does not seem to have any significance since "rotation around y-z" plane does not describe one either :) because y-z plane has an infinite number of normals in 4D and any rotation around any of them will be technically a rotation "around y-z plane" while a different rotation in 4D. I am sure you can find a 3D hyperplane in 4D for any composite rotation. Am I wrong?