A New Path to Equal-Angle Lines
quantamagazine.org
quantamagazine.org
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
“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”
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.)
However, in four dimensions, you have six regular polytopes: video [1] webpage [2] I only found this out relatively recently; while this is no secret it only seems to be recently getting around in the math general interest video channels and such.
After that, the regular case takes over and you only have 3 forever more; the tetrahedron analogue, the cube analog, and the octahedron analog. So here's a case where 0 through 4 dimensions is a special case, and 5 is the first that fits the general pattern.
In 4 dimensions, this is not true. A rotation in XY cannot compose with one in ZW so at least two rotations are required.
Here's an example that's easiest to sort-of-visualize:
1. Rotate the y-z plane of 4-d space by half a turn (180 degrees). That is, transform every point in 4-space according to w->w, x->x, y->-y, z->-z.
2. Rotate the w-x plane, also a half-turn. So w->-w, x->-x, y->y, z->z.
Composing these two transformations yields w->-w, x->-x, y->-y, z->-z. That composed transformation is an inversion, not a rotation. Is it clearer now? (It was late when I wrote last night, sorry. Let me try and rephrase how I defined a rotation: a linear transformation that carries one unit vector to a second unit vector, and leaves unchanged any vectors that are orthogonal to both. In my previous comment the plane I was talking about was the plane that contains both given vectors. Of course, the null rotation doesn't pick out any particular plane, though it's not otherwise special in these terms. IANAM, but would appreciate correction if I'm goofing.)
In 3d space, we have inverse-square laws. Think of a light source, the area that is covers at a given distance is the area of a sphere, so it makes sense that it would decrease with that area, or 4piR*R. Gravity is also inverse-square in 3D space (though relativity makes this much more complicated, I expect). Implication is that in a 4D space, it would be an inverse-cubed relation.
It turns out that stable orbits can only happen with inverse squared laws.
http://mathworld.wolfram.com/BorsuksConjecture.html
> Borsuk conjectured that it is possible to cut an n-dimensional shape of generalized diameter 1 into n+1 pieces each with diameter smaller than the original. It is true for n=2, 3 and when the boundary is "smooth." However, the minimum number of pieces required has been shown to increase as ∼1.1^(sqrt(n)). Since 1.1^(sqrt(n))>n+1 at n=9162, the conjecture becomes false at high dimensions.
> Kahn and Kalai (1993) found a counterexample in dimension 1326, Nilli (1994) a counterexample in dimension 946. Hinrichs and Richter (2003) showed that the conjecture is false for all n>297.
For visualizing within your own mind, I suggest that color makes a better 4th dimension than time does. (On paper, it's hard to see colors behind other colors, but it's not difficult to imagine this.)
At low dimensions, if you take a sphere of radius 1, and another sphere of radius 0.999999999999999 or so at the same origin, only a tiny fraction of the outer sphere's volume exists in the 'shell' between the two spheres.
At very high dimensions, all of the interior volume is in the spherical shell. The fraction of volume in the interior becomes insignificantly infinitesimally small.
Basically - at sufficiently-high dimensionality, a point chosen at random within the volume of an n-sphere will be on the surface.
* The volume of unit-radius sphere goes to zero as the dimension increases.
* All the mass of hypersphere is near the equator (d-1 dimensional disk with thickness ε).
* Alternatively almost all volume is close to surface like you said.
* Almost all the surface area of a high-dimensional sphere is near the equator.
* Distribution of distances between uniformly at random points concentrates around average distance as dimensionality grows.
* 0 centered unit cube and unit sphere in multiple dimensions: vertices of the cube escape far away from the sphere while the face centers of the cube stay at the distance of 1/2 from the center.
* Probability mass of Gaussians in multiple dimensions spreads all over the place.
Three dimensional spheres are invariant under rotation about any axis and the axes are not properties of the sphere.
Equators are surely not properties of spheres. So how does a hypersphere have an equator?
Or, there is a different meaning for some of these words than the ones I am used to; I didn't study maths at a higher level than that needed for a BSc. Physics forty years ago.
Obviously choosing the equator is totally arbitrary in symmetric object.
It's compiled from the same source code using GWT (and in fact the GWT / HTML5 version is now the mainstream release)