
A New Path to Equal-Angle Lines - jonbaer
https://www.quantamagazine.org/20170411-equiangular-lines-proof/
======
zellyn
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
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
Your last paragraph had me intrigued. I'm now wondering what other
observations we have that are simply special cases due to low dimensionality.

~~~
jerf
Well, here's another fun one for you then: Regular polyhedra/polytopes. In
zero dimensions the special case is that it's a nonsense question, which is
pretty common special case for 0D. In 1D you can't help but have a line be
equal to itself, so still a special case of it being nonsense. In 2D you have
the special case of being able to form an arbitrary number of regular
polygons. In 3D you get 5 regular polyhedra, the five Platonic solids, which
also turns out to be a special case.

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.

[1]:
[https://www.youtube.com/watch?v=oJ7uOj2LRso](https://www.youtube.com/watch?v=oJ7uOj2LRso)

[2]:
[http://math.ucr.edu/home/baez/platonic.html](http://math.ucr.edu/home/baez/platonic.html)

~~~
hashhar
Wow. This entire thread is amazing. Thanks a lot for the links to help me go
deeper into the rabbit hole.

------
mrcactu5
whoa -- I didn't know geogebra could be put on the web

~~~
murkle
We've had that working for a few years now :)
[https://help.geogebra.org/topic/welcome-to-the-
geogebraweb-b...](https://help.geogebra.org/topic/welcome-to-the-geogebraweb-
beta-release-forum)

It's compiled from the same source code using GWT (and in fact the GWT / HTML5
version is now the mainstream release)

