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Mathematicians discover new way for spheres to 'kiss'

(www.quantamagazine.org)
161 points isaacfrond | 1 comments | 16 Jan 25 09:57 UTC | HN request time: 1.393s | source
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danwills ◴[16 Jan 25 11:34 UTC] No.42724002[source]▶
I'd really love to know what the mathematicians are actually doing when they work this stuff out? Is it all on computers now? Can they somehow visualize 24-dimensional-sphere-packings in their minds? Are they maybe rigorously checking results of a 'test function' that tells them they found a correct/optimal packing? I would love to know more about what the day-to-day work involved in this type of research actually would be!
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iNic ◴[16 Jan 25 13:24 UTC] No.42724766[source]▶
The kind of intuition you gain for higher dimension tends not to be visual. It is more that you learn a bunch of tools and these in turn build intuition. For example high dimensional spheres are "pointy" and most of their volume are near their surface. These ideas can be defined rigorously and are important and useful. For medium dimension there are usually specific facts that you exploit. In my own work stuff like "How often do you expect random walks to intersect" is very important (and dependent on dimension).
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david-gpu ◴[16 Jan 25 14:07 UTC] No.42725261[source]▶
> For example high dimensional spheres are "pointy" and most of their volume are near their surface

I had a visceral reaction to this. In what sense can a sphere be considered pointy? Almost by definition, it is the volume that minimizes surface area, in any number of dimensions.

I can see how in higher dimensions e.g. a hypersphere has much lower volume than a hypercube. But that's not because the hypersphere became pointy, it's because the corners of the hypercube are increasingly more voluminous relative to the volume of the hypersphere, right?

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iNic ◴[16 Jan 25 16:09 UTC] No.42727180[source]▶
There is a standard thought experiment where you start with a hypercube of side-length 2, centered at the origin. You then place a radius 1 sphere on each vertex of this hypercube. The question then becomes: what is the largest sphere you can place at the origin so that it is "contained" by the other spheres. As it turns out in like dimension 6 or so the radius of the center sphere exceeds 1. It will actually poke out arbitrarily far (while still being restricted by the corner spheres).
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carltg_ ◴[16 Jan 25 20:34 UTC] No.42730526[source]▶
I hear this point parroted all of the time, but I think it is a misunderstanding and a poor visualization. Consider the same situation, but instead of focusing on the radius of the center sphere, focus on the distance between the spheres on the corners to the origin. For 1-dimension, these 'spheres' are unit intervals and so the distance is 1 (Central radius is 0). For 2-dimensions, these are circles at a distance of root(3) (Central radius is root(2)-1). 3-D: root(3) (Central radius is root(3)-1). Etc. So, it isn't the central circle getting more 'pointy' allowing the central radius to increase, but rather that the corner circles are getting further from the origin, allowing larger N-spheres (increasing proportional to the root of N). Thus, pointy is not the right way to conceptualize these spheres. For the more visual folk, I would recommend drawing this out and you can see this in action. More clearly, if a sphere became 'spikey' then the distance on the surface of the spike should be further than a neighboring point, which is NOT the case. Not trying to attack you, I just see this same point over and over and think that this warrants more thought
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1. iNic ◴[17 Jan 25 09:08 UTC] No.42735505[source]▶
Yes that is true, but there are other ways to see spiky-like behavior.

First, the volume of spheres (or balls rather) in higher dimensions goes to zero as the dimension grows. Said another way, to keep unit volume on a ball you need to grow the radius more and more (which I interpret as spiky).

Second, the volume of spherical caps grows like ~exp(- d h^2 /2), in particular the caps lose volume fast in higher dimensions. To interpret this as "spikyness" I like to visualize it as two balls intersecting (which is just 2x the cap volume). If they are of the same radius, but their centers are just slightly off their intersection volume goes to zero quickly!