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Mathematicians discover new way for spheres to 'kiss'

(www.quantamagazine.org)
161 points isaacfrond | 4 comments | 16 Jan 25 09:57 UTC | HN request time: 0.292s | 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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1. jochi427 ◴[16 Jan 25 18:55 UTC] No.42729341[source]▶
I remember learning about the probability of returning to the origin in a 2D random walk versus a 3D random walk when I took stochastic processes. After we proved with probability 1 you return to the origin in a 2D walk (and with probability 0 you return in 3D) my professor said "that's why you hand a drunk man the keys to a car and not an airplane when he leaves the bar". After checking wikipedia it looks like he riffed off this quote from Shizuo Kakutani: "A drunk man will find his way home, but a drunk bird may get lost forever".
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2. CamperBob2 ◴[16 Jan 25 20:31 UTC] No.42730494[source]▶
That's interesting, about the probability being zero in 3D. Is this on an integer lattice? The source that cannot be cited on HN without loss of karma says that the probability of returning to the origin in Z^3 is approximately 0.34.

I don't see how it could possibly be zero, even for reals, unless you're relying on the idea that the probability of any given real emerging from a uniform RNG is zero. That would seem to apply in 2D as well.

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3. jochi427 ◴[16 Jan 25 21:09 UTC] No.42730940[source]▶
I'm sure I am just misremembering -- it was definitely on Z^3 so I guess its actually 34%. Thanks for letting me know
4. penteract ◴[17 Jan 25 02:27 UTC] No.42733436[source]▶
Here's how to formulate the question in continuous space/time:

Random walks can be defined on continuous space and time as a probability distribution on functions R -> R^n (Brownian motion in n dimensions).

We can then ask whether Brownian motion beginning at the origin will ever revisit it i.e.

Given 2D Brownian motion X such that X(0)=(0,0), the probability that there exists t>0 such that X(t)=(0,0) is 1.

Given 3D Brownian motion X such that X(0)=(0,0,0), the probability that there exists t>0 such that X(t)=(0,0,0) is 0. (This is more clearly true when it doesn't begin at the origin, but it's almost certainly not at the origin at t=1, and you can divide the half open interval (0,1] into a countable number of intervals, each of which have 0 probability of passing through the origin.)

Random walks in 2D are space filling curves; random walks in 3D are not.