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Mathematicians discover new way for spheres to 'kiss'

quantamagazine.org

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Re: Mathematicians discover new way for spheres to 'kiss'

#41

Post by neom »

semi-extrinsic wrote: post #40
neom wrote: post #39

Earlier quoted context omitted.

wow. I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map i…

You're welcome :) The foundations for these concepts were laid by Piaget and Brissiaud, but most of their work is in french. In English, "Young children reinvent arithmetic" by Kamii is an excellent and practically oriented book based on Piaget's theories, that you may find useful. Although it is 250 pages. This approach has become mainstream in maths teaching today, but unfortunately often misunderstood by teachers.…

I should be able to chat with an llm about this paper, but my gut says you've given me the glimmer of where I need to go. This is something I've been deeply deeply frustrated about for 30 years now, I had really given up hope of ever being able to process mathematics (whatever they are) properly, it's a real task to figure out how to get someone to see how your brain work and then have them understand how to provide you with some framework to grasp what they know.

Once again I wanted to thank you for slowing down and taking the time to leave this thoughtful comment, if everyone took 5 minutes to try to understand what the other person is saying to see if they can help, the world would be a considerably better place. Thank you.

Re: Mathematicians discover new way for spheres to 'kiss'

#42

Post by iNic »

david-gpu wrote: post #31
iNic wrote: post #23

Earlier quoted context omitted.

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 "…

> 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…

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).

Re: Mathematicians discover new way for spheres to 'kiss'

#43

Post by rpigab »

quuxplusone wrote: post #27
rpigab wrote: post #21

> Mathematicians often visualize this problem in terms of spheres. You can think of each code word as a high-dimensional point at the center of a sphere. If an error-filled message (when represented as a high-dimensional point) lives inside a given sphere, you know that the code word at the sphere’s center was the intended message. You don’t want these spheres to overlap — otherwise, a received message might be inter…

> I want to understand this based on what the article says, but I can't. I can't represent error-filled messages as high-dimensional points. Well, start with an analogy. Let's say you and I want to communicate a message, which comes from a set of let's say 4 possible messages: "YES", "NO", "GOOD", and "BYE". Let's further suppose that the medium for this message (the "data channel") is going to be a single point sele…

This helped a lot, thanks! I now see a similiarity where I was missing the bridges between geometry and lossy information channels. It's really interesting, though it's a really complex problem.

Re: Mathematicians discover new way for spheres to 'kiss'

#44

Post by [deleted] »

david-gpu wrote: post #31
iNic wrote: post #23

Earlier quoted context omitted.

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 "…

> 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…

[deleted]

Re: Mathematicians discover new way for spheres to 'kiss'

#47

Post by dekhn »

I took a class taught by David Huffman (of Huffman coding) called Cybernetics (IIUC it was the UCSC equivalent of a class Wiener taught at MIT.

The very first day, he started out by talking about kissing spheres and concluded the lecture with "and that's why kissing spheres are easy in 7 dimensions" (or something like that).

Every lecture of his was like being placed in front of a window looking upon a wonderful new world, incomprehensible at first, but slowly becoming more and more clear as he explained. Sometimes I wish I could play in the garden of math.

Re: Mathematicians discover new way for spheres to 'kiss'

#48

Post by eszed »

neom wrote: post #39
semi-extrinsic wrote: post #37

Earlier quoted context omitted.

My two cents on this: I've done a lot of math, up to graduate courses in weird stuff like operator algebra. I've also read quite a bit of maths pedagogy. I've come to understand that the key thing that determines success in math is ability to compress concepts . When young children learn arithmetic, some are able to compress addition such that it takes almost zero effort, and then they can play around with the concep…

wow. I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map i…

Calculating 8x12 in my head relies on a trick / technique - they call it "chunking", I believe, in the Common Core maths curriculum that US parents get so angry about - that (I'm also in my 40s) was never demonstrated in schools when we were kids. (They tried to make me memorize the 12x table, which I couldn't, so I calculated it my way instead; took a little longer, but not so much that anyone caught on that I wasn't doing what the teacher said.) I'd like to think I was smart enough to work it out for myself, but I suspect my dad showed it to me.

I'll show it to you, but first: are you able to add 80 + 16 in your head? (There's another trick to learn for that.)

Re: Mathematicians discover new way for spheres to 'kiss'

#50

Post by jochi427 »

iNic wrote: post #23
danwills wrote: post #2

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 wo…

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 "…

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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