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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'

#31

Post by david-gpu »

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 "…

> 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?

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

#33

Post by evandrofisico »

davethedevguy wrote: post #4
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…

Likewise! In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them? Is that even a valid question, or does it just betray my inability to perceive higher dimensions? This is fascinating and I'm in awe of the people that do this work.

In my PhD I did study systems in higher dimensions (including fractal dimensions) and it is not a metaphor and no, I did not visualize them, it was more like defining a mathematical representation of the system geometry and working on top of it.

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

#34

Post by david-gpu »

Majromax wrote: post #22
close04 wrote: post #14

Earlier quoted context omitted.

> etc. That's handwaving the answer just as you were getting to the crux of the matter. "Are mathematicians really visualising spheres with physical space between them" in higher dimensions than 3 (or maybe 4)? From the experience of some of the bigger minds in mathematics I met during my PhD, they don't actually visualize a practical representation of the sphere in this case since that would be untenable especially…

Reportedly, Geoffrey Hinton said: “To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.”

My sister is a mathematican and she used to say that if you want to understand a 24-dimensional space, you start from a generalized n-dimensional space and then set n=24.

This wasn't atypical of her. She would also say that if your house is on fire then you call the firefighters, but if it is not on fire then you set it on fire, thereby reducing the problem to something that you have already solved.

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

#35

Post by btown »

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…

https://news.ycombinator.com/item?id=3995615 (both article and comments) describe various ways of looking at this - and there are many implications for machine learning e.g. https://news.ycombinator.com/item?id=3995964 !

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

#36

Post by awanderingmind »

crazygringo wrote: post #25

It's strange the article doesn't even mention just trying to simulate the problem computationally. Surely it's not too difficult to repeatedly place spheres around a central sphere in 17 dimensions, maximizing how many kiss for each new sphere added, until you get a number for how many fit? And add some randomness to the choices to get a range of answers Monte Carlo-style, to then get some idea of the lower bound? [E…

Here is an example of that sort of thing, using gradient descent as a starting point: https://arxiv.org/abs/math/0611451. It is technically about spherical codes rather than the kissing problem specifically, but they are closely related: https://en.wikipedia.org/wiki/Spherical_code

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

#37

Post by semi-extrinsic »

neom wrote: post #20
close04 wrote: post #14

Earlier quoted context omitted.

> etc. That's handwaving the answer just as you were getting to the crux of the matter. "Are mathematicians really visualising spheres with physical space between them" in higher dimensions than 3 (or maybe 4)? From the experience of some of the bigger minds in mathematics I met during my PhD, they don't actually visualize a practical representation of the sphere in this case since that would be untenable especially…

I have dyscalculia so I'm always studying how people who have "math minds" work, especially because I have an strong spacial visual thinking style, i thought i should be good at thinking about physical math. When I found out they're not visualizing the stuff but instead "visualized the equations together and imaging them into new ones" - I gave up my journey into math.

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 concept in their minds. For them, taking the next step to multiplication is almost trivial.

When a college math student learns the triangle inequality, >99.99% understand it on a superficial level. But compress it and play around with it in their minds, and can subsequently wield it like an elegant tool in surprising contexts. These are the people with "math minds".

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

#39

Post by neom »

semi-extrinsic wrote: post #37
neom wrote: post #20

Earlier quoted context omitted.

I have dyscalculia so I'm always studying how people who have "math minds" work, especially because I have an strong spacial visual thinking style, i thought i should be good at thinking about physical math. When I found out they're not visualizing the stuff but instead "visualized the equations together and imaging them into new ones" - I gave up my journey into math.

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 it? I also have dyslexia, and have not found a good strategy for phonics yet, and I'm now 40, so I'm not sure I ever will hehe :))

I even struggle with times tables because the lifting is really hard for me for some reason, it always amazes me people can do 8x12 in their heads.

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

#40

Post by semi-extrinsic »

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…

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. The point of using different strategies to arrive at the same answer in arithmetics is NOT that children should memorize different strategies, but that they should be given as many tools as possible to increase the chance that they are able to play around with and compress the concept being learned.

The clearest expression of the concept of compression is maybe in this paper, I don't know if it helps or if it's too academic.

https://files.eric.ed.gov/fulltext/EJ780177.pdf

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