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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-dime

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!

terminalbraid 2y ago
> Is it all on computers now? Most modern math is certainly not "all on computers" and in general not even "mostly on computers". There are definitely proofs for things like testing large spaces exhaustively which are sped up by computers (see the https://en.wikipedia.org/wiki/Four_color_theorem https://en.wikipedia.org/wiki/Four_color_theorem) and definitely for things like visualization (probably one of the oldest uses of computers for math), but usually the real work goes into how math has always been done: identifying patterns and abusing symmetries. For this one explicitly, if you read through the paper you'll find the statement that the main theorem presented here "does not depend on any computer calculations. However, we have made available files with explicit coordinates for our kissing configurations"
- viccis 2y ago
  It really depends though. Even in something like knot theory, that one might consider to be a very "pure" area, there's still a lot of computation involved that can be automated by computers.
davethedevguy 2y ago
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.
- bux93 2y ago
  I have a hard time visualizing even 3 dimension, but 4 dimensions and up, I just think of it as a spreadsheet where each thing has 4 or more columns of data rather than 3. Whether a 4th column is time, spin, color, smell or yet another coordinate.
  - nejsjsjsbsb 2y ago
    It sort of like the visualizable 3D "kissing spheres" is the story that makes it interesting, captivating and accessible and therefore competitive/social which makes it interesting even more, but basically at higher dims it's a bunch of equations as it is impossible to visualise on human wetware. You could do kissing starfish but no one cares as there is no lore. A bit like 125m world record doesn't matter. 100m is the thing. This is not a knock ... it is interesting how social / tradition based maths is. Another example is Fermat's Last Theorem. It had legendary status.
    - ndsipa_pomu 2y ago
      However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.
- aleph_minus_one 2y ago
  > 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? For such discrete geometry problems, high-dimensional spaces often behave "weirdly" - your geometric intuition from R^3 will often barely help you. You thus typically rather rely on ideas such as symmetry, or calculations whether "there is still space inbetween that you can fill", or sometimes stochastic/averaging arguments to show the existence of some configuration.
- jstanley 2y ago
  > just a visual metaphor It's not really a metaphor. An n-sphere is the set of all points that are the same distance away from the same centre, in (n+1)-dimensional space. That generalises perfectly well to any number of dimensions. In 1 dimension you get 2 points (0-sphere), in 2 dimensions you get a circle (1-sphere), in 3 dimensions you get a sphere (2-sphere), etc. EDIT: Also, if you slice a plane through a sphere, you get a circle. If you slice a line through a circle, you get 2 points. If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.
bell-cot 2y ago
I suspect that you have plenty of company...but from a journalism PoV, those kind of things are where it gets tricky. Explaining in detail, and at length, is a lot more work than this short article. Then there are the decisions - "just how much detail?", "just how long?", (worse) "how much mathematical background should we assume, in our readers?", and (worst) "how willing will our readers be, to slog through serious mathematics?". (I'm assuming you've already searched for math bloggers, and similar "labor of love" coverage of the topic.)
scythe 2y ago
In many cases you are "translating" the higher-dimensional geometry into something that is not geometric or which is much lower dimensional. You don't generally visualize 24 dimensions. You can get a decent intuition for 4 with practice but at some point this breaks down. For example, the 24-dimensional packing corresponds to the Leech lattice which itself corresponds to the Golay code: https://en.wikipedia.org/wiki/Leech_lattice https://en.wikipedia.org/wiki/Leech_lattice https://en.wikipedia.org/wiki/Binary_Golay_code https://en.wikipedia.org/wiki/Binary_Golay_code
iNic 2y ago
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).
- david-gpu 2y ago
  > 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?
  - btown 2y ago
    https://news.ycombinator.com/item?id=3995615 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 https://news.ycombinator.com/item?id=3995964 !
  - iNic 2y ago
    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).
    - 2y ago
jebarker 2y ago
They definitely don't visualize 24 dimensional spheres. When I did my PhD in pure math I found that gradually I just became comfortable working without any visual or spatial intuition and instead relying on the (algebraic and topological) machinery that had been put in place before me. Terence Tao had a nice essay [1] talking about how the final stage in becoming a professional mathematician is developing the intuition to know what is likely true or not in these very abstract spaces. I also never used a computer for anything other than latex. [1] https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...

terminalbraid 2y ago

> Is it all on computers now? Most modern math is certainly not "all on computers" and in general not even "mostly on computers". There are definitely proofs for things like testing large spaces exhaustively which are sped up by computers (see the https://en.wikipedia.org/wiki/Four_color_theorem https://en.wikipedia.org/wiki/Four_color_theorem) and definitely for things like visualization (probably one of the oldest uses of computers for math), but usually the real work goes into how math has always been done: identifying patterns and abusing symmetries. For this one explicitly, if you read through the paper you'll find the statement that the main theorem presented here "does not depend on any computer calculations. However, we have made available files with explicit coordinates for our kissing configurations"

viccis 2y ago
It really depends though. Even in something like knot theory, that one might consider to be a very "pure" area, there's still a lot of computation involved that can be automated by computers.

davethedevguy 2y ago

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.

bux93 2y ago
I have a hard time visualizing even 3 dimension, but 4 dimensions and up, I just think of it as a spreadsheet where each thing has 4 or more columns of data rather than 3. Whether a 4th column is time, spin, color, smell or yet another coordinate.
- nejsjsjsbsb 2y ago
  It sort of like the visualizable 3D "kissing spheres" is the story that makes it interesting, captivating and accessible and therefore competitive/social which makes it interesting even more, but basically at higher dims it's a bunch of equations as it is impossible to visualise on human wetware. You could do kissing starfish but no one cares as there is no lore. A bit like 125m world record doesn't matter. 100m is the thing. This is not a knock ... it is interesting how social / tradition based maths is. Another example is Fermat's Last Theorem. It had legendary status.
  - ndsipa_pomu 2y ago
    However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.
aleph_minus_one 2y ago
> 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? For such discrete geometry problems, high-dimensional spaces often behave "weirdly" - your geometric intuition from R^3 will often barely help you. You thus typically rather rely on ideas such as symmetry, or calculations whether "there is still space inbetween that you can fill", or sometimes stochastic/averaging arguments to show the existence of some configuration.
jstanley 2y ago
> just a visual metaphor It's not really a metaphor. An n-sphere is the set of all points that are the same distance away from the same centre, in (n+1)-dimensional space. That generalises perfectly well to any number of dimensions. In 1 dimension you get 2 points (0-sphere), in 2 dimensions you get a circle (1-sphere), in 3 dimensions you get a sphere (2-sphere), etc. EDIT: Also, if you slice a plane through a sphere, you get a circle. If you slice a line through a circle, you get 2 points. If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.

bell-cot 2y ago

I suspect that you have plenty of company...but from a journalism PoV, those kind of things are where it gets tricky. Explaining in detail, and at length, is a lot more work than this short article. Then there are the decisions - "just how much detail?", "just how long?", (worse) "how much mathematical background should we assume, in our readers?", and (worst) "how willing will our readers be, to slog through serious mathematics?". (I'm assuming you've already searched for math bloggers, and similar "labor of love" coverage of the topic.)

scythe 2y ago

In many cases you are "translating" the higher-dimensional geometry into something that is not geometric or which is much lower dimensional. You don't generally visualize 24 dimensions. You can get a decent intuition for 4 with practice but at some point this breaks down. For example, the 24-dimensional packing corresponds to the Leech lattice which itself corresponds to the Golay code: https://en.wikipedia.org/wiki/Leech_lattice https://en.wikipedia.org/wiki/Leech_lattice https://en.wikipedia.org/wiki/Binary_Golay_code https://en.wikipedia.org/wiki/Binary_Golay_code

iNic 2y ago

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).

david-gpu 2y ago
> 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?
- btown 2y ago
  https://news.ycombinator.com/item?id=3995615 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 https://news.ycombinator.com/item?id=3995964 !
- iNic 2y ago
  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).
  - 2y ago

jebarker 2y ago

They definitely don't visualize 24 dimensional spheres. When I did my PhD in pure math I found that gradually I just became comfortable working without any visual or spatial intuition and instead relying on the (algebraic and topological) machinery that had been put in place before me. Terence Tao had a nice essay [1] talking about how the final stage in becoming a professional mathematician is developing the intuition to know what is likely true or not in these very abstract spaces. I also never used a computer for anything other than latex. [1] https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...