Mathematicians discover new way for spheres to 'kiss'
quantamagazine.org
Mathematicians discover new way for spheres to 'kiss'
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Re: Mathematicians discover new way for spheres to 'kiss'
#2I'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!
Re: Mathematicians discover new way for spheres to 'kiss'
#3I'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…
> 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) 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"
Re: Mathematicians discover new way for spheres to 'kiss'
#4I'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.
Re: Mathematicians discover new way for spheres to 'kiss'
#5I'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.
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.
Re: Mathematicians discover new way for spheres to 'kiss'
#6I'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 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.
Re: Mathematicians discover new way for spheres to 'kiss'
#7I'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.
> 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.
Re: Mathematicians discover new way for spheres to 'kiss'
#8I'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…
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.)
Re: Mathematicians discover new way for spheres to 'kiss'
#9> “There may be structures without any symmetry at all,” said Gabriele Nebe (opens a new tab) of RWTH Aachen University in Germany. “And no good way to find them.”
She taught Lineare Algebra II when I took it! It was one of the toughest lectures I took during university. I remember looking to the person next to me and one of us asked "do you understand anything?" and the other said "no! I haven't understood anything for like 20 minutes" and we burst out laughing and couldn't get it together until we were asked to quiet down. Wadim if you hang out here, send me a mail or something!
Re: Mathematicians discover new way for spheres to 'kiss'
#10I'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…
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: