The interesting ta for me:
> Had she been one of his graduate students, he would have tried harder to convince her to work on something else. “If they work on something hopeless, it’ll be bad for their career,” he said.
Mathematicians discover new way for spheres to 'kiss'
11–20 of 74 posts
Re: Mathematicians discover new way for spheres to 'kiss'
#12Earlier quoted context omitted.
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.
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.
Re: Mathematicians discover new way for spheres to 'kiss'
#13> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern.
Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps?
> They look for ways to arrange spheres as symmetrically as possible. But there’s still a possibility that the best arrangements might look a lot weirder.
Like square packing for 11 looks just crazy (not same problem, but similar): https://en.wikipedia.org/wiki/Square_packing
Re: Mathematicians discover new way for spheres to 'kiss'
#14Earlier quoted context omitted.
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 sphe…
> 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 in much higher dimensions, like 24 (!). They all "visualized" the equations but in ways that gave them much more insight than you or I might imagine just by looking at the text.
Re: Mathematicians discover new way for spheres to 'kiss'
#15Earlier quoted context omitted.
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.
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 t…
However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.
Re: Mathematicians discover new way for spheres to 'kiss'
#16> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern. Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps? > They look for ways to arrange spheres as…
It would be fun to make that square packing for 11 from wood and give it to puzzle enthusiasts with this task: Rearrange the squares so you can add an additional 12th square. And then watch them struggle putting even those 11 squares back in.
Re: Mathematicians discover new way for spheres to 'kiss'
#17The interesting ta for me: > Had she been one of his graduate students, he would have tried harder to convince her to work on something else. “If they work on something hopeless, it’ll be bad for their career,” he said.
A small anecdote: my dad is a mathematician. For a significant portion of his postdoc/early career (in the 80's/90's) he worked on proving a particular conjecture. Eventually he abandoned it and went to be much more successful in other areas.
A few years ago someone found a counterexample. He was quite depressed for a few weeks at the thought of how much of his strongest research years had been devoted to something impossible.
Choosing a "good first problem" in math is quite difficult. It needs to be "novel," somewhat accessible, and possible to solve (which is an unknown when you're starting out)!
Re: Mathematicians discover new way for spheres to 'kiss'
#18two best spheres in a room; they might kiss :^)
Re: Mathematicians discover new way for spheres to 'kiss'
#19> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern. Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps? > They look for ways to arrange spheres as…
Three pennies form an equilateral triangle with (of course) 60 degree angles.
Six of those equilateral triangles will perfectly add to 360 degrees. Intuitive enough? (I'm being a little hand-wavey by skipping over the part where each penny triangle shares two pennies with a neighbor — why the answer is not 18 for example.)
For my mind though, the intuitiveness ends in dimension 2 though. ;-)
Re: Mathematicians discover new way for spheres to 'kiss'
#20Earlier quoted context omitted.
> 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 sphe…
> 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.