Comment by davethedevguy
2 days 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.
> 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.
> 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.
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.”
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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.
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Note that the solid set, all points within a certain distance of the center, is called a ball: https://en.wikipedia.org/wiki/Ball_(mathematics)
If the boundary is included, it's a closed ball, otherwise it's an open ball.
So the sphere is the "skin", the ball is the whole thing.
A bit different than common usage.
> If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.
Yep — and this will generally be the case, as the equation looks like: x1^2 + x2^2 + … + xn^2 = r^2. If you fix one dimension, you have a hyperplane perpendicular to that axis — and a sphere of one dimension lower in that hyperplane.
For four dimensions, you can sort of visualize that as x^2 + y^2 + z^2 + t^2 = r^2, where xyz are your normal 3D and t is time. From t=-r to t=r, you have it start as a point then spheres of growing size until you hit t=0, then the spheres shrink back to a point.
> 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.
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.
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.
However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.
A circle from a flat 2d manifold can be from a 3d sphere, cylinder, or other cross section.
Our mental models don't extend well beyond 3, possibly 4, dimensions, hence _all_ of our intuition starts to be doubtful after 3 dimensions.