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Likewise! In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres wit

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.
- close04 2y ago
  > 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.
  - neom 2y ago
    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.
    - semi-extrinsic 2y ago
      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 <0.01% 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".
tomrod 2y ago
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.
evandrofisico 2y ago
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.

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.

close04 2y ago
> 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.
- neom 2y ago
  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.
  - semi-extrinsic 2y ago
    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 <0.01% 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".

tomrod 2y ago

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.

evandrofisico 2y ago

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.