Mathematicians discover new way for spheres to 'kiss'

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!

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.”