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.
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.
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.
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".
I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map it? I also have dyslexia, and have not found a good strategy for phonics yet, and I'm now 40, so I'm not sure I ever will hehe :))
I even struggle with times tables because the lifting is really hard for me for some reason, it always amazes me people can do 8x12 in their heads.
In algebra, you learn that (a - b)(a + b) = a^2 - b^2. It's not too hard to spot this when it's all variables with a little practice but it's easy to overlook that you can apply this to arithmetic too anywhere that you can rewrite a problem as (a-b)(a+b). This happens when the difference between the two numbers you're trying to multiply is even.
For a, take the halfway point between the two numbers, and for b, take half the difference between the numbers. So a = (8 + 12) / 2 = 10. b = (12 - 8) / 2 = 2.
Here, 8 = 10 - 2 and 12 = 10 + 2. So you can do something like (10 - 2)(10 + 2) = 10^2 - 2^2 = 100 - 4 = 96.
It's kind of a tossup if it's more useful on these smaller problems but it can be pretty fun to apply it to something like 17 x 23 which looks daunting on its own but 17 x 23 = (20-3)(20+3) = 20^2 - 3^2 = 400 - 9 = 391