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.
I'll show it to you, but first: are you able to add 80 + 16 in your head? (There's another trick to learn for that.)
12 is made up of a 10 and a 2.
What's 8 x 10? 80.
What's 8 x 2? 16.
Add 'em up? 96, baby!
They teach you to do math on paper from right to left (ones column -> tens column, etc), I find chunking works best if you approach from left to right. Like, multiply the hundreds, then the tens (and add the extra digit to the hundreds-total you already derived), then the ones place (ditto).
It's limited by your short-term memory. I can do a single-digit times anything up to maybe five digits. Two-digits by two digits, mostly. Three-digits times three digits I don't have the working memory for.
Thank you kindly for taking the time to teach me this! This thread has been one of the most useful things in a long ass time that's for sure. If I can ever be helpful to you, email is in the bio. :)
A few years ago I had an in-depth conversation with a (then) sixth-grader of my acquaintance, and came away impressed with the "Common Core" way of teaching maths. His parents were frustrated with it, because it didn't match the paper-based methods of calculation they (and you and I) had been taught, but he'd learned a bunch of these sorts of tricks, and from them had derived a good (probably, if I'm honest, better than mine) intuition for arithmetic relationships.
8 × 12 = 8 × (10 + 2) = 8 × 10 + 8 × 2 = 80 + 16 = 96
8 × 12 = (10 − 2) × 12 = 10 × 12 − 2 × 12 = 120 − 24 = 96
8 × 12 = (10 − 2) × (10 + 2) = 10 × 10 − 2 × 2 = 100 − 4 = 96
8 × 12 = (5 + 3) × 12 = 5 × 12 + 3 × 12 = 60 + 36 = 96
8 × 12 = 4 × 24 = 2 × 48 = 96
8 × 12 = 2³ × (2² × 3) = 2⁵ × 3 = 32 × 3 = 96
etc.