Comment by bux93
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.