Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more.
How do mathematicians think about higher dimensions? Can intuitions about the meaning of dot-product, angles and lengths transfer from 2d geometry to a 100d?
If so, would it be enough to fully understand the higher dimesions, namely, could the same problem in 100d have properties\behaviours that are not seen in 2d\3d?