Higher dimensional geometry can show up in lower dimensional problems. This numberphile video (https://www.youtube.com/watch?v=6_yU9eJ0NxA) involves a puzzle about throwing darts at a dart board which is solved by using the volumes of 4+ dimensional spheres.
doesn't 4th just include time with a 3d object.
Would things like waves/pulses fall under that?
5th seems to be used in the Kaluza-Klein theory[0] for gravitation and electromagnetism.
Can you give an example of how we've made use of the 4th, 5th, 6th dimension, etc.?