Question

How can I visualise higher dimensional `n`-spheres, where the dimension, `D`, is larger than 3? How can I find the volume of such an object?

Answer 1

Unfortunately, we cannot “visualise” objects beyond our own 3-dimensional existence.

Explanation:

This is one reason that mathematics is so interesting – and even useful! We can represent, manipulate and use concepts (imaginary numbers are another one) that can impact our real lives, yet are beyond our physical ability to imagine, outside of mathematical formulas.

To grasp the limitations of higher-order visualization from a lower-order existence, I highly recommend both the book and the video “Flatland”. The interactions of 2-dimensional beings with a 3-dimensional space provide an excellent perspective on the implications of “dimensional” existence.

https://en.wikipedia.org/wiki/Flatland
http://flatlandthemovie.com/