Easiest way to start imagining four-dimensional things is by numbering the corners of a 4-cube.
First realize that the eight corners of a cube can be numbered “in binary” 000—001–010–100—110–101–011—111. Just like the four corners of a square can be numbered 00–10–01–11. (And just like the sixteen corners of a tesseract can be numbered as above.)
(Yes, there are combinatorics connections. Yes, there are computer logic connections. Yes, there are set theory connections.)
So the problem of comprehending higher dimensions reduces to adding more entries to a table. You can represent a 400-dimensional cube in Excel—and do calculations about it there, too.