Experience Other Dimensions
Flat Land and it's unofficial sequel Flatterland are good books to get a mathematical yet whimsical look into alternative dimensions. The Tesseract from A Wrinkle in Time was a hyper cube as well.
So we go from 0D (a point) to 1D (a line) to 2D (planes) to 3D (solid objects) and then what happens when you pull that plane into an unseen dimension? Well, use your imagination and watch this video:
6d-Hypercube from Tobby Lang on Vimeo.
Why Bother With 4D?
*(Corrected, thanks n8mansays)