The universe that we experience is 3 Dimensional, or, if you treat time as a limited dimension, 4 Dimensional.

There's a famous novella entitled "Flatland: A romance of Many Dimensions" primarily concerned with life in a 2-Dimensional universe, but also covering a 2-Dimensional being's adventures in 3 Dimensional and 0 Dimensional space. (It is available for free from Project Gutenberg)

The sci-fi novel "Diaspora" includes (spoiler alert) some intricately described adventures in a fully-thought-out 5-dimensional universe. Nuclear reactions are weak, chemical reactions are strong. Galaxies do not form, solar systems collapse into their star. Exploring the 5 dimensional surface of a single star is a nearly impossible task. Looking up at the sky at night is a truly mind-bending experience, even for our highly-evolved AI descendants.

If we are going to consider higher dimensions, it's good to start with 4, and make our way upward from there.

And we need to understand what is out of scope.

Spiritualism is not required. Writing of this sort is not mathematically practical:

FOURTH DIMENSIONAL SUB-PLANES
There are seven sub-planes of the fourth dimension. Each sub-plane resonates to a different frequency. The higher sub-planes resonate to higher frequencies and the lower sub-planes resonate to lower frequencies
—From Multidimensions.com

External Links

• Wikipedia: Flatland
• Gutenberg: Flatland: A romance of Many Dimensions
• Wikipedia: Diaspora
• Article: A Travel Guide for the Fourth Dimension