3d space may be described via cartesian: x,y,z or via polar: Theta, Phi and R co-ordinates.
A cubic or a spherical space.
Adding a 4th variable, simply adds another dimension.
To add an imaginary dimension one can use an imaginary value, commonly the square root of -1, signified by the letter i or j, known in the trade as a complex number, with both a real (x,y,z) and an imaginary counterpart (eg the 4th axis perpendicular to the three directions in space: x,y,z)

Duchamp became interested along with Jouffret in 4d mathematics as a counterpoint to his artworks, the Large Glass being accompanied by notes in the Green Box, describing..

If a 3d object casts a 2 dimensional shadow, how does a 3d shadow of a 4 dimensional object look?

Thomas Banchoff at the Geometry Centre, explores many of these mathematical twists and concepts, as does the Hyperspace Structures site in Leicester..

Multidimensional Modelling

The mathematics of modelling physics simulations, includes fields such as particle physics for the modelling of water, flames, gasses, fluids etc and tensor models for the simulation of fabrics, clay, and other flexible matter.

The Bristol site on Flexi matter

N-body and Particle modelling techniques