More than three dimensions

How would one sketch or visualise a 4- or 5-dimensional space?

I meam from a purely mathematical point of view (don't say time is the fourth dimension).

Just visualize n-dimensional space, then set n to 4 or 5.

they say a tesseract is a shadow of a 4-d cube. it is literally impossible to imagine such an object though

Vizualise a the values of a function as a point, a line, a plane or a 3-d space, and see how it changes as you move along the other axes.
In particular, visualizing a function in 4-d space by imagining a function in 3-d space that changes over time is pretty intuitive.
You don't need to visualize it, though, just play around with the formulas until the relationships themselves look concrete to you.

Just imagine more axes perpendicular to the normal 3.

grids in grids as a fractal

1) Use color as one or more dimensions. This takes practice, though.

2) Similarly to #1, use intensity. Brighter or better defined sections are closer, dimmer sections are farther.

3) Use the temporal dimension as a spatial dimension. An MRI displays a series of 2 dimensional slices of your 3 dimensional body in sequence from head to toe. Viewing them in order presents a complete 3 dimensional rendering.

Just off the top of my head.

Sometimes it is even difficult to imagine 3D (or lower) objects embedded in a higher dimension.  The Klein bottle is a classic example: it requires 4 dimensions to embed, but we immerse it in 3D anyway, and shove the self intersection under the rug.

S3 is a 3D object, but requires a 4th to embed.  However, after some thought, I realized that we can sort of embed the thing in 3 space.  We take the interiors of 2 S2s, and label them as two hemispheres of S3.  The boundaries of these 2 spheres is then identified as the central cross section of the S3.