Turning your head into the 4th dimension for a glome is probably the simplest example to do.
A tesseract will be much more interesting to look at. I've noticed already that thinking about a tesseract highlights the two types of movement that I think are important.
Just as we can view the base of a cube (that's sitting on it's base) along one to two edges the same applies to a tesseract.
We can view the base of a tesseract along one to three of what we would call edges. These extend vertically to form what we consider to be the volume of the faces we see.
For us 3Der's we are a little more limited on what we can view of those faces.
First we have to consider that, whereas we can only turn our head along left and right to observe the base of a cube in our world, we can angle our head a whole 360° of angles to follow the base of a tesseract.
But, not only can we tilt out head we can also spin in a circle to change our orientation. It makes it quite interesting to depict a tesseract using rotational projection.
I would rather have it be interactive as this is really what you need to appreciate the experience.
It did give rise to the the thought that I believe that 4D creatures would actually develop legs that not only propel them forward but that also are designed to allow them to change their orientation easily.
So they would need multi-perambulating legs that can run in cicles as well as forwards. Interesting thought.
Added 27/12/2014:
Here is a depiction of a tesseract in rotated projection. There are certainly things I want to improve in the depiction.
The important things it shows presently are that whereas we can see only three faces of a cube at any one time; using rotated projection you can see four faces of the tesseract at once: The three front cube faces: blue, pink and red and the top cube face: green. Most of these are shown ghosted off into the 4th dimension.
The things I would like to fix are: 1. I would like for each object its parts that are in the ana direction to be behind the object's current frame and parts that are in the kata direction to be in front of the object's current frame, 2. Full projection rather than stepped projection as was utilised here, 3. Less transparency for the denser parts in the 4th directions, 4. None of those black outlines.
As I mentioned this rotated projection style should more readily allows us to understand how the cube faces of a tesseract work.
Also a fully interactive model should give us a great feel for a tesseract allowing us to orient ourselves around it in ways we find more comfortable.