What would a tesseract actually look like? Say our world is actually 4-dimensional, and I have a tesseract sitting on my table. To a being only capable of perceiving 3 dimensions, what would it actually look like?
Would it just look like a cube?
I understand that the projection of a tesseract into 3D space looks like the image shown in this question (along with a bunch of images from Wikipedia), however I don't want to try and project a 4D cube into 3D space, I just want to look at a 4D object from 3D space.
Now let's say each side of the tesseract is painted a different colour. If it starts rotating (along the 4th dimension), would it appear as if faces were just appearing out of nowhere?
Now let's say that I can move along the 4th dimension, and do so (say 30 cm) around the tesseract. Ignoring the fact that my table might disappear, etc, if I look back at the spot where the tesseract is, would I still just see a cube? (Assuming I am still only capable of seeing in 3 dimensions).

The 3D slices of a tesseract will look like:
http://imgur.com/gallery/9rdLp
and, the re-worked animations, along with the 2D slices of a cube:
http://imgur.com/gallery/Frqrj
In the second gallery, I used a function that empties out the 3-cells of the 4D cube, which leaves behind just the skeletal 2D cells (analogous to a wireframe cube). Then, embed an infinite number of circles into that skeletal frame. The result is a bizarre surface, which when sliced in 3D, will come out as tesseract slices in the form of 3D wireframe models. 
