Just as with cubes, when you fill a tesseract (4-dimensional cube) with water, the surface of the water traces out a shape which depends on the orientation of the tesseract. Although we can’t properly visualize the 4-dimensional water, its “surface” will be a three-dimensional solid which we can visualize. Shown above are the surface of the water as the tesseract is filled while sitting on (i) a (cubical) face; (ii) a (square) ridge; (iii) an edge; (iv) a vertex. Click on the individual images for larger versions.
(For some reason, the animations above don’t always sync up on the dashboard, so try clicking through to the post if things look funny.)

Said another way, the shapes represented by the above animations each trace out tesseracts in four-dimensional spacetime; the only difference is how the tesseracts are oriented with respect to the time axis.

Notice also that with the tesseract balanced on a vertex the surface of the water forms a perfect octahedron when the tesseract is exactly half-full; we saw yesterday that for the cube the corresponding shape is the regular hexagon. This beauty persists in higher dimensions: the corresponding slice of the n-dimensional hypercube is the maximal centrally symmetric convex subset of the (n-1)-simplex (see this lovely paper of Chakerian and Logothetti for much more).