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.
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).