The tesseract in my avi cannot exist because it is an example of a Euclidian 4th dimensional cubic analog, which has a metric tensor with a signature of (-,+,+,+), but in reality the 4th dimension is a Minkowski continuum in which a cubic analog has a metric tensor with a signature of (+,+,+,+).
As for actually building one, I theoretically could, but the optics of the human eye are not capable of "seeing" dimensions higher then the third so there is no reason to. Our brains are capable of interpreting other dimensions, though, but it takes a lot of knowledge to do so. I had spent almost a full decade of studying advanced mathematics and physics before I could actually understand and conceptualize other dimensions. The best I can do is using optical illusions to simulate roughly what a 4th and other dimensional object looks like. It is kind of like drawing a "cube" on a piece of paper. It is actually a 2 dimensional object that looks a lot like a 3 dimensional object, but is not quite correct and is not capable of the same kind of motion. To show how 4th dimensional objects move, we have to use computer simulations like in my avi.