Does a hypercube really exist?
Maybe it is a philosophical question.
(1) I ask another question: if somebody prove that it actually exist he actually prove that our universe is a parallel universe?
According to Einstein the universe of us compose from 4-dimension: time, x, y, z.
(2) Is the hypercube can exists in our universe? [because both of them contain 4-dimension (?!)]
Phrasing it that way, then you need a way to correlate time measurements with distance. For instance, let's take one unit of distance. Your question is similar to asking if there is a unit of time whose magnitude is the same as one inch. If there is, then a one inch cube sitting in place for that amount of time would be a hypercube, would it not?
If there is no such comparison between magnitudes of time and space, then you can create what is known as a topological hypercube (any solid object with no holes over any amount time is homeomorphic to a hypercube).
A hypercube is the generalization of a cube to more than 3 dimensions. A tesseract is a 4 dimensional hypercube.
Putting one of these into Minkowski space-time (empty space-time... the space-time of Special Relativity) is going to give you some weird properties because of what Relativity does to the time axis. But I don't think there would be any problem with it, theoretically speaking. Note though: even if you could build one it would have a length of time to exist (the "time" edge of the tesseract has a length) so it would disappear on you at some point.
-Dan