Very neat stuff indeed! Seriously, this could be used for crafts and other neat projects even if you don’t care about the math.
This is a hyperboloid (also called a “hyperboloid of one sheet”). It is a smooth continuous surface that can be obtained by rotating a hyperbola around an axis. (Actually, this is just part of an infinite hyperboloid surface; the top and bottom circles are where I arbitrarily cut it off; a complete hyperboloid has no edges.) Surprisingly, it is a ruled surface, which means every point in it lies on a straight line that is embedded in the surface.