Imagine a solid 3-D cube. Now imagine that this cube is traveling close to the speed of light. To what degree will the spatial geometric properties of this object (or in general of any 3-D object) change (as predicted by the Lorentz transformations)? 

I assume that despite the changes to shape, some fundamental geometric features of 3-D objects are preserved no matter the frame of reference. I assume that a cube will preserve some of its fundamental geometry Namely, a cube undergoing Lorentz contraction will not become a torus or vice-versa. (I might be wrong about this) But what is the formal limit to these changes? Will the affine structure of a 3-D object undergoing Lorentz contractions change or would its topology also change? 

Thank you.