Hypercube

4D Cube in 3D
To fully understand I recommend reading this.



Instructions
To produce this object I used a common editor called blender.
First I created a basic cube with a smaller cube inside. I extruded the faces of the smaller cube to the outside of the larger cube while locking the extrusion to the axis the face was pointing to.
With the new object inside, I used the boolean tool to remove the area that it shared with the big cube and removed it. This left be with a solid "wireframe" like cube. I duplicated this object and shifted it using the blender grid.
Join these cubes into one object.
Onces that was done, I joined all of the corners using the edit tools and created joining vertices, then faces from those vertices. Don't fill the ends that are inside the cubes yet. (Its hard to get to them if you do)
The next bit is tricky. I highlight all of the new "joints" that aren't completely solid and unjoin them from the object. (Use the blender manual to attempt this) I filled in the empty ends where the cubes once existed without moving it on the scene!
You can now use boolean logic to join this into the cubes and clean up any faces that may become problematic using the edit tools.
*Note Dimensions don't matter much in this. You can scale my model or make your own with any sizes you like.