Adding a new axis complicates the rotation of the object. Be sure to check out milosz's hyprcube rotations to better understand the intricacies of the possible rotations by adding 1 extra dimension. Hidden within this hypercube, you can see 8 skewed cubes rotating (each cube is 1 side of the 8 sided, or 8-celled figure). I'll give you a hint, the very first can be seen at the bottom right of the image.
The image below shows an excellent colored hypercube. Each of the sides of the hypercube are colored differently. Thus, the individual cubes within the object can be better identified.
