How do you envision space in more than 3 dimensions, then rotate it to see what happens? (Strange Universe 2011)

Garrett Lisi, surfer and iconoclastic quantum physicist, gave a fascinating TED Talk. In it, he covered his controversial Theory of Everything. Interesting as that Theory is, that isn’t the focus of this reference. In his talk, he showed a number of views of the 226 known subatomic particles arrayed in 4 or more dimensions. He then rotated the views and showed that some fascinating patterns emerged. Interestingly, these patterns indicate that there must be some particles missing. The Higgs Boson is one of those suspected AWOL particles.

Enough of that. My question is how do you construct 4, 5, 6 or 8 dimensional space within a computer model so you can then rotate the space and see how things align along various axes.

This question is the first of the 2011 Strange Universe Series. The entire 2010 Series of 20 questions can be found from here