I asked this question on the main site. There was an answer, and I edited the question to be more accurate about what I'm asking.
Meanwhile, I did some searching on Wikipedia, and found "5-cell honeycomb." I guess this is what I was looking for, yet being an undergraduate student, I can't prove it.
I presume if anyone posts another answer, it's going to involve Coxeter-Dynkin diagrams. Is there any way to draw Coxeter-Dynkin diagrams here?