@Moderation Team Very cool.

Not sure, but maybe @Ed Pegg knows.

@Sam Carrettie Good point. In fact, skeletons of five of the six regular convex 4-polytopes are given in the RegularPolychoron class of GraphData. Strangely, when you run GraphData["RegularPolychoron"] the output is

{"HundredTwentyCellGraph", "SixHundredCellGraph", "SixteenCellGraph", "TesseractGraph", "TwentyFourCellGraph"}

meaning that PentatopeGraph (which is a valid named graph and is indeed the skeleton of the 5-cell) isn't in the RegularPolychoron class. Does anybody know why?

This is very nice, thanks. There is a good MathWorld article with some references to built in data about 16-cell. For example.

The vertices of the 16-cell with circumradius 1 and edge length $\sqrt2$ are the permutations of $(\pm1, 0, 0, 0)$. There are 2 distinct nonzero distances between vertices of the 16-cell in 4-space. The skeleton of the 16-cell is implemented in the Wolfram Language as GraphData["SixteenCellGraph"]. When embedded in three-space, the 16-cell skeleton is a cube with an "X" connecting diagonally opposite vertices on each face.

GraphData["SixteenCellGraph", "Graphs"]

GraphData["SixteenCellGraph", "Graphs3D"]