Only six

Pick a kind of cell and drag r: r cells stand round the gold edge in the middle (looking along it, it shrinks to a point; from the side it stands upright), each filling a wedge whose angle is the angle between its two faces. If the wedges come to less than a full turn, a coral gap is left; with at least three cells, that gap is what lets them fold up into the fourth dimension, and here they are shown laid out before the fold. If the wedges come to more than a full turn, they overlap, and the extra cells turn coral too. Count them: three, four or five tetrahedra leave a gap and six overlap; three cubes leave a gap and four close the turn exactly: cubes like that just fill space and never fold up into a polytope in four dimensions; octahedra and dodecahedra fit only three, the dodecahedra with a gap of only about 10°; three icosahedra already overlap. Each way of fitting three or more with a gap left over builds exactly one regular polytope in four dimensions, and the corner writes its {p, q, r}: six in all.