““There are only three Platinic solids in 5D. No problem; we'll check 6D. In six dimensions, there are still exactly the same regular shapes: the hypercube, simplex and cross polytope. Now that wasn't very exciting. We'll go on to 7D. Same three regukar shapes again. Nothing changes in 8D either. Dimension after dimension, only those three Platonic-solid shapes seem to appear. The search is futile: for every dimension from 5D onwards there are always the same three regular shapes. ............ It was Schlafli himself who proved this bleak Platonic-solid landscape for all dimensions, back in 1852.””
