Editing Platonic solid (section)

=== Higher dimensions ===
{{Further|List of regular polytopes}}
{| class="wikitable floatright" style="text-align:center; max-width: 22em"
|-
! {{nowrap|Number of}} dimensions
! {{nowrap|Number of convex}} regular polytopes
|-
| 0 || 1
|-
| 1 || 1
|-
| 2 || &infin;
|-
| '''3''' || '''5'''
|-
| 4 || 6
|-
| &gt; 4 || 3
|}

In more than three dimensions, polyhedra generalize to [[polytope]]s, with higher-dimensional convex [[regular polytope]]s being the equivalents of the three-dimensional Platonic solids.

In the mid-19th century the Swiss mathematician [[Ludwig Schläfli]] discovered the four-dimensional analogues of the Platonic solids, called [[convex regular 4-polytope]]s. There are exactly six of these figures; five are analogous to the Platonic solids : [[5-cell]] as {3,3,3}, [[16-cell]] as {3,3,4}, [[600-cell]] as {3,3,5}, [[tesseract]] as {4,3,3}, and [[120-cell]] as {5,3,3}, and a sixth one, the self-dual [[24-cell]], {3,4,3}.

In all dimensions higher than four, there are only three convex regular polytopes: the [[simplex]] as {3,3,...,3}, the [[hypercube]] as {4,3,...,3}, and the [[cross-polytope]] as {3,3,...,4}.{{sfn|Coxeter|1973|p=136}} In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.