In
three-dimensional space and below, the terms
semiregular polytope and
uniform polytope have identical meanings, because all uniform
polygons must be
regular. However, since not all
uniform polyhedra are
regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular
4-polytopes are the
rectified 5-cell,
snub 24-cell and
rectified 600-cell. The only semiregular polytopes in higher dimensions are the
k21 polytopes, where the rectified 5-cell is the special case of
k = 0. These were all listed by Gosset, but a proof of the completeness of this list was not published until the work of for four dimensions, and for higher dimensions. ;Gosset's 4-polytopes (with his names in parentheses): :
Rectified 5-cell (Tetroctahedric), :
Rectified 600-cell (Octicosahedric), :
Snub 24-cell (Tetricosahedric), , or ;
Semiregular E-polytopes in higher dimensions: :
5-demicube (5-ic semi-regular), a
5-polytope, ↔ :
221 polytope (6-ic semi-regular), a
6-polytope, or :
321 polytope (7-ic semi-regular), a
7-polytope, :
421 polytope (8-ic semi-regular), an
8-polytope, ==Euclidean honeycombs==