lpetrich
Contributor
For each polytope or tiling, one can construct a dual one, and that one has a reversed Schläfli symbol. Since duality is its own inverse, one has either self-dual polytopes or pairs of duality-related polytopes. Here's what's what.
1D: line segment -- self-dual
2D: regular polygons -- self-dual
Infinite families:
Simplex -- self-dual
Hypercube and cross-polytope -- duality pair
Hypercube lattice -- self-dual
Extras:
2D: hexagonal and triangular plane tilings -- duality pair
3D: dodecahedron and icosahedron -- duality pair
4D: 24-cell -- self-dual
4D: 120-cell and 600-cell -- duality pair
4D: 16-cell and 24-cell space lattices -- duality pair
There are some polytopes that have non-integer symbol members.
Star polygons have (fraction), where the numerator is the number of vertices, and the denominator is the number of vertices from a vertex to a vertex that it is connected to. Thus, a pentagram is (5/2).
In 3D: Kepler–Poinsot polyhedron
Small stellated dodecahedron: (5/2, 5)
Great stellated dodecahedron: (5/2, 3)
Great dodecahedron: (5, 5/2)
Great icosahedron: (3, 5/2)
In 4D: Regular 4-polytope
(5/2, 3, 3), (5/2, 3, 5), (5/2, 5, 3)
(3, 5/2, 5), (5, 5/2, 3), (5/2, 5, 5)
(3, 3, 5/2), (3, 5, 5/2), (5, 3, 5/2)
(5/2, 5, 5/2)
1D: line segment -- self-dual
2D: regular polygons -- self-dual
Infinite families:
Simplex -- self-dual
Hypercube and cross-polytope -- duality pair
Hypercube lattice -- self-dual
Extras:
2D: hexagonal and triangular plane tilings -- duality pair
3D: dodecahedron and icosahedron -- duality pair
4D: 24-cell -- self-dual
4D: 120-cell and 600-cell -- duality pair
4D: 16-cell and 24-cell space lattices -- duality pair
There are some polytopes that have non-integer symbol members.
Star polygons have (fraction), where the numerator is the number of vertices, and the denominator is the number of vertices from a vertex to a vertex that it is connected to. Thus, a pentagram is (5/2).
In 3D: Kepler–Poinsot polyhedron
Small stellated dodecahedron: (5/2, 5)
Great stellated dodecahedron: (5/2, 3)
Great dodecahedron: (5, 5/2)
Great icosahedron: (3, 5/2)
In 4D: Regular 4-polytope
(5/2, 3, 3), (5/2, 3, 5), (5/2, 5, 3)
(3, 5/2, 5), (5, 5/2, 3), (5/2, 5, 5)
(3, 3, 5/2), (3, 5, 5/2), (5, 3, 5/2)
(5/2, 5, 5/2)