[lpetrich]

I'll now consider regular polygons (2D) and their counterparts in greater numbers of dimensions: polyhedra (3D), polychora (4D), and polytopes in general.

In two dimensions, there is an infinite family of regular polygons, starting with equilateral triangles and squares. For n >= 3, they have n vertices and n edges. In the notation that I will use below:
n, n

In three dimensions, there are only the 5 Platonic solids. They are:
Tetrahedron: 4, 6, 4 (triangle)
Cube: 8, 12, 6 (square)
Octahedron: 6, 12, 8 (triangle)
Dodecahedron: 20, 30, 12 (pentagon)
Icosahedron: 12, 30, 20 (triangle)

In order: vertices, edges, faces. They have the following dualities, from reflecting the list of contents:
Tetrahedron (self), like polygons
Cube - Octahedron
Dodecahedron - Icosahedron

In four dimensions, there are 6 of them. They are:
5-cell: 5, 10, 10, 5 (triangle, tetrahedron)
Tesseract: 16, 32, 24, 8 (square, cube)
16-cell: 8, 24, 32, 16 (triangle, tetrahedron)
24-cell: 24, 96, 96, 24 (triangle, octahedron)
120-cell: 600, 1200, 720, 120 (pentagon, dodecahedron)
600-cell: 120, 720, 1200, 600 (triangle, icosahedron)

Dualities:
5-cell (self)
Tesseract - 16-cell
24-cell (self)
120-cell - 600-cell

Four more than four dimensions, there are 3 of them. Instead of a list, I will give a count of k-dimensional objects for n dimensions (point = 0, line = 1, ...).
Simplex: (n+1)!/((k+1)!*(n-k)!)
Hypercube: 2n-k*n!/(k!*(n-k)!)
Cross-polytrope or orthoplex: 2k+1*n!/((k+1)!*(n-k-1)!)

Duality:
Simplex (self)
Hypercube - Orthoplex

Simplex: triangle, tetrahedron, 5-cell, ...
Hypercube: square, cube, tesseract, ...
Orthoplex: square, octahedron, 16-cell, ...

So in summary, the number of regular polytopes:
infinite, 5, 6, 3, 3, 3, 3, 3, 3, 3, 3, ...