There are an infinite number of regular polygons in 2D-space including the equilateral triangle, the square, the regular pentagon, the regular hexagon and so on forever.

But there are only five regular polyhedrons in 3D-space.  They are called the Platonic solids and were discovered by the ancient Greeks.  They include the regular tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

Somewhat surprisingly there are exactly six such regular figures in 4D-space including this one—the hypercube.  Of course, this model has been “squashed down” in order to render it in 3D-space. The hypercube is the generalization of the square in 2D-space and the cube in 3D-space.  Analogous n-dimensional cubes exist in 5D-space, and in fact, in all higher dimensions.

Beyond 4 dimensions there are always just three regular figures. The three figures that exist in all higher dimensions are the analogs of the tetrahedron, the cube and the so-called duals of the higher cubes (the octahedron is the dual of the cube in 3D-space).  In this respect as least, higher dimensional spaces are kind of boring.