Below is a program which can display all possible three and four dimensional regular solids.
Euclid proved around 200 B.C. that there are exactly five regular solids in three dimensions. Ludwig Schlafli proved in 1901 that there are exactly six regular solids in four dimensions.
Schlafli also proved that the only regular solids in dimensions greater than or equal to five are the generalized tetrahedron, cube, and octahedron.