As stated in the previous post, there are five regular polyhedra. The tetrahedron, icosahedron, cube, dodecahedron, and octahedron. The Greek believed these to be the only regular polyhedra. After analyzing these polyhedra, multiple characteristics can be found that might help describe their uniqueness.
It helps to look at the basic 2D shapes that make up each of these polyhedra: the regular triangle for the tetrahedron, octahedron, and icosahedron; the square for the cube; and the pentagon for the dodecahedron. The regular triangle has 60 degree interior angles at all three of its vertices, The cube has 90 degree interior angles at all four of its vertices, and the pentagon has 108 degree interior angles at all five of its vertices.
It also helps to analyze how these shapes move from their two dimensional forms to their three dimensional forms. When looking at a cube, each of its vertices have three squares connected them.
When singling out a single vertex on a cube, we can analyze this in 2D by folding the 3 squares flat.
When thinking of this in another 2D scenario with 2 squares, folding this back into 3D would not be possible be because the two squares would simply fold flat onto each other. This helps show that, in order to make a regular polyhedron, three or more shapes must connect at a single vertex. This is seen from all of the five regular polyhedra listed. The tetrahedron, cube, and dodecahedron all have three of their respective shapes at each of their vertices while the icosohedron and octahedron have more of their respective shapes at their vertices.
So it is clear that the vertices of the regular polyhedra as well as the angles of the shapes surrounding the vertices contribute to the creation of the regular polyhedra. In further research, we will attempt to prove these to be the only regular polyhedra based on these characteristics they share.