While the perfect structure of the regular polyhedra are fascinating, there are only five that are currently known: the tetrahedron consisting of four triangular faces; the octahedron with eight triangular faces; the icosahedron with twenty triangular faces; the cube with six square faces, and the dodecahedron with twelve pentagonal faces. These five unique polyhedra have been recognized throughout the ages dating all the way back to the ancient Greeks. The Greeks believed that these five shapes were the only in existence and none others exist naturally. Through analyzing these five polyhedra, we have come to the conclusion that the Greeks were right in their theory.
Why is this, however? First, it helps to understand the differences between 2D and 3D. Think of the vertices of the faces on a 2D polygon. When connecting polygons at their vertices, each vertex has a 360 degree radius for which shapes can be placed. Once all of the degrees are occupied, the vertex can no longer hold any other shapes. The angles that center around a vertex are the interior angles of the polygons attached. Regular polygons have equal interior angles, and regular polygons make up regular polyhedra.
Now, what makes 2D different from 3D? In order for an object to shift from the second dimension to the third dimension, it must have a height. An object that has no height is simply flat, a 2D polygon. Therefore, in order to be a 3D object, the object must also have faces. As a result, a regular polygon cannot enter the third dimension on its own. There must be more polygons. In the case of the regular polyhedra, there must be more regular and equal polygons. These polygons are attached at their vertices and line up along their edges. Also notice that two regular polygons are not enough to create an enclosed object because the two polygons would simply fold onto one-another. A third polygon is needed, and as a result, a vertex must connect a minimum of three polygons. Notice that this is the case for all of the five regular polyhedra: the tetrahedron has three triangles connected at each vertex while the octahedron has four and the icosahedron has five. The cube has three squared at each vertex and the dodecagon has three pentagons at each vertex.
Remember that a vertice has 360 degrees to work with in two dimensions. Now, remember that a 3D object must have a height. In order for a polyhedra to obtain this height, its regular polygon faces must be at angles to one-another, or they must bend at their edges. When the faces bend, the vertices no longer have a 360 degree radius. In other words, the vertices can only connect a total number of polygons whose interior angles equal a total less that 360 degrees!
Notice that each for each of the five regular polyhedra, this is the case. The equilateral triangle has interior angles of 60 degrees. In a tetrahedron, each vertex needs only to hold 180 degrees. In an octagedron, each vertex needs only hold 240 degrees. In a icosahedron, each vertex needs only to hold 300 degrees of interior angles. However, notice that there exists no regular polyhedron where each vertex connects six triangles. This is because six triangles would require a vertex to hold 360 degrees, something possible in 2D but not in 3D. The same applies to the cube which has vertices connecting three squares with interior angles of 90. The sum of these three interior angles is 270 which is possible, but four squares would contain a sum of interior angles too high for a 3D vertex. The dodecagon has three pentagons at every vertex. Pentagons have interior angles of 108 degrees. Three pentagons would therefore take up 324 degrees on a vertex which is enough for the vertex to handle. Four, once again, is too much.
The next regular polygon, the hexagon, does not form a regular polyhedron. This is because the hexagon has interior angles of 120 degrees. Because at least three regular polygons are needed at each vertex in order to form a regular polyhedron, a vertex would need to be able to connect three hexagons. However, 120 multiplied by three is 360 which just barely exceeds the maximum angular capabilities of a vertex. Three are able to share a vertex in 2D but not 3D as shown in the figures below.
How do we know that no regular polygons besides triangles, squares, and pentagons can form regular polyhedra, through? Notice that, as sides are added to regular polygons, their interior angles are always increasing. Therefore, the interior angles of the hexagon are smaller that that of every other regular polygon with a larger number of sides. Therefore, their angles will never be small enough to form a regular polyhedra.