Platonic Solids are five and only five:
A cube has 6 squares, 3 squares meet at a vertex, angle at a vertex = 3 * 90 = 270 degrees
An octahedron has 8 triangles, 4 triangles meet at a vertex, angle at a vertex = 4 * 60 = 240 degrees
A Dodecahedron has 12 pentagon, 3 pentagons meet at a vertex, angle at a vertex = 3 * 108 = 324 degrees
A Icosahedron has 20 triangle, 5 triangles meet at a vertex, angle at a vertex = 5 * 60 = 300 degrees
In all the above cases:
angle at a vertex < 360 degrees.
So it is possible to form a polyheda. It is not possible with any other case. For proof read on the link which is very interesting. The proof is simple using Euler's formula.
Some more interesting facts on wiki:
From wikipedia - The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests he may have only been familiar with the tetrahedron, cube, and dodecahedron, and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that there are no other convex regular polyhedra.
The Platonic solids feature prominently in the philosophy of Plato for whom they are named. Plato wrote about them in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron.
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Coming up more of Origami Platonic solids on my blog...