- Natalija Budinski
Platonic solids are convex regular polyhedra. There only five of them: tetrahedron, cube, octahedron, dodecahedron and icosahedron.
Maybe you are wondering why are there only five Platonic solids? The proof is based on the fact that there are at least three faces at each solid vertex. In the process of subtracting internal angles that meet at the vertex, the sum must be less than 360. It is because at 360 the shape will be flattened. The Platonic solids consist of identical regular polygons, such as: triangles with 60 internal angles, squares with 90 internal angles, or pentagons with 108 internal angles. Hexagons cannot be the faces of Platonic solids because their internal angles are 120, and meeting three hexagons in the vertex will flatten the shape. So, when three regular triangles meet at the vertex, they constitute the 180 angle of the tetrahedron. Four regular triangles met in the vertex constitute the 240 angle of the icosahedron. Three squares met in the vertex constitute the 270 angle of the cube, and three pentagons met in the vertex constitute the 324 angle of the dodecahedron. Anything else would constitute an angle of 360 or more, which is impossible. For example, four regular pentagons would constitute a 432 angle (Budinski, N., Origami as a Tool for Exploring Properties of Platonic Solids, Proceedings of Bridges, 2016: Mathematics, Music, Art, Architecture, Education, Culture, pages 649-654).