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Magic Number: 2 times as many faces as vertices









Magic Number: Average vertex valence is 6








Magic Number: 5 platonic solids
A platonic solids with Schläfli symbol is a polygonal mesh fulfilling the following properties:

• 
• All faces are -gons: (where )
• All vertices have valence :   (where )

Number of edges cannot be negative:

Since and :


Gives only 5 solutions for :

• : Tetrahedron (4 triangular faces, self-dual)
• : Octahedron (8 triangular faces, dual to Cube)
• : Cube (6 square faces, dual to Octahedron)
• : Icosahedron (20 triangular faces, dual to Dodecahedron)
• : Dodecahedron (12 pentagonal faces, dual to Icosahedron)