Plato Solids

Why only 5 Plato solids ?

Plato Solid is: Regular Polyhedron 正多面体

• Each Face is n-sided polygon
• Each Vertex is common to m-edges (m ≥ 3)

Only 5 solids possible:
Tetrahedron (n,m)=(3,3) 正四面体
Hexahedron (or Cube) (n,m)=(4,3) 正六面体
Octahedron  (n,m)=(3,4)正八面体
Dodecahedron  (n,m)=(5,3)正十二面体
Icosahedron  (n,m)=(3,5)正二十面体

Proof:
Since each Edge (E) is common to 2 Faces (F)
=> n Faces counts double the edges
nF = 2E …(1)

Since each Vertex has m Edges, each Edge has 2 end-points (Vertex).
=> m Vertex counts double the edges
mV = 2E …(2)

(1) : E= n/2 F
(2): V= 2/m. E = n/m. F
(1) & (2) into Euler Formula: V -E + F = 2
(n/m. F) – (n/2.F) + F = 2
F.(2m + 2n – mn) = 4m

Since F>0 , m>0
=> (2m + 2n – mn) >0
=> – (mn -2n -2m) >…
=> (mn -2n -2m) <…
=>…