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platonic_solids
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) > 0
=> (mn -2n -2m) < 0
=> (mn -2m -2n) + 4 < 4
=> (m- 2).(n -2 ) < 4

(m,n) only 5 possibilities:
n= 3  3   3    4   5
m=3  4   5    3   3

Substitute into (1),(2):

F= 4  8   20  6  12
E= 6  12 30 12  30
V= 4  6   12  8   20

Tetrahedron 正四面体
(n,m)=(3,3) => (F,E ,V)=(4,6,4)

Cube or Hexahedron 正六面体(n,m)=(4,3) => (6,12,8)

Octahedron 正八面体
(n,m)=(3,4) => (8,12,6)

Dodecahedron 正十二面体
(n, m)=(5,3)=> (12,30,20)

Icosahedron 正二十面体
(n,m)=(3,5) => (20, 30,12)

The most complicated and the prettiest symmetric solid is:
Icosahedron 正二十面体

Icosahedron is the shape of the incurable HiV viruses.

Icosahedron is the symmetry of Galois Group, proved the unsolvable Quintic equations have no radical roots.

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