Platonic Solids and the Golden Ratio

There are five regular polyhedra. These are they: tetrahedron, cube/hexahedron, octahedron, dodecahedron and icosahedron. Here is a short summary in form of a table about the vertices, edges, faces and volumes of these solids:

Tetrahedron

Cube/ Hexahedron

Octahedron

Dodecahedron

Icosahedron

v vertices

4

8

6

20

12

e edges

6

12

12

30

30

f faces

4

6

8

12

20

V volume

√2 /12*a³

√2 /3*a³

(15+7*√5)/4*a³

(15+5*√5)/12*a³

We could group them into three groups:

• The triangular faced ones {tetrahedron, octahedron, icosahedron}

• The square faced one {cube}

• The pentagon faced one {dodecahedron}

If we put these polygons together like this, we can recognize the golden ratio in it: Here we add a triangle, square and a pentagon together in the following way. Then we connect the point A and H with each other – and exactly where this line crosses the square the golden ratio appears:

New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length.

Source
Also, if you didn’t know about the pentagram, even there appears the golden ratio:

Source: Wikipedia

And, two of the platonic solids have the golden ratio in themselves: the icosahderon and the dodecahedron.

Imagine 3 planes put together like this:

In a dodechedron the exact same planes can be put into it like this:

But if we want the edges of the planes to be on the edges of the dodecahedron, we have to use another ratio:

Then the ratio is 1 to phi².