SYMMETRY
OF PLATONIC SOLIDS AND THE GOLDEN MEAN
Symmetry of Platonic Solids
Cube & Octahedron

They both have the same number of edges,
being 12

The number of faces and vertices are
interchanged
o
Cube has 6 faces and 8 vertices
o
Octahedron has 8 faces and 6 vertices
Dodecahedron & Icosahedron

They both have the same number of edges,
being 30

The number of faces and vertices are
interchanged
o
Dodecahedron has 12 faces and 20 vertices
o
Icosahedron has 20 faces and 12 vertices
Symmetry
of Mapping One Solid Into
Its Reciprocal/Dual
If you connect the centre
of all the faces of the Cube you get the Octahedron
If you connect the centre
of all the faces of the Octahedron you get the Cube
If you connect the centre
of all the faces of the Icosahedron you get the Dodecahedron
If you connect the centre
of all the faces of the Dodecahedron you get the Icosahedron
The Tetrahedron is self
reciprocating.
That is, if you join
the centres of the Tetrahedrons 4 faces you get another
Tetrahedron
Golden Ratio
The ratio of the edge
lengths of 2 solids (Platonic Solids that is) embedded
in one another can be expressed in terms of the Golden
Ratio.
That is: Phi squared
over the square root of five.
Icosahedron and dodecahedron
are related to the Golden Ratio in more ways than
one.

12 vertices of any icosahedron can
be divided into 3 groups of 4 with vertices of each
group lying at corners of a Golden Rectangle (The
ration of length to width of the sides of the rectangle
are the Golden Ratio.

The rectangles are perpendicular to
each other and this common point if the centre of
the icosahedron
Also, and similarly:

The centres of 12 pentagonal faces
of the dodecahedron can be divided into 3 groups of
4. And each of these groups also forms a Golden Rectangle.