Platonic solids and duals

the five Platonic (Plato ~ 400 BCE) solids have one regular polygon as their faces:

image from GreatLittleMinds
which has nets for the solids











the dual of a polyhedron is obtained by joining the centres of each face:

• each face becomes a vertex
• each vertex is at the 'centre' of each face

 

what happens for the duals of the Platonic solids?

counting the numbers of edges and vertices for the dodecahedron (12 faces) and icosahedron (20 faces) isn't that easy







it can be helpful to think about the number of edges contributed by each face (regular polygon):
e.g. for a dodecahedron:

5 edges x 12 faces = 60 edges
each of these are counted twice
so 60/2 = 30 edges









and how many vertices each face contributes:

e.g. for an icosahedron:

3 vertices x 20 faces = 60 vertices
each of these are counted 5 times (number of polygons surrounding each point)
so 60/5 = 12 vertices









how can the number of edges be found when the number of vertices is known?
and vice versa?

which shape is a self-dual?

what is the relationship between the number of faces, edges and vertices of a platonic solid and its dual?

why is this so?

check that the following formula works, for each of the five platonic solids:

this formula can be derived from Euler's rule:
vertices + faces = edges + 2

how?