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?
it can be helpful to think about the number of edges contributed by each face (regular polygon):
e.g. for a dodecahedron:
each of these are counted twice
so 60/2 = 30 edges
and how many vertices each face contributes:
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