polyhedra: total angle sum

Euler's relationship connecting the numbers of faces, edges and vertices (F + V is close to E) for polyhedra is fairly well known

another (very neat) relationship for the 'total angle sum' was brought to my attention by Gordon Haigh, when he worked at Wolverhampton University

a powerpoint for this task gives the steps for proofs for a general prism and general pyramid, establishing Euler's  relationship for these polyhedra

animated platonic solids from Wikipedia:

itsphun

itsphun europe
itsphun

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?

Iles de Paix modular polyhedra

although there are many photos available I don't know whether or not these can be ordered?

Schlegel diagrams

Victor Schlegel (1843 ~ 1905)

if you make one of the faces of a polyhedra 'see through' and peer in
what you see, distorted and flattened, is a 2-D representation of the solid

alternatively, you could view it as a projection:

what are these (common) 3D shapes?

which solid goes with which Schlegel diagram?
















for the 2-D (flat) representations, what is the relationship between the numbers of :

• faces (enclosed spaces)
• edges (lines)
• vertices (nodes)?