If we take a look at our shapes you'll see they use lots of 2D shapes as faces. There are lots of ways for us to build up solids.

We can talk about Euler's Formula. The concept of face, edge and vertex (corner) should be easy to explain. Then get them to count up the number of each (hide the curvy edged shapes). Now consider the magic formula, V-E+F, if you keep on trying it you should always get two!

Platonic solids are special as all their faces are exactly the same, there's also not many of them. The faces are all regular (same angles and sides) and the same number of faces meet at each vertex. So we can start trying to build some. The first thing to focus on is around a vertex, what sort of shapes fit?

The five solids are the Tetrahedron, Cube, Octahedron, Dodecahedron and Isocohedron.

For simplicity of notion here I'm going to use Schlafli symbols {p,q} where p is number of edges each face has and q is how many faces meet at a vertex. These leave {3,3} {4,3}, {3,4}, {5,3} and {3,5} as our platonic solids.

Try building things up, at a vertex the things that meet must have angles less than 360 degrees or else they won't fill space. So we eliminate several options, in fact we get rid of anything with p>5 as q<3 makes no sense and {6,3} encodes a tessellation of the plane by hexagons. Similarly {4,4} and {3,6} are tessellations, leaving us with only five options. It turns out all of these work, you can show this by making them or getting out the dice.

This embodies a fairly basic proof which you can talk about.

Prisms, Antiprisms and Archimedean Solids

Archimedean Solids are the next step from platonic, removing the requirement that all the faces are the same. There are two infinite classes the prisms and Antiprisms (which have a belt of triangles and are slightly skew). These are sort of Archimedean but we exclude them as there's infinitly many of both. Then let's look at other ones. There's between 11-17 (roughly should double check) depending on your definition of 'the same', i.e. allowed isometries.

The easiest to find are the truncated platonic solids, you get these by slicing corners off the solids, this doubles the number of sides and creates new faces. This gives 5 Archimedean Solids.

You can also form some manageable ones with vertex patterns (3,4,5,4), (3,5,3,5),(8,6,4),(3,4,3,4),(4,4,4,3). There's also a very large one with pattern (10,6,4) notable for how large it is. Patterns describe the number of sides in each shape clockwise around a vertex.

Plus - There's a few weird 'Archimedean Solids' which depends on definition. There's two alignments of the shape (4,4,4,3) which has a belt of squares around the middle, this is formed with either the triangles alligned or not, the number of belts changes. There are also two snub shapes (3,3,3,3,4), (3,3,3,5), these expand the cube and dodecahedron with a belt of triangles. These two have a chirality which you can change by folding the net up and down.

PLUS - You can also do a topological proof using Schlafli symbols, from Euler's formula and the following fact 'pF=2E=qV' you can bound things by eliminating F and V in Euler and then getting 1/q + 1/p = 1/2 + 1/E > 1/2. Then as p,q >2 we can find the only five possibilities. Again this just shows there are at most five, by giving the shapes we show there are at least five. We need to check this as we could have introduced new false solutions when doing this manipulation or reasoning.

PLUS - Dual solids take faces <->verticies. We do this by placing a vertex in the centre of each face and an edge between these vertices if the faces touch. Tetrahedrons are self dual and then cubes and octahedron and iso and dodecs form dual pairs. This operation swaps p <-> q in the Schlafli symbol.

PLUS - You can link this to hexaflexagons by talking about symmetry groups. It may be easier to start with symmetry groups of faces then move to the 3D shapes. You can talk about order and also associativity.

PLUS - Talking about tessellations, there's a fairly interesting link as the platonic solids biject with tillings of the sphere (positive curvature) by projecting. The boundary cases {3,6}, {4,4} and {6,3} are plane tessellations and then any other {p,q} defines tessellations of hyperbolic surfaces (negative curvature).

Nets

Once you've built a shape, detach a few edges and flatten it out to form a net. A net is a 2D representation of a 3D shape.

There's a few interesting questions you can ask. How many different nets does a cube have? I think there are eleven, up to symmetries, but I may be wrong. Try the same for an open cube (one face missing).

History

The platonic solids have been historicaly important, with Greek philosophers believing they corresponded to the five elements (we now know of more than 5 elements). The tetrahedron, cube, octohedron, dodecahedron and iscohedron match to fire, earth, air, ether and water. Of these none are elements, especially ether which doesn't exist. Kepler also believed the platonic solids formed shells on which the planets orbited, while close, it's not true either.