Polygons and PolyhedraPlatonic Solids
At the beginning of this course we defined
In a regular polyhedron all
So what do the Platonic solids look like – and how many of them are there? To make a 3-dimensional shape, we need at least
If we create a polyhedron where three
If four equilateral triangles meet at every vertex, we get a different Platonic solid. It is called the Octahedron and has
And seven or more triangles at every vertex also don’t produce new polyhedra: there is not enough space around a vertex, to fit that many triangles.
This means we’ve found
Next, let’s try regular pentagons:
Like before, four or more pentagons
The next regular polygon to try are hexagons:
If three hexagons meet at every vertex, we immediately get a
The same also happens for all regular polygons with more than six sides. They don’t tessellate, and we certainly don’t get any 3-dimensional polygons.
This means that there are just
Notice how the number of faces and vertices are
We can turn a polyhedron into its dual, by “replacing” every face with a vertex, and every vertex with a face. These animations show how:
The tetrahedron is dual with itself. Since it has the same number of faces and vertices, swapping them wouldn’t change anything.