# Polygons and PolyhedraPlatonic Solids

At the beginning of this course we defined

In a *regular polyhedron* all **Platonic solids**

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 **Tetrahedron** and has *(“Tetra” means “four” in Greek).*

If four equilateral triangles meet at every vertex, we get a different Platonic solid. It is called the **Octahedron** and has *(“Octa” means “eight” in Greek. Just like “Octagon” means 8-sided shape, “Octahedron” means 8-faced solid.)*

If **Icosahedron**. It has *(“Icosa” means “twenty” in Greek.)*

If

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

If **cube**. Just like dice, it has *Hexahedron*, after the Greek word “hexa" for “six”.

If

Next, let’s try regular pentagons:

If **Dodecahedron**. It has * (“Dodeca” means “twelve” in Greek.)*

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

**Tetrahedron**

**Cube**

**Octahedron**

**Dodecahedron**

20 Vertices

30 Edges

**Icosahedron**

12 Vertices

30 Edges

Notice how the number of faces and vertices are **dual solids**

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.