# 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.

## Archimedean Solids

Platonic solids are particularly important polyhedra, but there are countless others.

**Archimedean solids**

## Applications

Plato was wrong in believing that all elements consists of Platonic solids. But regular polyhedra have many special properties that make them appear elsewhere in nature – and we can copy these properties in science and engineering.

Many **viruses**, **bacteria** and other small **organisms** are shaped like

Many **molecules** are shaped like regular polyhedra. The most famous example is

It was discovered in 1985 when scientists researched interstellar dust. They named it “Buckyball” (or Buckminsterfullerene) after the architect

Most **crystals** have their atoms arranged in a regular grid consisting of

Tetrahedra and octahedra are incredibly rigid and stable, which makes them very useful in **construction**. *Space frames* are polygonal structures that can support large roofs and heavy bridges.

Platonic solids are also used to create **dice**. because of their symmetry, every side has the

The