Platonic 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 three-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
If
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
If
Next, let’s try regular pentagons:
If
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 three-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
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.
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