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Polygons and PolyhedraPlatonic Solids

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At the beginning of this course we defined regular polygons as particularly “symmetric” polygons, where all sides and angles are the same. We can do something similar for polyhedra.

In a regular polyhedron all faces are all the same kind of regular polygon, and the same number of faces meet at every vertex. Polyhedra with these two properties are called Platonic solids, named after the Greek philosopher Plato.

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 faces to meet at every vertex. Let’s start systematically with the smallest regular polygon: equilateral triangles:

If we create a polyhedron where three equilateral triangles meet at every vertex, we get the shape on the left. It is called a Tetrahedron and has faces. (“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 faces. (“Octa” means “eight” in Greek. Just like “Octagon” means 8-sided shape, “Octahedron” means 8-faced solid.)

If triangles meet at every vertex, we get the Icosahedron. It has faces. (“Icosa” means “twenty” in Greek.)

If triangles meet at every vertex, something different happens: we simply get a tessellationa quadrilateralanother Icosahedron, instead of a 3-dimensional polyhedron.

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 Platonic solids consisting of triangles. Let’s move on to the next regular polygon: squares.

If squares meet at every vertex, we get the cube. Just like dice, it has faces. The cube is sometimes also called Hexahedron, after the Greek word “hexa" for “six”.

If squares meet at every vertex, we get another tessellationa tetrahedronanother cube. And like before, five or more squares also won’t work.

Next, let’s try regular pentagons:

If pentagons meet at every vertex, we get the Dodecahedron. It has faces. (“Dodeca” means “twelve” in Greek.)

Like before, four or more pentagons don’t workare possible because there is not enough space.

The next regular polygon to try are hexagons:

If three hexagons meet at every vertex, we immediately get a tessellationpolyhedronhexahedron. Since there is no space for more than three, it seems like there are no Platonic solids consisting of hexagons.

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 Platonic solids! Let’s have a look at all of them together:

Tetrahedron

Faces
Vertices
Edges

Cube

Faces
Vertices
Edges

Octahedron

Faces
Vertices
Edges

Dodecahedron

Faces
20 Vertices
30 Edges

Icosahedron

Faces
12 Vertices
30 Edges

Notice how the number of faces and vertices are swapped aroundthe same for cube and octahedron, as well as dodecahedron and icosahedron, while the number of edges stays the sameare different. These pairs of Platonic solids are called 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.

Plato believed that all matter in the Universe consists of four elements: Air, Earth, Water and Fire. He thought that every element correspond to one of the Platonic solids, while the fifth one would represent the universe as a whole. Today we know that there are more than 100 different elements which consist of spherical atoms, not polyhedra.

Images from Johannes Kepler’s book “Harmonices Mundi” (1619)

Archimedean Solids

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

Archimedean solids, for example, still have to be made up of regular polygons, but you can use multiple different types. They are named after another Greek mathematician, Archimedes of Syracuse, and there are 13 of them:

Truncated Tetrahedron
8 faces, 12 vertices, 18 edges

Cuboctahedron
14 faces, 12 vertices, 24 edges

Truncated Cube
14 faces, 24 vertices, 36 edges

Truncated Octahedron
14 faces, 24 vertices, 36 edges

Rhombicuboctahedron
26 faces, 24 vertices, 48 edges

Truncated Cuboctahedron
26 faces, 48 vertices, 72 edges

Snub Cube
38 faces, 24 vertices, 60 edges

Icosidodecahedron
32 faces, 30 vertices, 60 edges

Truncated Dodecahedron
32 faces, 60 vertices, 90 edges

Truncated Icosahedron
32 faces, 60 vertices, 90 edges

Rhombicosidodecahedron
62 faces, 60 vertices, 120 edges

Truncated Icosidodecahedron
62 faces, 120 vertices, 180 edges

Snub Dodecahedron
92 faces, 60 vertices, 150 edges

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.

Radiolaria skeleton

Icosahedral virus

Many viruses, bacteria and other small organisms are shaped like icosahedra. Viruses, for example, must enclose their genetic material inside a shell of many identical protein units. The icosahedron is the most efficient way to do this, because it consists of a few regular elements but is almost shaped like a sphere.

Buckyball molecule

Montreal Biosphere

Many molecules are shaped like regular polyhedra. The most famous example is C60 which consists of 60 carbon atoms arranged in the shape of a Truncated Icosahedron.

It was discovered in 1985 when scientists researched interstellar dust. They named it “Buckyball” (or Buckminsterfullerene) after the architect Buckminster Fuller, famous for constructing similar-looking buildings.

Fluorite octahedron

Pyrite cube

Most crystals have their atoms arranged in a regular grids consisting of tetrahedra, cubes or octahedra. When they crack or shatter, you can see these shapes on a larger scale.

Octagonal space frames

Louvre museum in Paris

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.

Football

Polygonal role-playing dice

Platonic solids are also used to create dice. because of their summetry, every side has the probability of landing facing up – so the dice are fair.

The Truncated Icosahedron is probably the most famous polyhedron in the world: it is the shape of the football.