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Polygons and PolyhedraMore on Polyhedra

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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 of Polyhedra

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.