Polygons and PolyhedraPlatonic Solids
At the beginning of this course we defined A regular polygon is a polygon in which all sides have the same length and all interior angles have the same size.
In a regular polyhedron all The faces of a polyhedron are the polygons which make up its surface. The “corners” of a polyhedron are called its vertices. A Platonic solid is a polyhedron where every face is a regular polygon with the same number of edges, and where the same number of faces meet at every vertex. There are only five different Platonic solids: the Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron. Plato (c. 425 – 347 BCE) was a philosopher in ancient Greece, and – together with his teacher Socrates and his student Aristotle – laid the very foundation of Western philosophy and science. Plato founded the Academy of Athens, the first higher learning institution in the Western world. His many writings on philosophy and theology, science and mathematics, politics and justice, make him one of the most influential thinkers of all time.
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 An equilateral triangle is a triangle in which all three sides have the same length.
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 TODO
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 (c. 425 – 347 BCE) was a philosopher in ancient Greece, and – together with his teacher Socrates and his student Aristotle – laid the very foundation of Western philosophy and science. Plato founded the Academy of Athens, the first higher learning institution in the Western world. His many writings on philosophy and theology, science and mathematics, politics and justice, make him one of the most influential thinkers of all time.
Archimedean Solids
Platonic solids are particularly important polyhedra, but there are countless others.
An Archimedean solid is a polyhedron made up of different kinds of regular polygons, that looks the same from every direction. There are 13 different Archimedean solids. A regular polygon is a polygon in which all sides have the same length and all interior angles have the same size. Archimedes (c. 287 – 212 BCE) was an ancient Greek scientist and engineer, and one of the greatest mathematicians of all time. He discovered many concepts of calculus and worked in geometry, analysis and mechanics. While taking a bath, Archimedes discovered a way to determine the volume of irregular objects using the amount of water they displaced when submerged. He was so excited by this discovery that he ran out on the street, still undressed, yelling “Eureka!” (Greek for “I have found it!”). As an engineer, he built ingenious defence machines during the siege of his home city Syracuse in Sicily. After two years, the Romans finally managed to enter, and Archimedes was killed. His last words were “Do not disturb my circles” – which he was studying at the time.
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 The icosahedron is a Platonic solid and consists of 20 faces that are all equilateral triangles. It has 12 vertices and 30 edges.
Many molecules are shaped like regular polyhedra. The most famous example is The truncated icosahedron is an Archimedean solid consisting of 12 regular pentagons and 20 regular hexagons.
It was discovered in 1985 when scientists researched interstellar dust. They named it “Buckyball” (or Buckminsterfullerene) after the architect Richard Buckminster “Bucky” Fuller (1895 – 1983) was an American architect, designer and inventor. He is famous for constructing geodesic dome – large, spherical structures. Similar looking Carbon molecules, the fullerenes, were later named after him.
Most crystals have their atoms arranged in a regular grid consisting of The tetrahedron is a Platonic solid and consists of four faces that are equilateral triangles. It has four vertices and six edges. A cube (sometimes called Hexahedron) is a regular polyhedron in which every face is a square. It is one of the five Platonic solids, and it has 6 faces, 8 vertices and 12 edges. The octahedron is a Platonic solid and consists of 8 faces that are equilateral triangles. It has 6 vertices and 12 edges.
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 A probability is number between 0 and 1 that indicates the likelihood of the occurrence of certain event. The probability of an event A is written as
The The truncated icosahedron is an Archimedean solid consisting of 12 regular pentagons and 20 regular hexagons.