In the previous lecture, we discussed how to draw triangles in the plane and on the unit sphere such that the triangles tile these surfaces. In this lecture, we discuss triangle groups in more detail by focusing on discrete symmetries of the unit sphere.
Cyclic Groups Revisited
In the previous lecture, we focused on positive integers , , and such that and . We showed that there always exists a spherical triangle with angles , , and . Unfortunately, these formulas from become degenerate when at least one of , , or is 1. Since for being the greatest common divisor of and , we may consider . In this case, we have , but we can interpret the isosceles triangle as having angles , , and . Such a region is sometimes called a lune. The cyclic group is generated by , , and satisfying :
We can give a different color to each copy of the triangle coming from . While in the plane the group is associated with the rotations of the regular -gon, the corresponding figure in is called a hosohedron.
Dihedral Groups Revisited
As a non-degenerate case, consider . Then we have and the isosceles triangle with angles , , and . The dihedral group is generated by , , and satisfying :
You may wish to compare these formulas for and and those in Lecture 6. We can give a different color to each copy of the triangle coming from . While in the plane the group is associated with the symmetries of the regular -gon, the corresponding figure in is called a bipyramid; such tilings of the sphere can be found below.
A Platonic solid is a regular, convex polyhedron. They are named after Plato (424 BC – 348 BC). Aside from the regular polygons, there are five such solids. The complete list of these regular solids can be found below.
|Platonic Solid||No. of Vertices||No. of Edges||No. of Faces||Edges at each Vertex|
|Tetrahedron||4 vertices||6 edges||4 triangular||3 edges|
|Cube||8 vertices||12 edges||6 square||3 edges|
|Octahedron||6 vertices||12 edges||8 triangular||4 edges|
|Dodecahedron||20 vertices||30 edges||12 pentagonal||3 edges|
|Icosahedron||12 vertices||30 edges||20 triangular||5 edges|
We have the following result.
- There are five Platonic Solids: The tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In particular, the cube and octahedron are duals of each other, while the dodecahedron and icosahedron are duals of each other.
- The Platonic Solids have the following groups as sets of symmetries:
- Each of the symmetry groups associated with the Platonic Solids is a triangle group. Explicitly,
- If , , and are positive integers such that , then .
Sketch of Proof: We sketch the ideas for the first two statements.
We consider only the tetrahedron, the octahedron, and the icosahedron since the rotation groups are the same for their duals. The tetrahedron has four vertices, so its rotation group is a subgroup of ; since any automorphism must be orientation preserving it is actually contained in . But each of the four vertices has edges attached, so there are at least automorphisms of the tetrahedron. Since , this must be all of them. The octahedron has four pairs of opposing triangular faces, so its rotation group is a subgroup of . Each of the six vertices has edges attached, so there are at least automorphisms of the octahedron. Since , this must be all of them. Finally, one can find an arrangement of the 20 triangular faces of the icosahedron so that their midpoints form five tetrahedra. The automorphisms of the icosahedron must be orientation preserving of this arrangement of the collections of midpoints, so the automorphisms form a subgroup of . Each of the twelve vertices has edges attached, so there are at least automorphisms of the icosahedron. Since , again this must be all of them.
Each vertex of the tetrahedron, octahedron, and icosahedron has edges attached for . We inscribe the vertices in the unit sphere by making the following choices:
where the angle satisfies
Form the triangle via the three vertices
in terms of the angles , and and the positive real number
Note that the affine point is the north pole, the affine point is the midpoint of the edge between and , while the point is the midpoint of the face formed by , , and . Hence the triangle group permutes around copies of this triangle on the sphere . Since the automorphism group of either the tetrahedron, the octahedron, and the icosahedron must permute the triangle as well, we see that . To be explicit, we have the matrices
For the final statement, the integers , , and , such that fit into the table below.
This completes the proof. .
Consider again the triangle where the affine point is the north pole, the affine point is the midpoint of the edge between and , while the point is the midpoint of the face formed by , , and . We can give a different color to each copy of the triangle coming from . Such tilings of the sphere can be found below.