Section 1
FINITE PLANE SYMMETRY GROUPS
Find three examples in nature that have different finite symmetry groups. Sketch these and give the specific elements in their symmetry groups.
Find the symmetry group for a square.
Find the symmetry group for a regular pentagon.
Show that the symmetry group for a regular $n$-gon must be finite.
Show that the symmetry group for a regular n-gon must be the dihedral group $D_n$.
Show that the dihedral group $D_n$ can be generated by two reflections, that is, any element of the group can be expressed as a product of terms involving only these two reflections.
Show that the number of symmetries of a regular $n$-gon is equal to the product of the number of symmetries fixing a side of the $n$-gon times the number of sides to which that particular side can be switched.
Find a formula for the number of symmetries of a regular polyhedron by generalizing the result of the last exercise. Use this to find the number of symmetries for a regular tetrahedron (four faces and four vertices) and a cube.