As Susan Goldstine explains [What the Origami Means] A symmetry of a polyhedron is a way of moving the polyhedron so that it occupies the same physical space as before it was moved. As an example, consider the symmetries of the cube. A cube can be rotated 90 degrees, 180 degrees, or 270 degrees around an axis passing through the centers of two opposite faces of the cube,

or it can be rotated 120 degrees or 240 degrees around an axis passing through two opposite vertices of the cube,

or it can be rotated 180 degrees around an axis passing through the midpoints of two opposite edges of the cube.

In fact, these comprise all of the symmetries of the cube except for the mundane but mathematically important move of leaving the cube where it is. The symmetries of a polyhedron reflect its structure and regularity.
  1. Describe the symmetries of the plane.
  2. Choose a regular polyhedron other than the cube, and describe the symmetries.
  3. Describe the symmetries of a sphere.
You may work alone or in a group. Be sure that your write up and pictures are your own group's, and give proper reference where it is due. Note: Describe the individual symmetries that preserve the space, like in the discussion of the cube. You do *not* need to address the structure of the symmetries (as a group).