In our first day back at the math circle, we discussed the motions of a shape (such as a triangle, square, pyramid, etc) which return the square to a fixed position. We found that these motions can be composed or “multiplied” like numbers, and investigated the nature of these compositions.

In more detail, began by cutting out a triangle, square, and pentagon from card stock and asked the following questions:

Task 1: Explore Motions of a Triangle

  • In what ways can you move the triangle, and still have it end up in the same place? (You are not allowed to bend, fold, or tear)
  • Did you count flipping it over? Did you count doing nothing at all?
  • How can you count these motions? Consider labelling the parts of the triangle.
  • How can you name these motions? If you have more than one way to refer to the same motion, be sure to resolve that.
  • How many motions are there and why?

Task 2: Composing the Motions of a Triangle

  • Can you make a "multiplication table" for these motions? Here, multiplying means doing one, followed by the other.
  • What patterns do you observe in the multiplication table?
  • How do these patterns compare with the patterns in a traditional multiplication table?
  • If you do motion X followed by motion Y, is that the same as doing motion Y followed by motion X? Sometimes? Why or why not?

Task 3: Explore Other Shapes

  • Now start over and do all the same things for the square.
  • How do the multiplication tables for the triangle and square compare?
  • What guesses can you make about the pentagon, hexagon, and so forth?

Next, we cut out and taped together a tetrahedron (some also did a cube).

Task 4: The Third Dimension...

  • Now build a tetrahedron (a figure with four triangular sides) and a cube.
  • What are the motions that leave the tetrahedron in the same place?
  • How can you count these motions?
  • How can you name these motions?
  • Make a multiplication table for the tetrahedron. (The cube is maybe too large)
  • What patterns do you observe?

Task 5: Extensions

  • Suppose you are allowed to tear the triangle. Can you create more permutations of the vertices than you could with non-tearing motions? Now answer the same question for the square.
  • Now consider the same question for the tetrahedron. Can you create more permutations of the vertices of the tetrahedron if you were allowed to tear it?
  • What if you have a mirror, and you allow configurations you see in the mirror to count as motions?
  • For the tetrahedron, some of the motions come from motions of the triangular faces. That is, they fix one vertex and rotate the other three. But some of the motions don't. How many do and how many don't?
  • Suppose I want to create the following permutation of the vertices of the tetrahedron: 1-->4, 2-->3, 3-->4, 4-->2. Is this possible? What about 1-->2, 2-->3, 3-->4,4-->1? Can you say anything about this general question?