 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?

• 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).

• 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?  