Wednesday Math, Vol. 77: Twelve positions, two generating moves, first example

Consider a regular hexagonal tile that has two sides. If you are putting such tiles into a pattern on the floor or on a counter-top, they will fit together nicely, like the pattern of a honeycomb in a beehive. Any tile you place down in a particular position will fit, but if you make marks to distinguish the corners from one another, there are twelve different positions for the tile that all fit equally well. I've numbered the "original position" with the numbers 1, 2, 3, 4, 5 and 6, starting with the 1 in the rightmost position and the numbers ordered counter-clockwise. If we think about this like a Rubik's Cube puzzle, albeit a really easy one, we can get from the original position to any other position by applying two generating moves in some specific order.

The first generating move is turning the tile 60 degrees. If I want to turn it farther than that, I can do the 60 degree turn twice (120 degrees) or three times (180 degrees) or four times (240 degrees) or five times (300 degrees). If I do the turn six times, we come back to the original position, so the 60 degree turn is said to have order 6.

The second generating move is to flip the tile over. There are actually six different ways to flip the tile over and still have the top and bottom edges remain horizontal, but we can pick any of the six and generate the others by flipping once then using the 60 degree rotation. I chose the flip that sends the leftmost corner to the rightmost corner as the generator. If we flip, then immediately flip again, we come back to the original position, so the flip has order 2.

A flip followed by a turn does not give us the same result as a turn followed by a flip. The study of all the ways these moves can interact with one another is an example of group theory, which is the mathematical distillation of the concept of symmetry. Because I have only a master's degree and not a doctorate, I teach lower division math classes, currently exclusively. Sadly, the introduction to group theory is a junior level collegiate math course, so I will probably never get to start a class by saying "Let G be a group, and H a subgroup of G." I would be very happy to teach this material someday, because it's really lovely.