Wednesday, July 15, 2009
Wednesday Math, Vol. 79: Twelve positions, two generating moves, third and last example
There are three different non-abelian groups with twelve elements. What this means in English is that there are puzzles with twelve different positions and moves to change from position to position, but with some kinds of moves, doing Move A first and Move B second does not give the same result as Move B first followed by Move A.
The first two puzzles were based on a hexagonal tile that could be flipped over to put the bottom side up, which is a completely different puzzle than the symmetries of a tetrahedron. The last non-abelian group is called T, for reasons unknown to me. I'm a little proud of the fact that I took the generating moves and turned into an object, since I'd never seen what puzzle the group T is supposed to represent.
The generating moves of this puzzle are 1) spin a gear 120 degrees and 2) turn the entire square 90 degrees. Doing this we can put any of the twelve letters from A through K in the upper left hand corner, and the letter in that position determines all the rest of the letters' positions. What makes this non-abelian is that turning a green gear clockwise makes the red gears next to it turn counter clockwise.
This is another way to represent T. Starting at any grey dot, making a black move, then a red move will put you at a dot different from making a red move first then a black move. For instance, starting in upper left, black-red sends you to the front row of the upper right, while red-black sends you to the upper right middle row.
This is a completely different set-up based on the hexagonal tile group D6. Notice that the black moves travel clockwise on the inside hexagon and counter-clockwise on the outside hexagon. This is what makes black-red different from red-black.
And the final generating diagram is of the symmetries of the tetrahedron, also known as A4. Here the black arrows form counter clockwise moves around on of the four triangles, while the red moves send you from one triangle to another.
I know some of my regular readers will have a hard time not dozing off reading this post. I may just be writing this for myself and a select few others, and I probably haven't done a good job of explaining the beauty of group theory, which of course deals with much more exotic puzzles than just these three simple objects that can be put in twelve different positions. For example, a Rubik's cube is solved by group theoretical operations. But this is the part of math that turned me from a computer science major to an honest to Lenny math major all those years ago, when I was introduced to the topic by the late, great Ted Tracewell.
I write this in his honor. After my grandmother, he's one of the first people I was close to that died, and I still miss him to this day. As the commenter Lockwood wrote when he sent condolences for my mom, the difficult emotions don't really go away, you just get used to them.