Wednesday, July 1, 2009
Wednesday Math, Vol 78: Twelve positions, two generating moves, second example
Last week, the topic was about a regular hexagonal tile with two sides, which in math is known as a dihedral six, D6 for short. This week, we have an honest to Plato three dimensional object, a regular tetrahedron, which has four faces, all of them equilateral triangles. What they have in common is that if we choose a particular position where we want to put the corners, we can rotate the object by some move and get the four corners in the same positions in space, but it's a different ordering of the corners.
For the tetrahedron, we can rotate the object by 120 degrees clockwise on the plane where that in this picture contains the vertices 2, 3 and 4, and the shape looks basically the same. The vertex 1 is still at the apex in our view, but listing the vertices clockwise from the upper left, we have 4, then 2, then 3 instead of 2, then 3 then 4. With vertex 1 at the apex, there are three symmetric positions for the other vertices, either stand still, rotate 120 degrees or rotate 240 degrees. We can also make another generating move that will change the number of the apex vertex. While it's a little hard to see in this two dimensional representation, a tetrahedron has an axis of symmetry by connecting the two midpoints of edges opposite one another, which is indicated by the red dotted line. If we turn the object 180 degrees through this axis, vertex 1 and vertex 4 switch positions, and likewise vertices 2 and 3.
So, any of the four vertices can be at the apex, and there are three variations of where the other vertices are, and because 4 x 3 = 12, this is another situation of a symmetry object that has twelve positions and two generating moves. The group of moves has a very different structure from D6. In the other group, one of the generators has order 6, which means a 60 degree turn must be repeated six times to give us 360 degrees, which is to say returning to the original position. In this group, moves have order 2 or order 3, except for the "do nothing" move, called the identity, which is in every group and always has order 1.
Looking at this as a puzzle, it's incredibly easy to solve a question of how do you move from one position to another with these generating moves, but looking deeper into the math, we have the ideas of subgroups and conjugates and mappings, the center, the centralizer, the normalizer, representation by matrices, representation by permutations, this list goes on and on.
For my taste, math and music are the prettiest things in the world, or at the very least they are at the same level of prettiness as hummingbirds in flight and Indira Varma's collarbone. The beauty of math takes some work to appreciate, while the other things listed here can be enjoyed by anyone who experiences them. Sometimes, the extra effort needed can make an enjoyable experience even sweeter, and that's certainly the way I feel about group theory, the mathematical distillation of symmetry.