## Saturday, 1 March 2008

### Close order Drill and the square root of -1

I’ve been thinking about groups lately, both the mathematical kind and the social kind. The thoughts were prompted by another trip with some of my kids to the Center for Mathematical Sciences in Cambridge to hear Professor Marcus de Sautoy speak on his new book, Finding Moonshine, which is soon to be released [the American title seems to be different] and is about symmetry and mathematical group theory. When mathematicians talk about symmetry, we talk about groups.

The social group that joins me on my trips seems diverse at first glance. There are a few seniors, a few juniors, and a smattering of sophomores. There are about the same number of boys and girls, and they are as different as high school boys and girls can be. The girls will giggle at themselves, amazed that they can be excited about going to a math lecture, while the boys grow more quiet than usual when they are out of their natural turf. We stand outside one of the lecture rooms and study a board covered with symbols that neither they nor I can decipher. They try to recognize some of the symbols, a “!” factorial symbol here, “and” and “or” logic symbols there. “Is that Sigma for summation?” Some one suggests its some kind of probability problem. “What’s that one, With the Pi in the parenthesis?” Grateful that I actually know one they don’t, I explain, “That’s a capital Gamma, a function like factorial that works for all real values, not just integers.” They say “Ahh” as if they understand. We’ll expand on that on another trip perhaps, a day when an expert can peal back another level of the mathematical mystery just a bit.

What makes some kids, some people, open their minds to the complexities of math, drawn to the cryptic symbols they don’t understand? It must be the same drive that led Champollion to decipher the Hieroglyphs. How is it that one kid can be blinded to the relationship of imaginary numbers by the simple hurdle of its name, while another wants to visualize a snowflake that exists in a universe with 196,883 dimensions.

All these thoughts wandered through my mind as I watched a group of ROTC students passing beneath my window and stop at attention in the open courtyard below. As they practices the simple stationary turns common to such formations, “A’ Ten Hut”, “Right Face”, About Face”, “Left Face”, I realized that I was watching them perform a physical demonstration of the same relations they would swear were too complex to understand in their algebra classes.

The mathematical term Isomorphism is from the Greek roots for “same body”. The parade ground moves represented an order four group that was isomorphic to the multiplicative relationships between the imaginary quantities they found so impossible to comprehend. The four activities on the drill pad could each be paired with the four primitive mathematical quantities, 1, -1, I, and –I, so that they each would produce the same result.