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.

One is the identity, Like “Attention” it keeps the position fixed. About face is the same as -1. If we think of right face as i, the square root of -1, then left face would be –i, the opposite of i. Any two actions on the pad operated like multiplication of its counterparts in the abstract number set. About face followed by about face was like Attention, not turning at all, just as multiplying -1 by -1 returned us to the mathematical identity. Right face followed by about face produced left face, and mathematically -1 x I gives –i. What about that mysterious i x i that so confused them in the algebra class. Right face followed by right face was just about face, the drill pad’s symbol for -1. No student in the small squad in front of me would have thought it difficult to imagine what his position would be if I asked him how he would be standing if he made back to back right faces.

The same abstractions that made math so powerful, that allowed us to represent the drill team, and the multiplication of complex numbers with the same symbols was a beauty they could not see. But some will twist their faces and squint against the dimensional curse of being born in three-space, hoping to get a glimpse of a four-space hypercube, one step closer to the Monster. Only 196,879 more dimensions to climb. But on the way home, they start with baby steps. Someone asks, “Can there be a group with just two elements?’ I was going to answer, when one conjectures, “Yeah, I think just one and negative one would be a group under multiplication.” They discuss it for awhile, ignoring the old man driving the car; after all, they learned tonight that “a mathematician seldom produces great work over the age of forty”, and if they don’t know exactly how old I am, they know that my age is much greater than forty; or as they would say, my age = .