I've finally gotten the noisy paint cans out of my car, entrusting them to the "Household Hazardous Waste" collectors of Fairfax County, so now I can turn my attention back to calculus. I am going very slowly, given that I only have enough time to learn perhaps one page's worth before I have to get back to art. Currently I'm finishing up the chapter on rules for derivatives. The rule for the product of two derivatives falls into a pattern which is known to classical scholars as chiasmus or chiastic, which in ordinary English means "crisscross." In the Greek alphabet, the letter X is called "chi," which is where those descriptive words come from. The idea is that a group of objects forms an X-shape, whether visually or in meaning. The simplest example of chiastic pattern I can think of would be something like this: "pizza, soda, soda, pizza." Or, in another possibility, "pizza, soda, pizza, soda." The two words are arranged in either a mirror or a sequential symmetry. Poets in Greek and Latin, and later in other languages including English, adopted this pattern in their verse.

Believe it or not, there is an entire website devoted to chiasm in words, proving, as if proof were needed, that whatever exists in the human world, no matter how esoteric or trivial, now has a website. But when I encountered the proof of the derivative product rule, I found it was chiastic. The definition of the product rule, without using math symbols, goes like this (quoted from the book):

"The derivative of a product of two functions equals the first function times the derivative of the second plus the second function times the derivative of the first."

In a more graphic interpretation, it could be phrased like this. "Derivative of (pizza x soda) = (pizza x "derivative of soda") + ("derivative of pizza" x soda)." This reminds me of my old high school math teacher Dr. Weinert, whose classes I ignored and nearly failed. One of the few things I remember is when Weinert, clad in his grubby chalk protector smock coat, would put his fist in front of an algebraic expression such as (a + b) and call whatever was in between the parentheses "Fist." Then he would do the same to another expression in parentheses and call it "Bushwah." So then he would multiply "Fist" times "Bushwah," and out would come something which would be beyond my comprehension until 2002, thirty-four years later.

When I got to quadratic equations and multiplying polynomials, I finally learned that you multiply them in a crisscross way. Mathematically, that's (a + b) x (c + d) = ac + bc + ad + bd, interweaving the components. It's a form of chiasmus. So now that I have managed to get to calculus, here it is again. Currently I have no idea what derivatives or limits actually do in what passes for the "real world." It's all abstract patterns. Mathematicians love them and don't even mind if it doesn't apply to anything in the world. But there is another application of "chiastic" forms, in religious literature, as not only a rhetorical device but as a signal of spiritual meaning. This religious site, from an Evangelical Protestant perspective, discusses chiastic structure in the Bible. This pattern existed in the Jewish scriptures long before Christianity, as this short article in Wikipedia attests. But the X form is particularly attractive to Christian writers, because X, the Greek Chi, is also the first letter of the Greek word for "Christos," and has been used since the beginning of the faith as a Christian symbol.

Posted at 4:13 am | link