How often have you used the word nondimensional in a sentence? There are occasions when I might have used it, had I known that was the word for what I was doing.
For example, when making calculations that involve physical quantities such as mass, length, time, velocity, and the like, it’s not enough to keep track of the numbers or variables as you calculate, you must also keep track of their units. As students know from bitter experience, it’s easy to confuse, or even lose, units along the way. I remember assignments returned with red ink circling my final result, beside which my teacher had posed the simple question: “Units?”
Had I known better, I might have answered that in fact I was employing a technique of mathematical analysis called nondimensionalization in which you deliberately drop the units from equations involving physical quantities. Among its other advantages, this technique can simplify problems. No kidding! It can also, as one MIT professor put it, “eliminate the possibility of reporting nonsense such as the logarithm of a kilogram.”
My own acquaintance with this word has humbler origins. This summer I decided to build steps from our deck to the ground. A total drop of just 32 inches. How hard could that be? Not very. Unless you use nondimensional lumber.
I inadvertently made nondimensionality part of my design the moment I had the bright idea to build the steps out of cedar logs lying around on the property, instead of using conventional, dimensional milled lumber. That one bright idea turned what could have been a weekend project into something more elaborate. For in sad contrast to nondimensional analysis, nondimensional lumber does not make calculations easier.
Practically speaking, for me nondimensional meant harder to measure, harder to calculate, harder to work with, harder to build. The roundness of the logs, their knottiness, and their continuously changing dimensions helped me appreciate why carpenters call unmilled wood nondimensional lumber. And why that’s not necessarily a compliment. Unlike your milled 2×8, who is a straight, reliable and measured fellow, your natural log is an irresolute, shifty, and dimensionless type who feels no compunction about his rough wavering character, nor gives a damn that his diameter is changing continuously from one end to the other.
I first heard of this exotic character—the more exotic for describing something as plain as lumber—the day my neighbor friend dropped off the cedar logs he’d split lengthwise down the middle for me. (He owns a sawmill.) Three of the four halves we unloaded from his truck eventually became my 12-foot-long steps, or treads, as they’re called in the trade.
But only after I’d done my dimensional analysis featuring nondimensional lumber, followed by headscratching and more than a little guesswork. Then I knew, roughly, how to size, cut and place the two stringers (the term of art for the logs that support the treads from below) that run from the underside of the deck down to the ground, where they’re fastened to footings set in concrete. After that, there was just the headache of measuring, notching, fitting, then re-measuring, re-notching, and re-fitting, before, finally, leveling and fastening the round (continuously changing) underside of the treads to the round (continuously changing) stringers, so that each step rises exactly 8 inches.
My work wasn’t finished until nearly a month after that day when my friend, a builder by trade, casually dropped nondimensional into the conversation. We’d hauled the treads from his truck to the worksite, and I was asking him what he thought of my design, and what advice he had for me. He looked at me from under his cap, a mischievous smile playing on his lips, and said, “Oh, so you’re gonna use nondimensional lumber. That will make things more interesting, but I’m sure they’ll turn out great. Just keep measuring and figuring, measuring and figuring.”
A tip of the hat to my father-in-law, who helped me get the treads on the stringers, and to my father, who helped me measure, mark and stake out the plan.