27 December 2007

A postal nomogram

Nomograms are collections of curves drawn on a sheet of paper that are used to do a specific calculation. See, for example, this Google Image Search. A typical example calculates some real-valued function of two variables by having three scales. Two of these scales, usually, are straight lines (although they don't have to be); these lines are marked with values of the two inputs to the function. To compute a value of the function f(a,b), you find the point marked "a" on the first scale, and the point marked "b" on the second scale; then you draw a straight line between those two points, and the marking of the point where that intersects the third scale is the value of f(a,b).

The simplest possible example would probably be the nomogram that computes the function f(a,b) = ab/(a+b); to obtain this, the "input scales" are just the x and y axes marked off as usual, and the "output scale" is the line y=x, with the marking z at the point (z,z). The line from (a, 0) to (0, b) intersects the line y = x at (ab/(a+b), ab/(a+b)). This is known as the "parallel resistance nomogram", since the resistance of an a-ohm resistor and a b-ohm resistor in parallel is ab/(a+b). More complicated computations can be done with more complicated curved scales.

You might think that these don't exist any more, since isn't it possible to compute things to any desired accuracy with modern computers? But today I saw an example of something similar at a United States Post Office. Apparently the Post Office has regulations saying that packages with volume under one cubic foot are to be treated differently than those with volumes over one cubic foot. You could ask the people behind the counter to do the multiplication, but that might be asking for trouble. So the postal service has made a mat which has the curves xy = 1728/h plotted on it, for various values of h; the curves are labelled with h. (So, for example, the curve passing through (16, 12) is marked 9, since a 16-by-12-by-9 package is one cubic foot, or 1728 cubic inches.) The postal employee can place a package on the mat, with one corner at (0, 0); the opposite corner will be at (l, w) where l and w are the length and width of the package. Then the curve which passes through the opposite corner is the maximum height allowed for the package. I wish I could find a picture of it online, but I can't. This isn't exactly a nomogram (at least under the definition I've seen), but it's very much the same principle. And it's quite interesting to see one "in the wild".


Efrique said...

I'm a big fan of nomograms and other calculation-graphics.

The post-office device sounds like a very handy idea.

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