GoKawiilWhat is a Nomogram and Why Would It Interest Me?¶
A nomogram or nomograph is a diagram that provides an easy, graphical way of calculating the result of a mathematical formula. Sometimes also called an alignment chart, a nomogram consists of a set of numbered scales, usually one for each variable in the formula, arranged so that a straightedge can be placed across known values to find the unknown value that solves the formula. Since an equation in two variables is usually represented by a graph, most nomograms represent formulas that involve three or more variables.
These graphical calculators were invented in 1880 by Philbert Maurice d’Ocagne and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Electronic calculators and computers have made nomograms much less common today, but when a fast, handy calculator of a particular formula is needed they can be very useful. The cost to produce one is a sheet of paper, and they are fun to design, easy to use, and can be beautiful designs that engage people.
For example, here’s a nomogram from 1920 that relates the variables \(l\), \(s\), \(r\) and \(\alpha\) for a slider-crank mechanism:
The equation that this solves is quite complicated:
\(s = r(1 - \cos \alpha) + l(1 - (1 - \lambda^2 \sin^2 \alpha)^{1/2})\) where \(\lambda = r/l\)
There is a sample isopleth line on the nomogram that solves the equation for one set of values, scaled by \(r\). For a value \(\lambda = r/l = 0.35\) and an angle \(\alpha = 75^{\circ}\), we find that \(s/2r \approx 0.455\), where we read off the same sides of the \(s/2r\) and \(\alpha\) scales. Note that in practice this nomogram would be drawn by a draftsman to a much larger scale for greater precision.
Try it out yourself! Pick a radius \(r\), a length \(l \geq 2r\) and an angle \(\alpha\), and find \(s\) on your calculator. Imagine an engineer solving this by hand for various parameters before calculators were invented. Then solve it on the nomogram here with a straightedge and compare your answers. When you’re finished, choose values of \(r\), \(l\) and \(s\) and solve for \(\alpha\). You’ll realize that a nomogram can solve even for implicit variables that cannot be isolated on one side of the equation!
How in the world was this nomogram designed? Somehow this layout of scales solves the equation for every combination of its values using just a straightedge. For the nine most common functional relationships, PyNomo generates vector-image nomograms in PDF form using simple but customizable scripts in which you provide the functions of the variables. Beyond this, experienced designers can use a tenth PyNomo option to draw nomograms with arbitrarily complicated layouts such as this one, and even linear and circular slide rules.
Designing nomograms is an enjoyable pursuit, much more so than in the past since PyNomo can provide the expert knowledge and also serve as the technical draftsman. And as described below, nomograms are very useful for a variety of applications even today.