Thanks for your comment, Chris! I really knew very little about nomograms when I started gathering references, and I was stunned by what I found. Since I’ve accumulated a lot of articles and books on nomography, I’m now planning on writing a blog essay that will just be a showcase of scans of the coolest nomograms I’ve run across. There are many that are a lot more inspirational than the simpler examples in my essays, and it would be a fun and easy article to write after I finish the one I’m working on now.

Nomography is an actively growing topic on the web. Sites on nomograms that have recently been updated include William Chung’s site, Leif Roschier’s Pynomo site with free software to generate nomograms (see the “Basics” and “Examples” links or the links under the Software Documentation area to see beautiful examples), and Eric Sumner’s newly-launched site. And there is the relatively young nomography forum as well for ongoing discussions. — Ron

It’s good to hear people are still designing these graphical calculators. I’m incorporating nomograms into some of my page designs for my Plans Unfolding paper organizer, such the one here (click on the image to see a high-res version).

I’ve truncated the 480 iterations of NOM here. I hadn’t even heard of this phenomenon before a few days ago—must be out of touch. 8^) — Ron

The Smith Chart at the top of the article is a nomogram I’ve used extensively. What’s absolutely fascinating about it is that the circular patterns are a particular type of transform called a conformal map, specifically the Smith Chart is a Moebius transform, one of the most fundamental conformal maps known in nature. Specifically: Gamma = (Z-Zo)/(Z+Zo) where all variables are complex. A perhaps more familiar conformal map is Z^2+C which is the generating function for a Mandelbrot/Julia set.

The Smith Chart specifically is a transform between the two major model spaces used by EEs: the space of lumped equivalent devices which includes resistors, capacitors and inductors and the space of distributed transmission lines which describes the realm of RF and Microwave circuits and systems. Every point on the interior curvy space represents a complex lumped impedance Z = R+jX, while the circumferential scales represents the equivalent phase delay on a transmission line, and the scalings on the bottom represent various equivalent forms of reflected power ratios.

Everything in the Smith Chart can be done trivially using complex algebra and a handful of equations but even though all that can be done with a slide rule even, the Smith Chart nomogram is faster and offers an intuitive model for what’s happening physically. Today’s microwave instruments still optionally display impedances measured using a Smith Chart graticule drawn on the screen - the nomographic scales are omitted however leaving only the Moebius transform itself. You can determine all sorts of stuff from such data displayed on a Smith Chart up to things like stability and match of a nonlinear device like a transistor amplifer simply by plottings its input impedance on a Smith Chart parametrically over frequency. (The pdf link describes all Smith charts and transmission lines more practically, the jpg is one of Agilent latest mm-wave analyzers - the photo sucks but the circular things on screen are Smith Charts - BTW I don’t work for Agilent but I used to work for HP and I still use Agilent products but also other companies’ also)

http://cp.literature.agilent.com/litweb/pdf/5965-7917E.pdf
http://cp.home.agilent.com/upload/cmc_upload/N5250A_large.jpg

I also collect old text books but of old or vintage electrical engineering (generally 1950s or earlier with favorites being 1910-1950 for vacuum tubes and prior to 1920 for crazy stuff like arc converters, spark gaps and such). These are full of nomograms as well, though none as elegant as the Smith Chart generally.

BTW the formality of modern EE (with standardization circuit analysis techniques, etc.) occurred in the mid-1930s. I know because I have a copy of pretty much every EE text book written in English from the first half of the 20th century. The prime mover was none of than Dr. Fred Terman who authored the first EE text book in 1933 which codified it all as we see it in today’s EE textbooks. Fred Terman was Bill Hewlett and Dave Packard’s mentor/advisor at Stanford and inventor of the Silicon Valley entrepreneurial business model though for reasons most people have no idea about (see YouTube Google talk link - very interesting). The back entrance to Agilent, nee HP T&M, corporate headquarters off Lawrence Expressway south in Santa Clara is called Terman Lane - named after the same.

http://en.wikipedia.org/wiki/Frederick_Terman
http://www.youtube.com/watch?v=hFSPHfZQpIQ

What folks today don’t seem to understand about those days is that today we have a false confidence in precision. Most of that resolution computers deliver doesn’t actually buy you as much as some today think it does. Most electrical measuring instruments, for example, are still limited by the total accuracy and resolution defined by the accuracy components like resistors which have only crept up to 3.5-4 digits even today from the 2-3 digits in the days of slide rules. You can be tricked by the instrument specifications if you look at them superficially.

There is a semiconductor analyzer family I use that operates from “20 fA to 2A” on the datasheet (”14 orders of magnitude! Wow! Gee, look progress! We’re good!”). But if you look closer it’s covering this range best-case by only 3.5 digits at a time by using automatic ranging. Like an automatic transmission in a car extends the total speed from 0 to 200 kph using only a narrow range of RPM values. So no individual measurement is actually better than 3.5 digits and more often is less (3.5 digits is at “long integration” which most folks find impossibly slow taking ~1 second per point). It’s possible to use timing accuracy, bandwidth control and feedback to go beyond 4 digits in one shot for a few, very specific measurements but not generally for any arbitrary type of measurement. Most stuff ranges narrowly like the above example.

Back in the days of nomograms and slide rules, the economies of calculations were such that you needed to be more aware of these limitations because it could mean the difference between hours and days of calculation time. The 2-3 digits of resolution you get from nomograms and slide rules was/is often enough to get the job done since that was what the accuracy of measurement was also. Exceeding the resolution beyond the accuracy *excessively* is simply garbage-in-garbage-out when it comes to end result accuracy but when computation gets cheap it’s too easy to ignore that. People today often think “the answer” really is all the digits in an IEEE 754 floating point number!! More often than not it should only be 2-4 digits based on the calculations and underlying measurement data.

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Wow, thank you for your comments, Jes–it was really fascinating reading. Although I’m a bit older than you, my fields were math and physics so I had seen the Smith Chart but virtually never used it. Another reader of this essay emailed me awhile back about a website collection of Smith Chart resources at http://www.sss-mag.com/smith.html. The presentation by Stephen D. Stearns (”Mysteries of the Smith Chart” at http://www.fars.k6ya.org/docs/smith_chart.pdf) was particularly recommended by him, and I think it’s just great.

So many thanks for taking time out for creating the site: I suggest the www-hierarchy should introduce “Hilaire Beloc Awards” - I nominate you for a gold

Thanks, Graham! When I started looking for information on nomography last December I was also surprised at how little information about the subject was available on the web, while there are certainly many websites on slide rules and the history of computing. I’m really gratified by responses on this topic that I’ve received by people such as yourself.

I read the Belloc quote ten years ago in a footnote of a sundial article by Fred Sawyer and I wrote it down because I liked it so much. Fred is now the President of the North American Sundial Society, and recently while I was looking up something on sundials for an essay I saw that he presents that same quote in the “About NASS” page of the NASS website, so I hope I’m not appropriating something out of turn. In any event, if there were such an award I’d nominate Fred for the incredible work he’s done over the years in preserving and extending the aesthetics and mathematics of sundials. — Ron

One thing I noticed was one of synonyms for nomograms, abacs. I am an occasional collector of French textbooks on groundwater. More precisely, I collect groundwater textbooks, and some of my favorites are in French. It has always puzzled me why design charts in these textbooks are referred to as “abaques”. Now I know why.

Thank you for your wonderful website.

Your welcome! I appreciate your thoughtful comments. — Ron