 As you may have noticed, the history of graphical computing (nomograms and the like) has become one of the major themes of this blog. I did not foresee this, as I knew virtually nothing about the subject before I started researching my first essays on nomography a couple of years ago. This topic is still one of my main pursuits, and I’m as astonished by what I find now as I was back then. To capture a bit of this spirit, I’ve created a free 2010 calendar titled The Age of Graphical Computing that is available for downloading and printing. The fun thing is that you can test the examples right on the calendar to show that they work!
There are two formats available: two-sided 8-1/2″ x 11″ sheets of paper printed in landscape mode that can be connected at their edge as shown in the photo on the left, and two-sided 11″ x 17″ sheets of paper printed in portrait mode with two pages per side that can be folded as a group and stapled in the middle. Either of these could be printed to fit on A4 or other sizes, I’m sure. White paper can be used, but the color scheme is really designed for a light beige or ivory paper and it looks so much more professional when it’s printed on paper of some color (gray might work). The stapled format requires no other binding. As you can see from the photo on the left, I printed the first (non-stapled) format and took the printed sheets (24 lb. Southworth ivory linen paper from OfficeMax) to a local office shop (Kinko’s FedEx) and had them add clear plastic sheets to the front and back and install a spiral wire (a 60-second job that costs $5). Drilling a hole in the center along the top to hang it completes the calendar. Using 3 rings through punched holes along the top may be a cheaper option.
Continue below to see thumbnail images and the download instructions.
Read the rest of this entry »
 8 Comments »
 [Updated October 19, 2009—see below]
Are you intrigued by nomograms but have no idea how to go about drawing them? PyNomo is an amazing, free software package for drawing precision nomograms. The output is in vector form in a PDF or EPS file, so it can be printed in any size and still retain its sharpness. PyNomo directly supports 9 basic types of nomograms based simply on the format of the equation, so for these types there is no need to convert the equation to the standard nomographic determinant or use geometric relations. But it also supports compound nomograms as well as more complicated equations that have been cast into general determinant form, so it can produce output for any equation that can be plotted as a nomogram.
When I started writing an essay on using PyNomo my plans were to show three examples of nomograms. But I had so much fun making really cool nomograms that the essay turned out to be more of a user’s manual, with examples of all the supported types and descriptions of the many parameters you can use to customize your nomograms. Leif Roschier, the author of the software, spent a great deal of time reviewing draft versions of the essay and making software updates for new features that were rolled into it, so the essay is comprehensive in scope and quite complete in details and practical advice. PyNomo is clearly my choice for drawing nomograms going forward, and I think you will find it as uniquely wonderful as I have.
The essay is too long and the example nomograms too detailed to be rendered in HTML here. The PDF version of the essay can be found here. The PyNomo website, which also contains many examples, is found here.
——————-
Updated October 19, 2009, to Version 1.1 for the new features of PyNomo Release 0.2.2:
- Automatic spacing of tick marks along scales—more tick marks where space is available and less where it’s crowded.
- Drawing of sample isopleths between specified values on scales.
- Printing of only significant digits of scale values by default, producing a cleaner-looking nomogram overall.
5 Comments »
 William Thomson called them “beautiful and ingenious geometrical constructions,” and in variance to their rather humdrum name dygograms are certainly charming to the eye. But these geometric constructions can conveniently generate and then calculate the magnetic deviation of a ship compass at a location.
With our electronic calculators and computers, we take for granted the effortless arithmetic and trigonometric calculations that so vexed our ancestors. Pre-calculated tables for roots and circular functions, generated through hard work, were often used to create tables of magnetic deviations for specific ships and locations. To reduce the chance of misreading these tables, a few types of graphical diagrams, not just dygograms, were invented to provide fast and accurate readings of magnetic deviation. These graphical calculators are the focus of this part of the essay.
Read the rest of this entry »
4 Comments »
 The Scottish mathematician and lawyer Archibald Smith first published in 1843 his equations for the magnetic deviation of a ship, or in other words, the error in the ship’s compasses from permanent and induced magnetic fields in the iron of the ship itself. This effect had been noticed in mostly wooden ships for centuries, and broad attempts to minimize it were implemented. But the advent of ships with iron hulls and steam engines in the early 1800s created a real crisis. A mathematical formulation of the deviation for all compass courses and locations at sea was needed in order to understand and compensate for it, and Smith became the preeminent expert in this sphere of activity. With Capt. Frederick J. Evans he extended his mathematical treatment to detailed procedures for measuring the magnetic parameters for a ship, and he also invented graphical methods for quickly calculating the magnetic deviation for any ship’s course once these parameters were found, constructions called dynamo-gonio-grams (force-angle diagrams), or dygograms for short.
Today, radio navigational systems such as LORAN and GPS, and inertial navigation systems with ring and fiber-optic gyros, gyrocompasses and the like have reduced the use of a ship’s compass to worst-case scenarios. But this triumph of mathematics and physics over the mysteries of magnetic deviation, entered into at a time when magnetic forces were barely understood and set against the backdrop of hundreds of shipwrecks and thousands of lost lives, is an enriching chapter in the history of science. Part I of this essay presents a brief sketch of the problem and the analysis and solutions that were developed to overcome it. Part II sets out with a discussion of Smith’s graphical methods of computing the magnetic deviation and concludes with a list of the references cited in the essay.
Read the rest of this entry »
2 Comments »
You are invited to participate in a new forum established to share ideas and information related to lost art in the mathematical sciences. If you have feedback related to a specific essay or its user comments, please continue to provide comments at the end of the post. Otherwise, for general comments or suggestions for future essays, and in fact for wide-ranging discussions on erstwhile discoveries in mathematics and science, please feel free to post entries on the forum here, specifically on the discussion board for this blog. I will be posting essays here as I always have, and I’m still soliciting guest essays for this blog—the forum is simply a separate but related enterprise that involves more people and opens up more topics.
No Comments »
 In a clear breach of this blog’s charter, I’d like to announce the release of free software I developed for creating convenient, pocket-sized paper organizers. Using LaTeX as a typesetting engine, a high quality PDF file is generated of 16 mini-pages, which is then printed on both sides of a sheet of letter or A4 paper and folded to create a small booklet that can fit in your pocket. The Windows interface directly supports several types of standard pages (List, Text, Calendars, Contacts, etc.) and maintains all user data between sessions. It also provides page types not seen in conventional organizers, such as a Vigenere Cipher page for on-the-go encrypted text and an Astronomy page with a calculated planisphere of current star/planet/moon locations along with other astronomical data. Beyond this, custom user-designed pages can be easily written in LaTeX script and shared in the Plans Unfolding forum and galleries. For more information, please visit the Plans Unfolding home page here. Now back to the subject at hand—thanks for your indulgence.
2 Comments »
 Mental calculators of yesteryear were usually described in magazines, newspapers and books in ways that can be startling in our more cynical age. But even today newspaper articles, documentaries and television features on modern lightning calculators appear almost regularly, often with a “hook” such as diminished capabilities in other areas (the “Einstein” effect). Surely there must be some reports that try to be objective, but I haven’t found them. At best they are naively written by people with little mathematical background; at worst they use considerable license (deception, really, if only by omission) to present a better story. This part of the essay is not directly related to the historical art of mental calculation itself, but I think it serves as a cautionary tale in evaluating articles on it.
Read the rest of this entry »
2 Comments »
 The types of calculations performed by lightning calculators were historically quite limited, notable mainly for the size of the numbers and the speed at which they were manipulated. But remember that the questioner had to verify every calculation by hand, making higher powers and roots (particularly inexact roots) much less feasible. The dawn of calculators and computers propelled some of these tasks into hitherto uncharted territories such as 13th or 23rd roots, deep roots of inexact powers, and so forth, much of it supported by more sophisticated mathematics. Here we will review the methods of calculation used in the past, many of them not commonly known, as well as other techniques that are relatively new.
Read the rest of this entry »
1 Comment »
 Individuals with preternatural abilities to calculate arithmetic results without pen, paper or other instruments, and to do so at astonishing speed, are the stuff of mathematical and psychological lore. These “lightning calculators” were sometimes of limited mental ability, sometimes illiterate but of average intelligence, and sometimes exceptionally bright, this despite the popular notion of the idiot savant. The techniques used by these people are not generally well known. In fact, despite claims by educators that acquiring a mental facility with arithmetic operations is essential to a student’s mathematics education, I see little in the textbooks other than simple estimations based on rounding values, surely the most basic and least interesting mental task. The field of mental calculation may not be a lost art per se, but in this digital age it most certainly is a neglected one.
Part I of this essay attempts to take a fresh look at both historical and modern lightning calculators. Part II describes classic and modern methods of mental calculation. And finally, Part III demonstrates as a cautionary tale the shallow and deceptive nature of most media coverage of lightning calculators, an important consideration in analyzing reports on them.
Read the rest of this entry »
2 Comments »
by Liunian Li 李留念 and Ron Doerfler
Designing a nomogram for an equation containing more than three variables is difficult. The most common nomogram of this sort implements pivot points, requiring the user to create a series of isopleths to arrive at the solution. In this guest essay, Liunian Li describes the ingenious design of a nomogram that requires just a single isopleth to solve a 4-variable equation. For convenience the method is described in both English and Chinese.
Read the rest of this entry »
7 Comments »
In Part III of my essay on The Art of Nomography, I mentioned the use of Weierstrass’ Elliptic Functions to create a nomogram composed of three variable scales overlaid onto a single curve. In particular, Epstein describes using this family of functions to create a nomogram for the equation u + v + w = 0, adding that the formula can be generalized for functions of these variables. This topic generated some interest, and it certainly is interesting to me, so I’ve explored it in more detail by designing a single-curve nomogram based on functions of u, v and w. This essay describes the procedure I followed to create a “fish” nomogram (found here) manifesting the formula for the oxygen consumption of rainbow trout as a function of weight and water temperature—a modest attempt to blend art with artifice.
Read the rest of this entry »
3 Comments »
 I’ve been fascinated by astrolabes for a very long time, roughly 20 years. It was this avocation that led to my interest in sundials and, because they share museum space, my interest in clocks. When I lived in Rockford, Illinois, I would haunt the Time Museum, an institution that produced the most beautiful book on astrolabes. Adler Planetarium in nearby Chicago has one of the best astrolabe collections in the entire world, producing another beautiful book solely on Western astrolabes and a gorgeous book on antique scientific instruments in general. None of these provide the mathematical details of astrolabe design beyond a description of stereographic projection, and indeed this kind of detailed information is rarely found. The Astrolabe, a new book by James E. Morrison, is an absolutely unique and wonderful book on the mathematics needed to create accurate, beautiful designs of astrolabes, quadrants and other related instruments. I can’t recommend it enough to those who share the interests of this blog.
Read the rest of this entry »
6 Comments »
 In addition to providing sophisticated nomograms, the use of determinants as described in the previous Part II offers one other huge advantage. Often the scaling factors of variables have to be manipulated to get a nomogram that uses all the available area and yet stretches portions of the curves that are most in need of accuracy; alternatively, there may be a need to bring distant points (even at infinity) into a compact nomogram. This can be done by morphing the nomogram with any transformation that maps points into points and lines into lines. It is also intriguing to consider the aesthetics of such transformations, creating eye-catching nomograms as an artistic process.
This final part of the essay reviews the types of transformations that can be performed on a nomogram, and it concludes by considering the roles of nomograms in the modern world and providing references for further information.
Read the rest of this entry »
9 Comments »
 The previous Part I of this essay described the construction of straight-line nomograms using simple geometric relationships. Beyond this, a brief knowledge of determinants offers a powerful way of designing very elegant and sophisticated nomograms. A few basics of determinants are presented here that require no previous knowledge of them, and their use in the construction of straight line nomograms is demonstrated. Then we will see how these determinants can be manipulated to create extraordinary nomograms.
Read the rest of this entry »
8 Comments »
 Nomography, truly a forgotten art, is the graphical representation of mathematical relationships or laws (the Greek word for law is nomos). These graphs are variously called nomograms (the term used here), nomographs, alignment charts, and abacs. This area of practical and theoretical mathematics was invented in 1880 by Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision.
Along with the mathematics involved, a great deal of ingenuity went into the design of these nomograms to increase their utility as well as their precision. Many books were written on nomography and then driven out of print with the spread of computers and calculators, and it can be difficult to find these books today even in libraries. Every once in a while a nomogram appears in a modern setting, and it seems odd and strangely old-fashioned—the multi-faceted Smith Chart for transmission line calculations is still sometimes observed in the wild. The theory of nomograms “draws on every aspect of analytic, descriptive, and projective geometries, the several fields of algebra, and other mathematical fields” [Douglass].
This essay is an overview of how nomograms work and how they are constructed from scratch. Part I of this essay is concerned with straight-scale designs, Part II additionally addresses nomograms having one or more curved scales, and Part III describes how nomograms can be transformed into different shapes, the status of nomograms today, and the nomographic references I consulted.
Read the rest of this entry »
20 Comments »
|
Entries (RSS)