I have moved my Dead Reckonings blog from its original home to this site to improve performance and bring a new look to the blog. I hope you like the new style. I tried out some newer typefaces but reverted to the old Georgia because I simply love the look and readability of numbers in this font, and there are a lot of numbers in this blog. I will be publishing new content soon, so please stay tuned.
Ron
I am happy to announce Sliver, a free software application I wrote over the last two years for multivariate data visualization. Sliver includes parallel coordinate (PC) plots, PC plot matrices, various types of configurable 2D and 3D scatterplots, and plots overlaid on Google Earth, all fully linked by color brushing. Transparency (alpha blending) is supported as seen at left. Sliver also offers data animation, including Google Earth animation as well as the Grand Tour, a rotation through n-space that reveals correlations and structures in multidimensional data. Have a look!
This modern application may seem at odds with the theme of my blog, but in fact it was Maurice d’Ocagne who coined the term parallel coordinates in regard to his parallel scale nomograms. A hundred years later Alfred Inselberg extended this idea of parallel scales as a way of visually analyzing multi-dimensional data. Inselberg and others make use of some of d’Ocagne’s work on point-line duality to characterize functional relationships between variables as structures and envelopes of the lines drawn between their axes.
The final completion of my 2013 calendar, Graphical Astronomy, has been delayed, so at this point I am going to update the dates for 2014 and post it this fall. As partial compensation, I’ve created a Valentine’s Day card for mathematically-inclined people that can be downloaded, printed and folded. It is appropriate whether the person giving the card or receiving it is interested in math, or both, and in fact it’s not Valentine’s Day specific so it can be used for birthdays or anytime at all.
Last month Joe Marasco, Leif Roschier and I published an article on Bayes’ Theorem in The UMAP Journal that included a foldout of large circular nomograms for calculating the results from it. The article, Doc, What Are My Chances?, can be freely downloaded from the Modern Nomograms webpage, which also offers commercial posters of the two nomograms used to calculate Bayes’ Theorem (one for common cases and one optimized for calculating rare cases).
I want to announce that my fellow collaborators in nomography, Joe Marasco and Leif Roschier, and I have a new website called Modern Nomograms to offer posters of new nomograms that we hope will interest people. Our initial posters are nomograms for calculating results from Bayes’ Theorem as described in the next post here, but we expect more will follow.
This project does not in any way affect the content here—essays will continue to be written as usual on lost arts in the mathematical sciences, including nomography. This is simply an outlet to provide an option for nomograms in poster form.
by Ron Doerfler and Miles Forster
Part I of this essay provided background information that demonstrates the difficulty of the problem of mental extraction of 13th roots and the efforts of calculators to master it. But can it be possible for us to extract 13th roots of 100-digit numbers without devoting portions of our life to it? With a basic talent in mental arithmetic and some study, it can certainly be done, even if not in record time. We present a new method that involves no logarithms, no antilogarithms and no factoring, one that works with 13th powers that end in 1, 3, 7 or 9 (the cases attempted by record holders). The memorization consists of one table and a few formulas. A printer-friendly PDF version of Parts I and II is linked at the end of this essay.
by Ron Doerfler and Miles Forster
Mental calculators of note (so-called “lightning calculators”) developed areas of expertise in performing calculations that seem astonishing, even unbelievable, to the rest of us. One such specialty is calculating the 8-digit root of a 13th power of 100 digits. Achieving record times historically required massive memorization and calculating speed, racing through a procedure that remains a mystery to most people. Part I of this essay provides a historical overview of the extraction of 13th roots, including the methods used by a few mental calculators, methods that largely rely on a mix of intensive mental calculation and large-scale rote memorization. It demonstrates the creativity and drive of these marvelous people.
In Part II of this essay we will propose a new method for 13th roots like those posed to lightning calculators that is relatively easy to learn, one that makes this feat feasible for those of us with basic mental math capabilities and a desire to do something amazing. A printer-friendly PDF version of Parts I and II is linked at the end of this essay.
Those of us who enjoy our expeditions through the lost world of nomography quickly discover that many original sources that are still considered masterpieces in the field have never been translated into English. In my case this is a serious handicap, and I find myself struggling through the texts trying to understand the rationale behind the beautiful nomograms I see on the printed page. Nomography and its predecessors were invented in France and only later spread to such places as Germany, the U.S, Britain, Russia and Poland. Today it seems to me that most research appears in Czech journals. In addition to the language difficulties, many of the original sources are in obscure, hard-to-find journals.
In 1982 H.A. Evesham produced his doctoral thesis, a review of the important discoveries in nomography. It is often cited in other works, but unless you know someone who knows someone, it is very difficult to find—I have never been able to locate a copy. However, just recently Mr. Evesham’s thesis has been professionally typeset and released as a book (click here or here) by Docent Press under the aegis of Scott Guthery. If you have an interest in the theoretical aspects of nomography beyond the basic construction techniques of most books and of my earlier essays, you will appreciate this book as much as I do. Mr. Evesham does a wonderful job of weaving mathematical discoveries in nomography from many contributors into a readable but scholarly work.
I have researched and written about methods of mental calculation over the years, and I’m often surprised at the ingenuity evident in the mathematical methods developed specifically for it. Based on my Lightning Calculator series of essays here, I’ve created a new 2011 calendar titled Lightning Calculation. It’s a unique, interactive calendar for developing abilities in mental calculation. You can download a free PDF file for printing on your computer, or if you prefer, order it for delivery through Lulu.com. I also think it might be a nice thing to make as a Christmas gift for someone interested in this sort of thing, or for displaying in a math classroom.
Part I of this essay described and analyzed Charles Lallemand’s L’Abaque Triomphe, the first published example of a graphical computer called a hexagonal chart. Here we discuss the mathematical principles behind hexagonal charts and provide examples of these charts from the literature of the time. The related development of triangular coordinate systems is also covered in this second part. Trilinear diagrams, a simpler offshoot of triangular coordinate systems, are still seen today in fields such as geology, physical chemistry and metallurgy (as shown to the left). A list of references is provided as well. A printer-friendly Word/PDF version with more detailed images is linked at the end, with a hexagonal overlay in the appendix that can be used to exercise these charts.
Dead Reckonings 