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.
Archive for the “mathematics” Category
Bayes’ Theorem is a statistical technique that calculates a final posttest probability based on an initial pretest probability and the results of a test of a given discriminating power. Thomas Bayes (1701-1761) first suggested this method, and Pierre-Simon Laplace published it in its modern form in 1812. It has generated quite a bit of controversy from frequentists (who work from a null hypothesis rather than an initial posited probability), but this technique has become much more popular in modern times. Among many other applications, this is a common technique used in evidence-based medicine, in which statistical methods are used to analyze the results of diagnostic tests. For example, a diagnostic test might have a sensitivity of 98%, or in other words, the test will return a positive result 98% of the time for a person having that disease. It might have a specificity of 95%, which means it will return a negative result 95% of the time for a person who does not have the disease. For a disease that has a prevalence (a pretest probability) of 1% in the general population, Bayes’ Theorem provides, say, the probability of a person having the disease with a positive test result. Pretty darn likely, right? Well, it turns out it’s about 16% because the false positives from the 99% who do not have the disease overwhelm the true positives from the 1% who have it. This is the basis for recent recommendations to stop PSA screening in men, for example, as expensive and counterproductive. But all this is described in detail in the article. Enjoy! [Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]
Mar
11
2012
![]() ![]() New Modern Nomograms WebsitePosted by: Ron D. in administrative, mathematics, off-topic
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. [Please visit the new home for Dead Reckonings: http://www.deadreckonings.com] by Ron Doerfler and Miles Forster
by Ron Doerfler and Miles Forster
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. As always, a printer-friendly PDF version of Parts I and II is linked at the end of this essay.
Feb
12
2011
![]() ![]() Book Review: The History and Development of Nomography, by H.A. EveshamPosted by: Ron D. in mathematics 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.
NOTE: If you choose to have the calendar professionally printed by Lulu.com rather than on your own printer, through December 31 you can enter the code REMARKABLEYEAR305 at checkout for 20% off the price of $17.60. The Lulu site is also posting daily coupon codes in December here.
May
05
2010
![]() ![]() Lallemand’s L’Abaque Triomphe, Hexagonal Charts, and Triangular Coordinate Systems (Part II)Posted by: Ron D. in mathematics
May
05
2010
![]() ![]() Lallemand’s L’Abaque Triomphe, Hexagonal Charts, and Triangular Coordinate Systems (Part I)Posted by: Ron D. in mathematics
The article, a significantly revised version of my blog essay, has now been published in the UMAP Journal, and per the standard agreement I can post the PDF of the article here for anyone to download. More information and a link to the article are below.
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. 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 (Version 1.1) can be found here. The PyNomo website, which also contains many examples, is found here. [NEW – December 21, 2011]: Detailed instructions on downloading and installing all required software applications onto a Windows XP or Windows 7 PC can be found here. ——————- Updated October 19, 2009, to Version 1.1 for the new features of PyNomo Release 0.2.2:
[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]
Apr
18
2009
![]() ![]() Magnetic Deviation: Comprehension, Compensation and Computation (Part II)Posted by: Ron D. in mathematics, physics
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.
Apr
18
2009
![]() ![]() Magnetic Deviation: Comprehension, Compensation and Computation (Part I)Posted by: Ron D. in mathematics, physics
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.
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.
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. 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.
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.
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.
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