This is one of those postings that I expect about half of my blog friends to skip over - those who have no reason to know what a partial differential equation is or why they would *want *to know. I understand, and think none the less of those who feel that way. As a freshman in college, I was among that group. However, the next year, as a physics major, I was required to start thinking about thermodynamics. I would bet even money that you have heard of the First Law of Thermodynamics and can even tell me that it has to do with conservation of energy. (See how smart we are?)

A 2011 comic take on the game version of the laws by American chemical engineer Rich Byrnes

Byrnes, Rich. (2011). “Boil’s Laws: the Laws of Thermodynamics and Casinos”, *Funnybone*, ChEnected, Apr. 08

By my second semester junior year, I was required to take a course in thermodynamics. Although I don't recall the exact title of the book we used in Introduction to Thermodynamics, I recall that it was a thin book with a grey cover. As I recall (but, I wouldn't bet too much money on this, it was in 1958, after all) I believe that little book was by Sears (*not* & Roebuck!) That was a difficult course for me. As it turned out, the course and book were super-saturated with partial differential equations, a course that, at that time, I had not gotten around to attempting. Somehow, my math always seemed to be behind my physics coursework. I needed matrix algebras long before I got to a math course that included them.

One of the more interesting things about my taking thermo was that Elder Brother and Hunky Husband were *also* taking thermodynamics - and - we were all living together (along with EB's late wife). Funny thing was, the three of us could not compare notes or even converse cogently about our respective classes. EB took thermo in the chemical engineering department and HH took thermo in either the mechanics department or the mechanical engineering department. HH wanted to talk steam tables, I wanted to talk partial differential equations (PDE), and who recalls what a chemical engineer wanted to talk about? I do recall the two guys playing catch (thereby destroying) my clay model that I had built of the pressure/volume/temperature (better known as a PVT) diagram for water. Grr!

Cop Car's Roomies - Elder Brother, Hunky Husband, Expert Seamstress - Spring 1958

Fast forward to 1981 at which time I interviewed for a job at Tyndall AFB, Florida, working on USAF's HAVE BOUNCE program - part of the *Rapid Runway Repair program *to which *BDM* was contracted. (This was actually my second interview - BDM had first asked me to come to McLean VA, to their corporate headquarters.) I recall someone's bragging to me that the USAF computer being used for aircraft simulations would do xxxx fast Fourier transforms (FFT) per second. "What's a fast Fourier transform", you may ask. Well, it is a faster way of computing the same answers as one gets using the old, standard Fourier transform - which is now called a discrete Fourier transform (DFT). Did I hear you mutter, "Ask a silly question...."? Either FFT or DFT should give the same answer in solving (drum roll) differential equations, including PDEs!

*Fourier Transform (FT)*

Why did HAVE BOUNCE need to use FFTs at all? Because the HAVE BOUNCE program involved testing of various aircraft (F-4, C-130, C-141, C-5, B-1, A-10, Boeing 747, Douglas DC-10 are the ones which I recall) to measure their response to operations over a surface that was shaped as a 1-cos wave of known maximum amplitude. The measurements were made in the time domain, that is, one would plot out the data along a time axis. We needed to know the response as a function of frequency - plots along a frequency axis.

BTW: In case I didn't mention it, FFT solutions/second was/is used in comparing speeds of computers in practical terms.

Why did I bring up the whole subject? Because, I visited the *Improbable Research* website this morning only to find the following posting.

September 19th, 2018

You may be wondering how aesthetically pleasing is your country’s diffraction pattern. This new physics study proves that Albert F. Rigosi shares your mental hobby:

“Analysis of Fraunhofer Diffraction Patterns’ Entropy Based on Apertures Shaped as National Borders,” Albert F. Rigosi, *Optik*, vol. 172, November 2018, pp. 1019-1025. *(Thanks to John Ng for bringing this to our attention.)* The author, at Columbia University and the National Institute of Standards and Technology, reports:

“How aesthetically pleasing is your country’s diffraction pattern? This work summarizes the calculated and experimental Fraunhofer diffraction patterns obtained from using apertures lithographically formed into shapes of national borders. Calculations are made based on the fast Fourier transform of the aperture images. The entropy of a diffraction pattern image, based on its two-dimensional gradient, for each of 113 nations has also been computed. Results suggest that most nations’ diffraction patterns fall under one of two prominent trends forming as a function of geographical area, with one trend being less entropic than the other.”

The top images here shows shows a diagram of the experimental setup. The bottom collection of images show: “Three example nations. (a) The aperture for the continental USA is depicted. (b) is the FFT calculation of the aperture above, and the corresponding experimental data is shown below in (c). (d) The aperture for Egypt is depicted, along with its FFT and experimental data in (e) and(f), respectively. (g) The aperture for Papua New Guinea is shown with its (h) calculated FFT and (i) experimental diﬀraction data.”

Additional everything can be found in an appendix.

posted by **Marc Abrahams** in Arts and Science, Research News | Comments Off on How Aesthetically Pleasing Is Your Country’s Diffraction Pattern?

I have zero interest in fractal patterns of the outlines of various countries. They have no need to be pleasing as far as I am concerned; but, perhaps you can understand how it sent my mind off in a mathematical direction?

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