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Guilloche Patterns

Ptolemy Lives!

One of the reasons I like old stock certificates so much -- and currency banknotes too, for that matter -- is the intricate pattern of lines that forms the borders. These delicate, lacy interwoven lines are a real visual trip, and there are virtually an endless variety of engraving patterns seen from one certificate to the next.

I learned that these patterns were made using something called a "geometric lathe". This machine was invented in the 19th century -- back when people still knew how to design and build cool mechanical devices! The contraption pulled an engraving tool over a steel printing plate using a series of interacting cam wheels whose individual settings created a wonderfully complex sinusoidal pattern. If you ever used or saw a kid's plastic Spirographtm toy, I imagine it worked something like that: Wheels turning within wheels.  Look at this example of the border lathework on an old Studebaker-Worthington stock certificate:

Now, I knew that a point on the edge of one small wheel revolving around the perimeter of a larger circle describes a figure called an epicycloid. I have read where some of the geometric lathes using for banknote printing used as many as 10 separate cam wheels, each with their own settings, and each influencing the resulting movement all the other wheels -- and, ultimately, of the engraving tool itself. Figures made from multiple wheels spinning around each other in this fashion are known as Guilloche patterns.

I played around with creating Guilloche patterns using VisualBasictm. The heart of the code was the following pair of functions for describing the point of a drawing pen, as the wheels - 4 of them in my version -- made their simultaneous turn around a larger "hidden" circle with radius=rada. Each of the smaller "cam" circles had their own unique radii -- radb, radc, radd, and rade. The coefficients b, c, d, e served to further "size" the wheels. The angle, given as "i", is finely incremented around the main circle from 0 to 6.28 (radians, which is the normal angular unit for Basic code).

x = xorg + (rada + radb) * Cos(i) + b * Cos(((rada + radb) / radb) * i) + c * Cos(((rada + radc) / radc) * i) + d * Cos(((rada + radd) / radd) * i) + e * Cos(((rada + rade) / rade) * i)

y = yorg - (rada + radb) * Sin(i) - b * Sin(((rada + radb) / radb) * i) - c * Sin(((rada + radc) / radc) * i) - d * Sin(((rada + radd) / radd) * i) - e * Sin(((rada + rade) / rade) * i)

I made my code so I could adjust the cam wheel radii and coefficients using slider bars -- in my mind's eye I imagined those 19th century engraving machine operators turning big metal cranks to do the same thing. I also added the capability of repeating the drawing cycles a desired number of times, with all the radii (including the main circle radius) shrinking at each pass.

Here are a few typical program outputs, showing a first image with single pass around the central "hidden" circle, and an image adjacent to it showing multiple passes around the circle with shrinking radii on each pass. Looking at the results, I reckon you might could also call this a "doily making" programů

Postscript:  Ptolemy (aka Claudius Ptolemaeus) was a Greek who lived in Alexandria, Egypt from approx. 87 -150 AD. He developed a system of epicycles -- small circles revolving around larger circles -- to account for the apparent planetary motions (particularly retrograde motion) in an earth-centered model of the solar system. Ptolemy's system used at least 80 epicycles to explain the motions of the Sun, the Moon, and the five planets known in his time. It remained the accepted wisdom until Copernicus and Kepler finally developed the heliocentric model.

Ptolemy's system was actually more accurate than Copernicus' original 1543 version, and his model continued to give the best results until Kepler later introduced the (proper) notion of elliptical planetary paths in 1609.  Hmmm, that's a theory that stood up well for almost 1500 years.  Short of Aristotle, I don't know of many other human beings that can boast of that accomplishment.  I'll wager not many modern scientific theories will remain that intact by the year 3500.  (It'd be nice to be around to collect the bets...)

 Back to Recreational Computing... 

GIF on this page from MathWorld -- A Wolfram Web Resource, on Eric W. Weisstein contribution page "Hypotrochoid".


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