Graphing to Guilloché, Functions to Fabergé (yes… as in eggs):

An Examination of Transformational Coefficients in Curve Sketching

Elliott Ostler

Introduction
American secondary mathematics education has been accused of it for years; breadth without depth, and lack of connections to anything meaningful.  Certainly the problem isn’t that mathematics doesn’t naturally connect to other things, but rather that we’ve not put great effort into teaching about those meaningful connections.  Confounding this problem is that some of the most impressive applications are not necessarily supported by traditionally “important” curriculum topics (those that allow students to score well on standardized tests), so they are often considered “enrichment” and put off until such time as when everything else has been adequately covered.  The silver lining in this cloud is that meaningful (and digestible) applications for traditionally important topics in secondary level mathematics are actually plentiful for anyone who cares to do a bit of research… so, here is a bit of research.  Additionally, since the evolution of these mathematical applications is often as interesting as the applications themselves, it is perhaps appropriate to share some of that as well, particularly when it provides insight into unique uses of a concept.

For now, we will leave specific applications unstated, and instead begin with a simple curve sketching example using straight lines, a concept which will become increasingly more relevant as we progress.  Equations of lines and their respective graphs are commonly seen in a variety of applications from unit conversions to inferential statistics, and although these applications are most certainly important, the frequency of their use, both in classrooms and otherwise, makes them appear somewhat contrived and generally not very interesting.  There are, however some interesting applications in the graphing of lines and other functions that are not so well known.  We will begin by examining some numeric patterns which translate to geometric patterns when they are graphed.

Anyone who has ever investigated straight line geometry knows that curves can be created with a series of straight lines.  These lines can be considered tangents to an actual curve at given points if you wish, but primarily we just want to look closely at the transformational coefficients, those values that provide information about the location, shape, and direction of curves.  In this example, we will limit our graph to points only in the first quadrant. To create the geometric pattern we will predictably adjust the slope variables up and down in a series of lines, and simultaneously decrease the y-axis intercepts.  The result looks like a curve.  Consider the following equations:

Lines:                           Pattern Graph:

The Evolution of Practical Connections with a Rich History
In the last example (which can be easily shown with a spreadsheet graphing option or graphing calculator), predictable manipulation of the slope coefficient and the y-axis intercept constant creates a neat overlapping spider-web pattern.  These patterns are fun to look at maybe, but not really applicable to much. So what is this all leading up to?  Well, oddly enough this kind of graphing is a first stage in some very complex and useful functions that produce what are known as Guilloché patterns.  The word Guilloché (pronounced by some as Ga-Lowsh, and by others as Gee-o-shay) is actually a French word for a painted or carved kind of ornament.  The patterns usually consist of a series of circles that are intricately overlapping and woven together to create some rather unique “spirograph” type patterns.  These overlapping curved patterns can be observed in Greek, Assyrian, Roman, French, and English architecture and art.  The more advanced and contemporary definitions and applications of Guilloché patterns however, are a little more subtle, and quite a bit more complicated.  In point of fact, you’ve probably touched one recently.  Can you guess what it is?

While you’re thinking about where you may have encountered a Guilloché pattern, I’ll give you some additional background about how they became popular in the United States.  In 1962, a mechanical engineer from England named Denys Fischer was designing bomb detonators for NATO, and in the process invented spriograph, a concept that became one of the most popular toys in America in the late 1960’s.  Certainly the transition in thinking that takes a person from bomb detonators to spirograph is quite a leap; but the truth is, spirograph is the geometric manifestation of some very complex mathematics.  The patterns the toy creates are called hypotrochoids, which fall into a class of functions known as roulettes… indicating functions not unlike a roulette wheel.  Are you ready for the application?  The wheel within a wheel function of the spirograph, which is used to inscribe an overlapping curve, is exactly what is used to create the patterns on paper money (only there are more wheels used).  Did you guess that money was what contained Guilloché patterns?  Look at a dollar bill and you will see the intricate Guilloché patterns near the perimeter on both the front and back.

These patterns can be seen on the paper currency in virtually every country in the world.  According to the United States Bureau of Printing and Engraving, there was a time that anyone with \$50,000 to spare could start a bank and issue banknotes.  Of course, if the bank failed, the notes they issued would become worthless, so it was necessary for the banks to protect their currency… enter the Guilloché patterns.  As it happened, the larger (and I’m sure richer) banks could employ more talented artists, who in turn, could produce more sophisticated Guilloché patterns on the money.  Why?  Because more complex patterns were more difficult to replicate, thus reducing the possibility of forgery.  Today, the colored watermark, ultraviolet, and infrared printing techniques add another layer of security that make unauthorized duplication of the bills very difficult.

Let’s now take a look at the transformational coefficients that affect the shape, direction, and vertex points of another common algebraic function.  By doing so, we will create patterns slightly more like those seen on the dollar bill.  Using the same basic idea of predictably changing the coefficients as was done in the example using lines, see if you can determine what each coefficient affects for the base parabolic function:

Parabolic Curves          Pattern Graph:

In this example, only two of the coefficients were manipulated, but the pattern created is a little more interesting than if we had just used lines.  What would happen if the ‘h’ value were to be manipulated as well… perhaps in a pattern that changed from positive to negative each time?  The geometric patterns emerging from creative manipulation of the variables ‘a’, ‘h’, and ‘k’ could produce virtually limitless options for a security pattern or document decoration.  We also know that these coefficients act similarly for different kinds of functions, from rational to trigonometric.  Trigonometric functions in particular allow for the types of designs that represent truer Guilloché patterns because of the natural curves.   You will find yourself closer yet to the patterns on the dollar bill if you try manipulating the variables from the base function:

Now, because they tend to better represent the patterns we want to investigate, those on the dollar bill, let’s look at some trigonometric patterns.  One of the most popular, but probably over simplified, representations of a sine function is to roll a disk along a straight edge while mapping the curve that follows a fixed point somewhere between the center and edge of the rolling disk.  Though this is not actually a sine wave, it does illustrate the oscillation factor of many trigonometric functions, and in particular those that create our Guilloché patterns.  This curve is actually a cycloid-type curve.  The more complex patterns are then created by rolling the same disk along a curve, circle, or ellipse… or at least some figure other than a straight edge.  The most complicated patterns are created by having several different disks of varying size rolling along or inside each other simultaneously.  These types of Guilloché patterns were, at one time, constructed with very complicated machinery; the most notable application perhaps being the decorating of the famous Fabergé Eggs.

In many Eastern European cultures, eggs were decorated as a celebration of the onset of Spring.  For instance, in Russia during the late 19th and early 20th centuries, Czar Alexander III and then Nicholas II, who continued the tradition, annually commissioned jeweled eggs to be fashioned by Karl Fabergé for the Czars.  The decoration technique used by Fabergé included Guilloché machining which turned the egg on a lathe-type device in order to engrave the design on the metallic surface.  The complexity of the pattern was determined by calibrating the size and rotational coefficients for the gears.  Basically, calibrating the gear size on the lathe is analogous to how we have changed the formula coefficients in our examples to create unique designs on a flat surface.  The difficulty in Fabergé’s process is that etching a pattern on an ellipsoidal surface is somewhat different than the patterns we’ve been producing on a plane.  The calculations necessary for Fabergé’s work can be best described with the construction of cycloid patterns in spherical geometry.  I’ll bet you had never thought of Karl Fabergé as a mathematician!

Let’s now take another step forward by looking at some other cycloid patterns.  The kinds of cycloid curves that can be graphed on planes can typically be modeled mathematically by functions that add trigonometric terms of the type  a sin (bt) and c cos (dt) and  where t is an iterative variable.  For example, points (x, y) on a cycloid curve can be parametrically represented as follows:

Note that by incrementing ‘t’, each successive (x, y) point is modified slightly even when the translational, amplitude, or wavelength coefficients ‘a’ through ‘h’ are held constant.  This is generally true with any planar function, though in our previous examples we graphed a series of functions rather than adjusting a ‘t’ value.  This allowed us to follow more closely the differences in the coefficients.  If we graduate to a more advanced parametric function for determining each successive (x, y) point as ‘t’ increases, we may create an example such as the following:

Conclusion

The Guilloché pattern above is entitled “The Slinky” and can be seen along with others on the web at http://www.atsweb.neu.edu/math/cp/blog/Guilloche/Guilloche.htm.  These types of dynamic geometric patters are actually fairly predictable once one becomes comfortable with how the various transformational coefficients affect the shape of a given equation’s graph.  Like anything else, practice producing and interpreting the graphs that emerge from manipulating transformational coefficients makes understanding come more quickly.

Secondary level mathematics students will probably never produce patterns that are as sophisticated as those seen on paper money, but by looking at the various geometric patterns a base function can produce, one should be able to identify how the coefficients are being manipulated.  Students should also be encouraged to create their own patterns, and by experimenting with the coefficients, be able to bring their own flavor to the patterns they generate.  Because of the consistency in how these coefficients affect the shape, location, and direction of various base functions, students learning about pre-calculus mathematics should begin to gain an excellent sense of curve sketching in a very short time.  Who knows, they may even learn something about connecting other mathematical topics to the real world through creative outlets like Guilloché patterns.

References

Gardner, Martin. (2001). Toroidal Currency: A Gardner’s Workout. AK Peters, Ltd.

Vakil, Ravi. (1997).  A Mathematical Mosaic: Patterns and Problem Solving.  Brendon Kelly Publishing Inc.

Stegenga, Wil. (2002). Geometric Patterns and Designs for Artists and Craftspeople (Dover Pictorial Archive Series).  Dover Publications