Clamp a straightedge on a piece of paper and take a small disk made of plastic with a small hole in its center. Lay the disk flat on the paper, put a pencil point through the hole, and roll the disk along the paper. The pencil point will trace out a straight line on the paper, whose distance from the straight edge will equal the radius of the disk. Not very interesting. Now, instead of using a hole in the center of the disk, make one about 1/4 of the way out, and put the pencil point throught that hole. Now as the disk rolls along the straightedge, the pencil point will rotate around the center, and you'll get a curve that looks like this:

This is a cycloid-type curve, the simplest example of a guilloché (ghee-oh-shay) design. More complicated examples are obtained by having the disk roll along a curve other than a straight line, or having the disk roll along another disk which is rolling on along a curve. In fact, truly complicated designs are constructed by have disks of many different sizes rolling along (or inside) each other, and having the pencil point placed at different positions along the radius or even along an extension of the radius. These guilloché designs at one time were constructed with the aid of a very complicated machine. Their most well-known application was in the creation of the famous Fabergé eggs.

It turns out that curves of these types can be modeled mathematically using functions constructed by adding terms of the type and . For example, one might plot many points where, say, and :

These days guilloché figures are created using computers. You can find out more about them by searching the Web. One good site is:

http://www.maa.org/editorial/mathgames/mathgames_02_09_04.html.

Below are some guilloché figures created by the late Stan Miller (an old friend of mine).

Number 1, "The Slinky":

Number 2, "The Ribbon":

Number 3, "The Fish":