Guilloche Pattern Generator

Tilt an engine-turned watch dial under a lamp and bands of light run across it. Nobody drew those lines. A lathe cut them, one after another, each a copy of the last moved over by a hair.

That is the whole idea of guilloche, and it is why the patterns reward a closer look than they usually get. They are among the few kinds of ornament that can be written down completely as a handful of numbers. Change one number and the lace reorganises itself. Understand the numbers and the mystery goes, though the pleasure, oddly, stays.

A word with a doubtful father

The word is French, written with an acute accent on its last letter, and it has been in use since at least the 1770s. It is often said to honour an engineer named Guillot who invented a turning tool or machine. No first name and no dates have ever been attached to him, and dictionaries treat the story with suspicion.

English uses the plain spelling in a second, narrower way: an architectural band in which two ribbons interlace around a row of central spaces. That motif is far older than any lathe. The name was carried back onto it because the old stone braids resemble what the machines later made, which is a rare case of a pattern being named after its own descendant.

In the workshop the plainer term is engine turning. It means mechanical decorative engraving done on a rose engine or a straight-line machine, and it has been applied to metals, clay, glass, bone and wood.

The machine that wobbles on purpose

An ordinary lathe spins the work about a fixed axis, so a tool held against it can only cut circles. A rose engine breaks that rule. On its spindle sits a stack of cams called rosettes, each with a wavy edge, and a follower, known as a rubber or a touch, is held against one of them by a spring. As the spindle turns, the rosette pushes the headstock to and fro, and the work rocks with it. The tool therefore cuts a circle with a wave riding on it. A museum description of a French rose engine dated 1768 records this arrangement, with serrated rosettes producing wavy lines that vary in pitch or depth.

One wavy circle is not yet a pattern. The pattern comes from repetition with a shift.

Between cuts the turner moves the tool toward the center by a small amount and shifts the pattern round a little, an adjustment the trade calls phasing. The shift makes each cut cross over its neighbours, and the crossings produce the optical illusions for which the craft is known. With the same rosette and the same rubber, different phasing gives visibly different patterns. The cam supplies only the wave. The step and the phase supply the design.

The technique is old. Ornamental lathes of this kind were developed in the sixteenth century and worked at first on soft materials such as ivory and wood. A description of the rose engine appeared in English in 1683, and by the eighteenth century the engine turner was a tradesman producing fine line work on metal. From about 1790 engine-turned objects came into popular circulation, and the craft found its widest popularity in the early nineteenth century on watch dials, cases and movements. Conservators who study it call the characteristic flash of the highly reflective cut surface arguably its most important tangible property.

Why paper money wears lace

Printers adopted the same machines for a harder purpose. In the decade before 1820 detected cases of counterfeit notes in Britain rose by more than 500 percent, and engineers looked for images that a forger with a graver could not match. One answer, patented in 1821 as compound-plate printing, used plates in two interlocking parts engraved while fitted together; a rose engine lathe built in London in 1821 to make such plates survives in a museum. An 1819 patent already covered a machine to engine turn security notes.

The logic is simple. A geometric lathe cutting a metal plate lays down fine lines that cross at controlled places and never wander. A hand copyist must reproduce every crossing; the machine only has to keep turning. A central bank museum notes that the notes of its country in the 1970s still carried such patterns, made with geometric lathes, as security features meant to make reproduction extremely difficult. Currency engraving remains a matter of fine lines, dots and dashes cut into steel.

That history is also why this ornament sits near a legal line, and the line should be stated plainly. Decorative lace is free for anyone to draw. The designs of actual banknotes are not free to copy.

In the United States, federal law permits color illustrations of currency only if they are under three quarters or over one and a half times the true size, one sided, and if the working files are destroyed after final use. The European Central Bank allows reproductions that meet published criteria, such as a one-sided copy at 125 percent or more, or 75 percent or less, of the note. In the United Kingdom the Forgery and Counterfeiting Act 1981 makes it an offence to reproduce any British currency note, or any part of one, without prior written consent.

Rosettes for a certificate, a label or a book cover are ornament. A copy of a note is something else.

Two circles, a ratio and two fences

The mathematics is the mathematics of a small circle rolling on a large one. Fix a circle of radius a. Roll a circle of radius b around the inside of it, and attach a pen at distance h from the centre of the rolling circle. The pen draws a hypotrochoid, with x = (a - b) cos t + h cos(((a - b) / b) t) and y = (a - b) sin t - h sin(((a - b) / b) t). Roll the small circle around the outside instead and the differences become sums: an epitrochoid, a family that appears in a book by Albrecht Durer in 1525, where he called such curves spider lines.

The interesting question is when the pen comes home. The curve repeats only when both angles in the formula have made whole turns at once. If a and b are whole numbers, that first happens when t reaches 2 pi b / gcd(a, b), where gcd is the greatest common divisor.

Take wheels of 96 and 36. Their greatest common divisor is 12, so the rolling circle goes round 36 / 12 = 3 times, the parameter runs to 6 pi, and the finished curve has 96 / 12 = 8 lobes. In short: 8 lobes, 3 turns. Change 36 to 35, which shares no divisor with 96, and the pen must go round 35 times and draw 96 lobes before it closes. One tooth of difference turns a sturdy eight-pointed figure into dense lace. This is the lever a designer pulls most often.

Set h equal to a - b and the hypotrochoid collapses into a rose curve, r = a cos(n theta). Its petal count has a famous quirk: n or 2n petals, n when n is odd and 2n when n is even, because for odd n the second half turn retraces the first. A rose with n written as a fraction p / q closes at theta = pi q when p q is odd and at 2 pi q when it is even.

An irrational ratio never closes at all. The line goes on filling its ring for as long as anyone cares to draw.

The banded patterns on borders are built differently. Choose two envelope curves, an inner fence and an outer one: two circles, two ellipses, or the two edges of a rounded frame. Send a sine wave around between them so that it touches each fence once in every lobe. Then draw the same wave again, shifted by a fraction of a lobe, and again. Six strands a sixth of a lobe apart give a braid; two strands half a lobe apart give a chain of eyes. Give the wave a fractional number of lobes per circuit, say 77 lobes spread over 6 circuits, and one line winds round six times before it closes, weaving with itself.

Everything in such a band is counting. Lobes, strands, phase step, circuits. Corners need one decision more: if lobes are spaced evenly along the middle of a rounded frame, they crowd on the inside of each bend and open on the outside, which is how printed borders have always turned.

Printing a line that thin

A drawing file can describe these curves far more finely than most devices can show them. In the vector format used on the web a stroke has a width in the drawing’s own units and a path is a list of moves and segments, so a file can state a line of 0.1 mm exactly. A printer that lays 600 dots to the inch has dots of 25.4 / 600 = 0.0423 mm, so a 0.1 mm line is about two and a third dots wide. On a screen the same line may be less than half a pixel. An honest preview draws it paler, in proportion to the area it covers, instead of fattening it.

Several limits follow, and they are worth saying outright. A curve handed over as straight chords is only as round as its tolerance, and that tolerance should be stated, not hidden. A line narrower than one dot cannot be laid down at its true width by that device, whatever the file says. Arithmetic can tell you how many dots wide a line is; it cannot tell you how a particular press, ink and paper will render it, and only a proof print settles that. None of this is a defect in the mathematics. It is the difference between a curve and a mark.

A computed rosette is also not a security feature. Real currency relies on specially made inks and paper and on engraved steel, not on a pattern that anyone can calculate.

What remains, once the mystery has been taken out, is a small and exact pleasure. A ratio of two whole numbers, a fence on either side, a phase step: write those down and a machine, or a few lines of arithmetic, will fill a circle with lace that closes on itself to the last hair.