n 1996, Europa Star published a critical assessment of the tourbillon by Jean-Claude Nicolet, a professor at La Chaux-de-Fonds School of Watchmaking. His analysis came to the scathing conclusion that “in creating the tourbillon, Breguet thought he was eliminating the effects of gravity. It was an error on his part. He succeeded only in masking them a little”. Nicolet then compared the ingenious watchmaker to a “conjuror who causes an elephant to vanish”.

This was tantamount to blasphemy, and the reaction of watchmakers to this critical assessment was equally scathing. No one disputed the scientific argument; the principal accusation was that we were out to “kill the golden goose” that the tourbillon was in the process of becoming. At the time, in 1996, it was estimated that around one thousand tourbillons had been manufactured over the course of two centuries, such was their rarity. Since then, and despite the reservations expressed as to the true chronometric benefits of a tourbillon in a wristwatch, tourbillon fever has taken over. Today, as many tourbillons are probably produced in one year as in the previous two centuries.

Since then, however, watchmakers have become increasingly conscious of the limitations of the tourbillon, especially in wristwatches, and have been exercising their ingenuity to overcome them. The result has been new configurations, double tourbillons, gyrotourbillons, triple-axis tourbillons, tourbillons inclined at 30° (which, let it be said, achieved the best results in the history of the International Chronometry Competition) to name just a few examples.

But no one has yet succeeded in designing the perfect tourbillon for a wristwatch, because to even out all the deviations in rate, its axis would have to sweep all directions uniformly. And that’s a headache, because all the mechanisms available to date have one intrinsic, mathematical defect. We will come back to this.

No one has yet succeeded in designing the perfect tourbillon for a wristwatch, because to even out all the deviations in rate, its axis would have to sweep all directions uniformly. And that’s a headache.

The team at Instant-Lab EPFL researching a mathematical design for the tourbillon

Teamwork

Noémie Mandon is a young Frenchwoman who recently graduated from the Cluny Arts and Crafts Institute of Technology (which teaches mainly mechanical, industrial and power engineering). For her final year of studies, an academic exchange gave her the opportunity to study mechanical engineering at the EPFL in Lausanne, Switzerland. She was drawn to micromechanics, and more particularly to mechanical watchmaking, for its beauty and sophistication. She was also driven by an interest in research and innovation.

She arrived to Instant-Lab EPFL directed by Professor Simon Henein in Neuchâtel, to work with Ilan Vardi (see our article Breaking the Second Barrier). At their first meeting, Ilan Vardi revived a subject that had been lying dormant for quite some time: the geometry of the tourbillon. It became the subject of her Master’s research project.

The goal of her research was to construct a theoretical object using mathematical design, the purpose being to correct the intrinsic defect of tourbillons in order to offset “almost totally” the effect of gravity: to average it out, as in Breguet’s pocket watches, but for all directions. But to do this also meant coming up with a means of varying the tourbillon’s speed of rotation, which in the tourbillons available today is constant for each axis. In mathematical design, a theorem is implemented mechanically.

Ilan Vardi, the head of the project, brought together a talented team to assist Noémie Mandon (he emphasises the collective, team aspect of the work). It included Roland Bitterli, a scientist at Instant-Lab, in charge of “making sure the object did not become intractably complex”, Patrick Flückiger, a PhD student who in 2019 won the Omega Student Award with a Master’s project on a new type of Foucault’s pendulum, and Quentin Gubler, an engineer and watchmaker from Ulysse Nardin, where he worked on several projects related to compliant mechanisms.

Does the tourbillon really serve no purpose?

Invented in 1795 to offset the effects of the earth’s gravity on a pocket watch – that is, a a watch positioned vertically in a jacket pocket – the purpose of the tourbillon is to average out the vertical positions, thereby improving the watch’s accuracy. In this sense, it achieves its purpose. “Proof of this,” Ilan Vardi adds, “is that tourbillons automatically pass the COSC tests in all four vertical positions."

But things start to get more complicated in wristwatches. On the wrist, the tourbillon’s position relative to gravity is changing continually in all directions. “Useful only when the balance axis is perpendicular to the direction of gravity”, the tourbillon no longer serves its original purpose in a wristwatch. Because, while it can average out positions on a single plane (vertical), it cannot do so in three-dimensional space.

To solve this problem, watchmakers gradually turned to multiaxis tourbillons, starting with Anthony Randall, who invented the first double-axis tourbillon in the late 1970s. Numerous other propositions followed, including the Double Tourbillon 30° by Greubel Forsey, which enables the tourbillon’s axis to sweep a circle at a constant latitude, then the Gyrotourbillon invented by Eric Coudray for Jaeger-LeCoultre, after which came the triple-axis tourbillon, versions of which have proliferated.

Watchmakers have become increasingly conscious of the limitations of the tourbillon, especially in wristwatches, and have been exercising their ingenuity to overcome them, via double tourbillons, gyrotourbillons, triple-axis tourbillons, tourbillons inclined at 30°...

In search of uniform distribution of points in a sphere

But, as Noémie Mandon explains, we know that “to average out the effects of the orientation of the balance in relation to gravity in a wristwatch, the distribution of the points on the sphere swept by the axis of the balance has to be as uniform as possible”.

The uniform distribution of points on a sphere is an important mathematical problem, and has been the subject of numerous scientific papers. While we know how to distribute points uniformly on a rectangle, “for spheres there is no theoretical solution”. As will be explained below, the simplest method for distributing points on a sphere leads to an overconcentration at the poles. To understand the consequences of this phenomenon for multi-axis tourbillons, a friend of Anthony Randalll, the English engineer Guthrie Easten, highlighted this lack of uniformity in 1985. He found a cogent image to illustrate it.

The simplest method for distributing points on a sphere leads to an overconcentration at the poles. This lack of uniformity in the distribution of the points implies “a failure to compensate for the effect of the tourbillon’s position in relation to gravity”.

Imagine a tourbillon set at the centre of a Chinese paper lantern. Its axis is fitted with an ink gun which, at each oscillation of the escapement, shoots a drop of ink inside the lantern. If the effect of the multi-axis tourbillon were perfect, the drops of ink would be spaced out evenly across the sphere’s surface. But at the poles we unfortunately find an abnormal concentration of drops.

Why? Because the rotations of a multi-axis tourbillon are constant in latitude and longitude, they do not cover the same longitudinal distance, depending on the latitude at which they are located. When it arrives at the pole, the distance travelled is cancelled out. This lack of uniformity in the distribution of the points implies “a failure to compensate for the effect of the tourbillon’s position in relation to gravity”.

Variations in speed?

So, what can be done to remedy this? As Anthony Randall explains, in order to cancel out rate differences completely or maintain a constant rate in every possible position, “it is crucial that the double-axis tourbillon should vary in speed,” specifically when approaching the poles, where concentration increases.

Watchmakers know how to vary speed, for example in a chronograph. What solutions exist for varying speed according to the position of the tourbillon? Is it better to do massive amounts of calculations, or to apply a theorem and create a theoretical object?

The team opted for the second method: mathematical design.

Watchmakers know how to vary speed, for example in a chronograph. What solutions exist for varying speed according to the position of the tourbillon?

Mathematical design

The avenues proposed by Guthrie Easten to solve the problem of variable speeds consisted of using non-circular gears or cams to vary the speed of the balance axis. But quite apart from the fact that this approach requires a vast amount of skilful calculation, designing a non-circular gear is a particularly complex undertaking.

So Instant-Lab chose another approach. “Mathematical design” may sound very modern, but in fact,the designers of the Antikythera mechanism (an astronomical calculator driven by a crank) were performing mathematical design in the first century BC, although they did not use the term. They “simply” produced mechanical implementations of the mathematical models of the sun and moon proposed by Hipparchus (c. 190–c. 120 BC).

But the primary inspiration of the scientists at Instant-Lab was the work of another great mathematician of antiquity, Archimedes. In one of his best-known theorems, Archimedes established the correspondence between a sphere and its circumscribing cylinder. He found that “the area of any surface of the cylinder is equal to the area of its projection on the sphere”. One well-known example illustrates this perfectly.

The world map drawn using a projection of the earth in uniform latitudes (a) corresponds to the current multi-axis tourbillons. As we can see, the territories close to the poles appear far bigger than they are in reality – the equivalent of the denser projection of points in our Chinese lantern. The lateral projection of the earth onto a cylinder (b) preserves the surface areas of the territories and continents.

Following Archimedes’ lead we can therefore state that, unlike the map showing the projection of a sphere onto a cylinder, one way of uniformly distributing the points on a sphere is to distribute them uniformly on a rectangle which is rolled up into a cylinder – and then project them onto the sphere. As we can see, using this method the projection of the points at the poles remains uniform.

But let us return to the rectangular map of the world corresponding to a uniform latitudinal projection, but with deformation at the poles. The speed of the axis of rotation of the multi-axis tourbillon is constant latitudinally, causing an “accumulation” of positions at the poles. But this lateral projection preserves the longitude. In that case, it suffices to modify the speed of the axis of rotation of the balance wheel latitudinally on the sphere, i.e. make it more rapid when sweep- ing the upper part of the projection rectangle. “Which simply comes down to projecting a vertical line onto a circle”, the researchers explain.

One way of uniformly distributing the points on a sphere is to distribute them uniformly on a rectangle which is rolled up into a cylinder – and then project them onto the sphere.

From theorem to mechanical model

The mathematical design, then, consists of making a mechanical model of this projection, placed on a table turning at a constant speed longitudinally.

The critical point of this mechanism is the projection of a vertical line at constant speed onto a circle. This lateral projection of a vertical line onto the circle allows a faster speed to be obtained at the poles (where it theoretically attains infinite speed), which entails a lesser concentration of uniformly distributed points at those locations, theorem states.

To achieve this, it is necessary to mechanise an alternating rectilinear movement (i.e. alternately rising and falling) at constant speed. The constant speed is generated by a gear, which is also rotating at constant speed.

Just such an alternating rectilinear mechanism is described under #114 in Henry Brown’s 1868 book entitled 507 Mechanical Movements. The researchers were inspired by a number of historical examples. One of them was the Antikythera, and more particularly the mechanical expression of the non-uniform movement of the moon or sun. Without going into all the details, the model of Hipparchus’ work consists of taking an eccentric circle, representing a virtual sun, turning at a constant speed and projecting its positions onto an offset circle, centred on the earth. This gives rise to a non-constant speed on that geocentric circle. In concrete terms, the Antikythera achieves this projection thanks to a system consisting of a pin and slot, a device reproduced in the Instant-Lab mechanism.

The Antikythera achieves this projection thanks to a system consisting of a pin and slot, a device reproduced in the Instant-Lab mechanism.

Exploded reconstruction of the Antikythera Mechanism, a 2,000-year-old device often referred to as the world’s oldest “computer”.

©Tony Freeth

Polar problems

But there was still one big problem, as the first models showed. The mechanism tended to stick at the poles (due to the progressive loss of alignment of the pin with its slot, against which it rubbed). Moreover, once at the pole, the mechanism’s wheel could just as well start turning in a clockwise or anti-clockwise direction, whereas it was crucial that it should always turn in the same direction.

This problem necessitated the addition of a supplementary mechanism that takes over at the passage through the poles (through the intermediary of an extra two-toothed gear). As a result of this, the distribution of the positions of the tourbillon would no longer be homogeneous, but this defect could be minimised by designs increasingly faithful to the theory.

A supplementary mechanism takes over at the passage through the poles (through the intermediary of an extra two-toothed gear).

Over to the watchmakers

During our visit to Instant-Lab, we were able to see for ourselves how these mechanisms work using a theoretical projection of a vertical line onto a circle. The researchers have succeeded simply and effectively in varying the speed of rotation according to the latitude. With a longitudinal axis of rotation, the distribution of the positions in three-dimensional space is uniform, with a much reduced discrepancy at the poles.

For the moment, this mechanism is mounted on a frame and driven manually by a crank. But the addition of the balance spring and escapement “will be relatively straightforward for master-watchmakers,” the team explains. Applying it to double and triple-axis tourbillons will enable the effects of gravity to be offset “almost totally” – a fundamental step towards achieving the “perfect” tourbillon that will at last be capable of fulfilling all its chronometric promises.

Now the ball is in the watchmakers’ court. Because, as Ilan Vardi says, “our business is concepts. We offer simple, elegant, clear conceptual solutions. Our prototypes are there to inspire of exploration, but we have no vocation to become watchmakers ourselves. We solve one problem, then move on.” So, a challenge to watchmakers: the perfect – or near-perfect – tourbillon is within reach. It is up to them to actually produce it.

The researchers have succeeded simply and effectively in varying the speed of rotation according to the latitude. For the moment, this mechanism is mounted on a frame and driven manually by a crank. But the addition of the balance spring and escapement “will be relatively straightforward for master-watchmakers,” the team explains.

One final remark: this work of mathematical design, applied here to watchmaking, also opens up numerous avenues of exploration in other domains where the effects of gravity must be counteracted, or which seek to make an object rotate in every possible plane and direction in a uniform manner. For example, what is the human eye if not simply a rotating sphere?

Just one demonstration that innovation in mechanical watchmaking can open up new vistas in numerous other fields.

A challenge to watchmakers: the perfect – or near-perfect – tourbillon is within reach. It is up to them to actually produce it.