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When Is an Ellipse Not an Ellipse?

A picture I took of a 1992 Patek Philippe catalogue
Let's get the potentially controversial part of this out of the way: Ellipse is a misnomer when it comes to the Patek Philippe references 3788, 3930 and related versions. More specifically, the watch case in these designs is not rendered in the geometrical version of an ellipse that is frequently taught in high school. I'm honestly a little embarrassed to have only recently come to this conclusion, but geometry was never my strong suit. I won't take all the blame. In most discussions of the Ellipse (the watch), inevitably the "golden ratio" comes up. I think I was distracted by that, because the golden ratio is a pretty interesting topic. Nevertheless, I'd like to set the record straight, or at least straighter, with this post.

An ellipse is basically an oblong shape.
The Ellipse in Washington, DC, a grass area in President's Park.
Compared to a canonical ellipse, the Patek Ellipse has sides that are too straight leading to "corners" that arch too quickly. Real ellipses are also defined by an equation, like:
A graph of this equation is below.
The graph of the ellipse equation.
As I'll show in just a bit, the case profile of the Ellipse watch is also defined by an equation, but it is meaningfully different from an actual ellipse. The Cartier Bagnoire is much closer to an ellipse than the Patek Ellipse.

I came to this realization after flipping through the pages of a 1992 Patek catalogue I received as a holiday gift.
Another picture of a page from the 1992 Patek catalogue.
There, I saw a designer's drawing of the Ellipse watch, which I share here. I'd never seen this drawing before. It features a crosshair at the center surrounded by four overlapping circles. I later learned that these four overlapping circles are called a quatrefoil, a shape that is important in a number of religions. Some claim that in Christianity it represents the four gospel authors in the Bible. The design also appears in mosques. Van Cleef and Arpels famously adopted a quatrefoil in their Alhambra bracelet.

I spent a bit of time trying to figure out how a combination of the quatrefoil and the golden ratio could result in the case design we find in the Ellipse.
Example of a cloverleaf highway design in Michigan, US.
That effort, by and large, went nowhere. I will note that the quatrefoil drawing presented in the 1992 Patek catalogue does lend credence to the claim on John Riordan's Collectibility that one possible designer of the Patek Ellipse was inspired by the interwoven "cloverleaf" offramp pattern frequently featured on American highways.

I began to turn my attention away from the quatrefoil / golden rule and started concentrating on the end result of the design.
A very early tube television, image rotated 90 degrees. The design similarity with the Ellipse watch is clear.
The Ellipse is basically a rectangle with rounded corners and curved edges. The shape was vaguely familiar. It looks like a very early tube television. In fact, some software refers to this design as the "TV Screen Shape." Eventually, I identified the actual shape of the Ellipse because I'd discussed it in an earlier post on the shape of wrists. It is a Lamé Curve, also known as a superellipse.

Gabriel Lamé was a 19th century mathematician who made significant contributions to the field of partial differential equations.
A Patek Philippe Ellipse ref. 3748 (L) and Hein's Sergels torg in Stockholm (R).
In 1818 he offered the first known analysis of his eponymous curve, which was later dubbed a "superellipse" by Dutch polymath Piet Hein. In 1959, planners in Stockholm, Sweden held a competition for the redesign of a downtown public square. Hein's proposal for a Lamé curve (superellipse) fountain won the competition. The fountain stands to this day (see photo). It bears a striking resemblance to the Ellipse reference 3748, in which diamond hour markers evoke the circular "windows" at the base of Hein's fountain.

Interestingly, it is entirely possible that Patek's Ellipse designer was inspired by Hein's fountain, rather than the golden ratio. The timing is consistent. Hein's design was presented to Stockholm planners in 1960. The Patek Ellipse debuted in 1968.

Regardless, once I realized the Patek Ellipse followed a Lamé curve (superellipse), I decided I'd try to "reverse engineer" the equation behind the Patek Ellipse case. Here is what I settled on:
In the image presented here, you can see that the gaph of the above equation matches a photo of the Patek Ellipse almost perfectly.
A graph of the equation defining the shape of Patek's Ellipse (L) and the graphed equation overlaying a Patek Ellipse (R).
You can also see that the actual equation for the Ellipse watch is meaningfully different from the ellipse equation I presented earlier. Interestingly, we now know exactly where the golden ratio fits into the design of the Patek Ellipse. If you look in the exponent of the equation just above, it rounds to 2.62. This is one plus a rounded value of the golden ration (1.62).

It appears that marketers may have enjoyed some unconstrained freedom when naming and describing the design of the Patek Philippe Ellipse. Precision is important in engineering and horology, but calling a Lamé curve (superellipse) an ellipse muddies the waters. I can certainly understand why Patek did not want to call the watch the "Lamé," because doing so could lead to jokes among English speakers about how "lame" the watch is. The brand probably also took a pass on Superellipse because that is perhaps a little less "serious" than most reference names we typically see from the maison. It is fun, though, to note that the superellipse shape of the Ellipse case is a decent proxy for the actual shape of the human wrist in cross section, a point I make here. From that perspective, we can think of the Patek Ellipse as the opportunity to display wrist geometry atop your own wrist.
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