## The Hercules Number: How a Dimensionless Physical Parameter Got Its Name

I did not conceive or give birth to the Hercules Number. But I did name it. Here’s the story.

In science and mathematics, we often get to name things. To help with exposition, sometimes we essentially have to name them, because it can help us do a better job of explaining things. In fact, we also enjoy naming things. To borrow some words from the computer game Beyond Zork, we often want a scientific idea to “bask in the glow of a new-forged synonym.”

And we get to name all sorts of things—concepts, theorems, examples, equations, groups, graphs, manifolds, numbers, physical parameters, and more. Sometimes we name them after people—ideally after somebody other than ourselves, and occasionally even after the person who actually had the idea first—and other times we get to be more creative. Naturally, the same object can go by more than one name, especially when multiple scientific fields are involved. On occasion, we fail miserably in our naming adventures. In abstract algebra, for example, there are so-called extra special groups. (My abstract algebra professor at Caltech couldn’t tell us about them without giggling over the name, so that is how you should read the previous sentence.)

One of the peculiar traditions in continuum mechanics (and especially in fluid mechanics) is the incessant naming of dimensionless physical parameters: the Reynolds number, the Rayleigh number, the Prandtl number, the Péclet number, and myriad others. If one states all of these numbers quickly one after another, one would almost have a George Carlin routine, except with a lot less cussing.

Illustration of (a) why it is really hard to separate two interleaved books and (b) the dimensionless parameter now known as the Hercules number. [This illustration is Figure 2, and its associated caption, from a quick study in Phys. Today 69, 6, 74 (2016).]

A recently-introduced dimensionless parameter that caught my eye was the Repunzel number from ponytail physics (the subject of the 2012 Ig Nobel Prize in physics). And the naming of this number was very much on my mind a few months ago when I wrote an entry in the Improbable Research blog about a very cool new paper in Physical Review Letters (PRL) on how hard it is to pull apart two interleaved phone books. The authors of the paper had introduced a dimensionless parameter, but it didn’t have a name. And it clearly required a Herculean effort to pull apart those phone books, so I knew what name I wanted to attach to that dimensionless parameter. So with inspiration that was part Herculean, part Oxonian (I have spent the last 9 years in Oxford surrounded by applied mathematicians who study continuum mechanics), and part Repunzelian, I wrote the following sentence:

Restagno and colleagues also fit the data to a curve of force versus a dimensionless amplification parameter—following the continuum-mechanics tradition of using cute names for dimensionless parameters, let’s call it the “Hercules number”—that depends on the number of pages, the page thickness, and the size of the overlap region between the books.

The authors of the PRL paper enjoyed the new monicker—one might even say that the Hercules number was basking in the glow of its new-forged synonym—and they mentioned it to me via Twitter. And now Kari Dalnoki-VeressThomas Salez, and Frédéric Restagno (three of the authors of the original PRL paper) have written a “quick study” in the June 2016 issue of Physics Today. As you’ll notice, they use the name Hercules number, and of course I am very pleased about that. I am always happy to contribute to mathematics and physics with my wit and snark.

Bonus: When it comes to naming a scientific idea after a person (and whether or not one has chosen the correct one), I would like to invoke the “Three Laws of Discovery” that are listed among the quotations on Ig Nobel laureate Michael Berry’s website:

1. Discoveries are rarely attributed to the correct person. (Arnold’s Law, which is of course self-referential)
2. Nothing is ever discovered for the first time. (Berry’s Law)
3. Everything of importance has been said before by someone who did not discover it. (Whitehead’s Law, though I am not sure which Whitehead it is. I assume it is one of the mathematical ones.)

Another Bonus: The story of the penguin diagram in nuclear physics is absolutely lovely. (That name came about substantially through the efforts of Melissa Franklin, whom you may have heard on Improbable Research podcasts.)

A Third Bonus: In mathematics, there is a problem called the Ten Martini Problem.