In Proposition 3 of On the Measurement of the Circle, Archimedes asserts, based on calculations involving regular polygons circumscribed around and inscribed in a circle, that “the ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71”. He thereby strongly reinforced, if he did not actually create, the tradition of considering that ratio, two millennia later referred to as π, to be fundamental.

Pondering π
Pondering π

Was Archimedes wrong?

When I was in elementary school, my teachers told me that π was the most important number in mathematics, which scared me into cramming as much of it as I could into my brain. I got to all of 35 places. This many places actually turned out to be harmful to my science career, for whenever I used all of them in calculations in high-school physics, I was penalized for something my teacher called “misplaced precision”.

But is π the most appropriate number? Are there not beauties and economies to be had by using the ratio of circumference to radius, rather than the ratio of circumference to diameter? After all, there are so many instances in science and mathematics where forests of 2π occur — from Maxwell’s equations to Fourier series formulae — that it would surely make life much simpler by defining a new constant — let’s call it ψ for the sake of argument — to be 2π, which is the number of radians around a unit circle. Would we lose any beauty and economy by using this new constant?

I posed this question to the Princeton University mathematician John Conway, one of the most creative mathematicians working today. Conway, it turned out, had strong feelings on the subject. “2π is obviously the correct constant!” he told me immediately — although he also told me of arguments, which he did not find persuasive, for a third option, π/2.

I mentioned Euler’s formula eiπ + 1 = 0, which readers of this magazine once voted as one of the greatest equations in science (see “The greatest equations ever”). Would not the beauty and economy of this equation — which contains five of the most fundamental concepts of mathematics and four operators each exactly once — be diminished if it had to become eiψ/2 + 1 = 0? Conway’s response was to mention another formula of which Euler’s is a special case: e2π/n = n√1. This formula, Conway maintained, is preferable and more economical than Euler’s because of its generality: it takes account of the two possible square roots of 1, namely +1 and –1.

In 2001 the Mathematical Intelligencer published an article entitled “π is wrong!” (21 7). Its author, the University of Utah mathematician Bob Palais, mentioned numerous formulae that use 2π, including Stirling’s approximation and Cauchy’s integral formula, to show how much easier they are with a redefined constant. He concluded by noting that our species has broadcast π via radio telescopes to possible extraterrestrials, and expressed alarm about “what the lifeforms who receive it will do after they stop laughing”. Evidently, not only the beauty and economy of our expressions, but our very reputation for science literacy in the universe is at stake here.

Practically speaking, shifting from π to ψ would involve going against more than taste but tradition, and would be as difficult as changing to the metric system. Still, if π is vulnerable to being changed, what other constants are safe?

Was Planck wrong?

In physics, an obvious candidate for renovation is Planck’s constant, h. Given the swarms of ℏ’s, ℏ =  h/2π, that appear in equations, would it not have been simpler to have defined it that way in the first place? Are there situations that would make us prefer h to ℏ? Was Planck right or wrong?

I approached Fred Goldhaber — a colleague with a broad grasp of physics and a gift for explanation — with this question. “It depends on the problem you are addressing,” he said, with characteristic common sense. “I can think of at least two cases where it is simpler to use h than ℏ.” One is in phase-space diagrams where position, say, is on the x-axis and momentum on the y-axis. In classical mechanics, every point on the diagram is a possible position. In quantum mechanics, however, Planck’s constant is an irreducible unit. Possible positions on the diagram are not points but blocks, each of which has a total area equal to h.

The second case he mentioned was the Aharonov–Bohm effect. This involves shifts in the interference pattern when two particles pass on either side of a space, insulated from the particles, where a magnetic field is present. The significance of the effect is that classical particles are unaffected by the magnetic field, whereas quantum particles are affected. Goldhaber pointed out that the interference pattern shifts by one fringe every time the magnetic flux changes by one quantum of flux. The formula for the flux quantum is h/e, where e is the charge on the electron; if the unit were ℏ, it would be the less economical 2πℏ/e.

The critical point

Some constants seem destined to live forever. We can do little, I suppose, about G, the universal gravitational constant, or about the fine-structure constant α, a measure of the strength of the electromagnetic interaction — even though its value might be varying with time. Still, it might be possible to find other constants that might be changed for the purposes of beauty and economy.

I invite you to e-mail me with candidates for constants that are not expressed with maximum efficiency. I shall report on the results in a future issue.