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.

### 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 e^{iπ} + 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 e^{iψ/2} + 1 = 0?
Conway’s response was to mention another
formula of which Euler’s is a special case:
e^{2π/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.

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