In my column in February, I discussed several
fundamental constants, such as π and
Planck’s constant, *h*, that, I thought, might
not be expressed with maximum beauty and
efficiency. I asked readers for other candidates
and received dozens of replies. Many
of you were particularly concerned about
whether it would be better to define π as the
ratio of the circumference of a circle to its
radius, rather than its diameter.

In February I cited an article from 2001
called “π is wrong!” by University of Utah
mathematician Bob Palais that was published
in Mathematical Intelligencer (**21** 7),
which identified formulas that would be
simpler with such a redefined π. In response
to my column, Palais wrote that he started
thinking about π and possible simplifications
when he noticed, to his bafflement, that his
students would reach for their calculators to
work out cos(π/2) or sin(π/2), when he had
gone to the trouble to devise questions for
which they did not have to do so.

But Palais noted that his sense of urgency
was not widely shared. He mentioned an
entry from August 2007 by the computer scientist
Bill Gasarch on the blog *Computational
Complexity* entitled
“Is pi defined in the best way?”. It looks at
two examples from mathematics. One concerns
the expansion of the zeta function
ζ(*n*) = Σ*r*^{–n}, which is a little, but not much,
simpler if 2π rather than π is used. The other
involves calculating the formulas for the volume
and surface area of an *n*-dimensional
sphere, for which it is only a matter of taste
whether formulas with 2π are better.

### A base affair

Indeed, most respondents seemed less aroused than intrigued that π, and other constants of mathematics and science, could be amended at all. Some proposed different mathematical structures that might be profitably changed, such as bases. Richard Hoptroff, a physicist who works for the software firm HexWax in London, wrote “Don’t you think the use of base 10 has passed its sell-by date? It’s a bit arbitrary now that we don’t need to count on our fingers any more. How about following the computing world’s example and switching to the far simpler 2?” Using base 2, he noted, would eliminate suspicious irrational behaviour such as thinking that 1101 is unlucky, or that 1010011010 is the number of the beast.

And Aasim Azooz of Mosul University in
Iraq said he had wondered if the sacred time
variable, *t*, which can only be defined in
terms of two successive events, could be
replaced with a variable such as entropy —
and how such a substitution might change
physics. But he confessed he had been too
distracted by other events in his country to
focus on the issue.

Those who preferred one version of a
constant usually admitted it to be based on
convenience. As Robert Olley from the University
of Reading noted, “Even the difference
between the two versions of Planck’s
constant *h* and * h* depends on whether one is
thinking physically in terms of frequency ν or
mathematically in terms of angular momentum
ω. Physics is not applied mathematics!”

Meanwhile, Igor Zolnerkevic, a former physics graduate student and now a science writer in Brazil, observed that a maximally efficient theory only requires two dimensional constants. He cited a paper by George Matsas from the Universidade Estadual Paulista in Brazil and colleagues, entitled “The number of dimensional fundamental constants” (arXiv:0711.4276), which has implications for what a brutally efficient approach to constants would look like, and suggests that certain constants are more fundamental than others.

But Matthew Thompson of the Naval
Research Laboratory in Washington, DC
pointed out that we rarely prefer such brutal
efficiency, citing cases involving “constant
pairs” where science needs only one constant
but uses two. He mentioned the Nernst
equation in electrochemistry *R/F* = *k/E*,
where *R* is the gas constant, *k* is the Boltzmann
constant, *E* is electric potential and
*F* is the Faraday constant, i.e. the electric
charge carried by a mole of electrons.
“Why”, asked Thompson, “do you need *F*
and *R*? We keep them since those formulas
are a bit more efficient in real-world terms
than their microscopic brethren.”

My colleague at Stony Brook Fred Goldhaber
explained to me why convenience in
constants fails to stir physicists. “Making the
units more efficient may simplify people’s
thinking,” he suggests. “It may simplify
teaching. But it doesn’t make physics progress.
It does not help us achieve a new level
of understanding of the world. What physics
are we doing today that we could not do with
old non-SI units? And with computers to do
the calculations, it no longer even matters
how stupid the units are. What *does* matter
is that people working on the same project
agree on their choice of units, as shown by
the Mars Climate Orbiter mission failure!”

### The critical point

The subject of changing constants triggered
several letters about units, and the issue generated
much more passion than mathematical
constants. I received half a dozen letters
from metrologists, for instance, about the
movement to tether the SI base unit, the
kilogram, not to an artefact but to constants
of nature, such as Planck’s constant or
Avogadro’s number. In the words of a 2006
article by Ian Mills from Reading and colleagues
(Metrologia **43** 227), “In the 21st
century, why should a piece of platinum–iridium alloy forged in the 19th century that
sits in a vault in Sèvres restrict our knowledge
of the values of *h* and *m*_{e} [the mass of
the electron]?”

The passion seemed stimulated by two factors: the prospect of bringing additional precision to the kilogram over the long term; and the sense that an SI unit’s role is more completely fulfilled if it is tied to a true invariant of nature. But other respondents wondered whether such a shift really amounts to an achievement of new knowledge or understanding, or only to reshuffling our conventions. That is a discussion for another time.

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