Earlier this year I asked readers to send me their shortlists of great equations. I also asked them to explain why their nominations belonged on the list and why, if at all, the topic matters (Physics World May p19).

James Clerk Maxwell
James Clerk Maxwell

I received about 120 responses -- including single candidates as well as lists -- proposing about 50 different equations. They ranged from obvious classics to "overlooked" candidates, personal favourites and equations invented by the respondents themselves.

Several people inquired about the difference between formulae, theorems and equations -- and which I meant. Generally, I think of a formula as something that obeys the rules of a syntax. In this sense, E = mc2 is a formula, but so is E = mc3. A theorem, in contrast, is a conclusion derived from more basic principles -- Pythagoras's theorem being a good example. An equation proper is generally a formula that states observed facts and is thus empirically true. The equation that describes the Balmer series of lines in the visible spectrum is a good example, as are chemical equations that embody observations about reactions seen in a laboratory.

However, these distinctions are not really so neat. Many classic physics equations -- including E = mc2 and Schrödinger's equation -- were not conclusions drawn from statements about observations. Rather, they were conclusions based on reasoning from other equations and information; they are therefore more like theorems. And theorems can be equation-like for their strong empirical content and value.

It thus makes sense to classify both kinds as equations, which is exactly what respondent David Walton from the University of Manchester did. He distinguished between equations (such as F = ma) that comprise axiomatic models that "define the interrelationships between various observables for all circumstances" and equations that are approximate models (such as Hooke's law), which define "the interrelationships between the various observables over a defined range and within a defined accuracy". I therefore interpreted the term "equation" loosely.

Simplicity

Respondents had many different criteria for greatness in equations. Half a dozen people were so impressed with simplicity that they proposed 1 + 1 = 2.

"I know that other equations have done more, express greater power [and have a] broader understanding of the universe," wrote Richard Harrison from Calgary in Canada, "but there's something to be said for the beauty of the simplest things of their kind." He then recalled how 1 + 1 = 2 was the first equation he taught his son. "I remember [him] holding up the index finger of each hand as he learned the expression, and the moment of wonder when he saw that the two fingers, separated by his whole body, could be joined in a single concept in his mind."

Neil Blackie also voted for 1 + 1 = 2. "For this equation to come into being there had to be the invention of a method for representing a physical reality, quantities had to be given names and symbols," he argued. "There had to be a system to show how these quantities could be grouped together or taken apart. The writing down of this equation gave us the ability to present ideas, to discuss concepts, which led to an ever-expanding sphere of knowledge."

Other simple equations that were proposed included v = H0d, which Edwin Hubble composed in 1929 to describe the fact that the galaxies are moving away from us at a speed, v, that is proportional to their distance, d, where H0 is the Hubble constant. Balagoj Petrusev, an undergraduate student at the Institute of Physics in Skopje, Macedonia, suggested the Hamiltonian variational principle in the form δS = 0. A proper selection of the form of SPhysics World last month (September p64). articulates "a universal principle that stands true in classical mechanics, classical electrodynamics, relativistic mechanics, non-relativistic quantum mechanics and so on". In fact, Andy Hone from the University of Kent wrote a eulogy to this equation in

The unifying power of a great equation is not as simple a criterion as it sounds. A great equation does more than set out a fundamental property of the universe, delivering information like a signpost, but works hard to wrest something from nature. As Michael Berry from Bristol University once said of the Dirac equation for the electron: "Any great physical theory gives back more than is put into it, in the sense that as well as solving the problem that inspired its construction, it explains more and predicts new things" (Physics World February 1998 p38).

Great equations change the way we perceive the world. They reorchestrate the world -- transforming and reintegrating our perception by redefining what belongs together with what. Light and waves. Energy and mass. Probability and position. And they do so in a way that often seems unexpected and even strange.

For this reason, several respondents proposed equations that linked two or more disparate concepts, concrete and abstract things, the visible and the invisible. They included Boltzmann's equation S = k InW.

It relates entropy, S, which emerged as a concept during the development of thermodynamics early in the 19th century, and a purely abstract quantity, W, that emerged from the statistical treatment of systems with many degrees of freedom. Bragg's equation (nλ = 2dsinϑ)wrote another respondent, "links diffraction spots (visible reality) with the underlying crystal structure (invisible reality) and can be easily visualized with a standard textbook picture."

One of the most frequently mentioned equations was Euler's equation, eiπ + 1 = 0. Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics -- once and only once -- in a single expression. These are: e (the base of natural logarithms); the exponent operation;π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.

Practicality

Many respondents were impressed by equations that have a practical influence on human life. These included: the compound-interest equation, the implications of which from the Renaissance to the present are "obvious, staggering and unwelcome"; income-tax formulae; the simple ratio a/b = c/d, which is basic to construction, surveying and so forth; simple electrical equations, such as V = IR; basic mechanical equations, such as work done = force x distance; Shannon's capacity equation, which relates to the modern world through the Internet and digital communication; and, last but not least, Pythagoras's theorem.

Roger Bailey nominated the "sunrise equation" cos(time) = -tan(lat) x tan(dec), which identifies the time of sunrise or sunset as a function of latitude and solar declination. This, he pointed out, is "fundamental to our sense of time" and it "fits on a T-shirt". Engineer John Wilcher suggested the ideal-gas law, PV = nRT, pointing out that "the relation of pressure, volume and temperature is relevant to almost everything we do", including common but often overlooked uses such as car tyres, angioplasty procedures and oil drilling.

Meanwhile, Iain Christison, an emeritus professor of animal agriculture at the University of Saskatchewan, Canada, suggested EM = H + P -- metabolized energy equals heat plus product. It describes the fact that "all of the useful energy consumed by animals, including people, is released as heat or stored as product". The equation, he added, "carries within it an intricate balance of cause and effect that influences all of us with every mouthful and with every step".

Historical relevance

Some respondents proposed equations that played key roles in the history of science. For example, Alan Denham proposed the Balmer series 1/λ = R (1/n12 + 1/n22). The long history of this equation stretches from Fraunhofer's studies of the spectrum of sunlight in 1814, to Kirchhoff's suggestion in 1859 that each atomic species has a unique spectrum, to Angstrom's publication of the wavelengths of a thousand Fraunhofer lines in 1868, to schoolteacher Johann Balmer, who in 1885 noticed that the frequencies of light emitted by hydrogen atoms were mathematically related. The equation's history was continued by Lyman's observations in the ultraviolet region and by others in the infrared, by Rydberg (who gave his name to the constant R in the equation), and by Bohr, whose work in 1913 explained the equation. "As soon as I saw Balmer's formula the whole thing was immediately clear to me," Bohr once said.




These equations are listed in order of the number of people who proposed them. The first two received about 20 mentions each out of a total of about 120; the rest received between two and 10 each. Equations are given, where appropriate, in their most common form.

"Thus this century-long story," Denham wrote, "involving the theoretical and practical investigation of science by some of its most distinguished practitioners, would be incomplete without giving due honour to the contribution of a secondary-schoolmaster who spotted that the published scientific data conformed to a pattern that none of the scientists of his day were aware of."

Maxwell's equations

The responses suggested that there is no single criterion for greatness, and that a truly great equation ranks high in each of the above criteria. However, most votes were given to Euler's equation and to Maxwell's equations, which describe how an electromagnetic field varies in space and time. Although Maxwell's equations are relatively simple, they daringly reorganize our perception of nature, unifying electricity and magnetism and linking geometry, topology and physics. They are essential to understanding the surrounding world. And as the first field equations, they not only showed scientists a new way of approaching physics but also took them on the first step towards a unification of the fundamental forces of nature. A firm called Ocean Optics in Florida even sells T-shirts with Maxwell's equations on.

Tony Watkins recalled how he learned the equations during his second year as an undergraduate at Southampton University almost 20 years ago. "I still vividly remember the day I was introduced to Maxwell's equations in vector notation," he wrote. "That these four equations should describe so much was extraordinary...For the first time I understood what people meant when they talked about elegance and beauty in mathematics or physics. It was spine-tingling and a turning point in my undergraduate career. After a year of rapidly dwindling interest in physics (and rapidly decreasing results!), my passion was reignited by four lines of symbols." He even renamed his next bicycle Maxwell in honour of the great man, having previously ridden on his Carnot Cycle. Sadly for him, he never got round to learning tensors to see Maxwell's equations expressed even more simply.

The critical point

Nobody accepted my invitation to discuss why greatness in equations matters, which leaves me free to address the topic myself. Debating the issue has drawbacks, for it can foster the idea that equations are independent tools rather than embedded in networks of other equations, practices and information. Nevertheless, it helps us to recollect, among other things, what Richard Harrison called that "moment of wonder" that was apparent in his son's contemplation of 1 + 1 = 2.

As adults, we lose that wonder. We come to think of equations as just another set of tools that lie about ready-at-hand in the world. We lose our appreciation for their origin, thinking that they are not really of human origin: on the eighth day, God created equations as the blueprint for His recent work. As Galileo wrote -- disingenuously, polemically -- the Book of Nature is written in mathematical symbols. That's untrue, of course. We write, and continually rewrite, the book of nature.

As the philosopher Immanuel Kant once wrote: "When we discover that two or more heterogeneous empirical laws of nature can be unified under one principle that comprises them both, the discovery does give rise to a noticeable pleasure...even an admiration that does not cease when we have become fairly familiar with its object". This delight is more than having our expectations fulfilled or surprised, more than about the domination and control of nature, more than a biological product. The pleasure, Kant continued, is a feature of the exercise of the human intellect. "Even the commonest experience would be impossible without it," he wrote, which is why we "gradually come to mix it with mere cognition and no longer take any special notice of it."

In reawakening that sense of wonder, debating what makes equations great therefore re-educates us about the fundamental nature of science, and knowledge, itself.