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Home » Page 1786

The greatest equations ever

In It Must Be Beautiful: Great Equations of Modern Science, Graham Farmelo assembled and edited essays on 11 great equations of the 20th century. Six are from physics: E = hf, E = mc2, Einstein’s general-relativity equation, the Schrödinger wave equation, the Dirac equation and the Yang-Mills equation. The other five include the Drake equation on the likelihood of us forming radio contact with extraterrestrial life, and Shannon’s equations on information transmission.

Farmelo shuns defining greatness in equations, but compares them with poems. Both are composed of abstractions with which we address the world, even though many individual terms do not refer to anything specific. While “poetry is the most concise and highly charged form of language”, he says, equations are “the most succinct form of understanding of the aspect of physical reality they describe”. We sense greatness in equations as well as poems, even though we do not have an objective measure for it.

Wardrobe numbers and overcoats

Whether a particular equation is “great” obviously has something to do with the properties of the equation itself, such as simplicity and symmetry. This would seem to favour three-letter equations like F = ma, E = hf and E = mc2.

But clearly other criteria come into play as well. For example, we have to consider the relationship of the equation with the world – otherwise it would just remain hieroglyphs. Uniquely, equations do not refer directly to things but to quantities measured from special situations staged in the laboratory. Force, energy, time or acceleration do not lie around like ordinary objects; measurements of these quantities have to be “read off” from events that have been specially conceived, prepared and systematized. The relationship between a measurement and what it measures is thus not like that of a word to an object, but – as Einstein once remarked – more like that of “wardrobe number to overcoat”. Equations, as it were, link the wardrobe numbers to one another – what links wardrobe numbers to overcoats is laboratory preparation and measurement.

To use an analogy of the science philosopher Patrick Heelan, a laboratory is like a garden where special kinds of things are grown in an environment that is isolated (although never completely) from the life outside. The special things that emerge within the laboratory walls are thus artifacts – like greenhouse orchids – which may exist only momentarily, but their properties help us to understand and explain that wider and wilder external life. The laboratory creates the conditions under which special things appear that show themselves as structures of the world. Neither the world inside nor outside the laboratory is a static environment, however; both are mediated by technology and continually changing, which allows new things, concepts and interests.

Whether an equation is great, it seems to me, has something to do not only with the properties of the equation, but also with the scope and depth of the phenomena to which it refers. This would favour equations dealing with fundamental things such as space, time, fields and energy. Great equations can seem to be “wiser even than their discoverers” about such fundamental things, as Hertz said of Maxwell’s equations, for “we get more out of them than was originally put into them”. This is why, Hertz felt, that “mathematical formulae have an independent existence of their own”.

Cultural flesh

Inside and outside science, furthermore, equations can acquire what one might call a cultural “flesh”. That is, equations are more than bare and abstract scientific tools but can develop a lore, history and meaning of their own. This can happen to even the

most elementary of equations. In US high schools, students are often reminded of Ohm’s law using the phrase “Rhode Island equals Vermont”. During the Second World War, faced with the urgency to churn out radio operators speedily, one radar school outside Chicago taught it in three versions – V = IR, I = V/R and R = V/I – because it was faster to teach these equations than it was to teach trainees how to manipulate equations.

Equations can exemplify epistemological and moral lessons about science, and appear to be signposts toward entry into nature’s deeper mysteries. The convoluted story of Schrödinger’s equation, and its competition with Heisenberg’s matrix mechanics, is sometimes used to instruct physics students on the different ways of practising science.

In popular culture, meanwhile, E = mc2 has come to stand for science – even human knowledge – itself. It is a staple of popular cartoons and images of science, and even turned up in the recent movie School of Rock, in which a washed-out rock musician is stuck teaching junior-high-school kids.

The French intellectual Roland Barthes observed that while photographs of Einstein often show him next to a blackboard covered with impenetrable symbols and equations, cartoons often portray him, chalk in hand, next to a clean blackboard on which he has written down this particular formula as if it had just come to him. Barthes observed that this equation restores the image of “knowledge reduced to a formula…science entirely contained in a few letters”. It has become a Gnostic image symbolizing knowledge at once ultimate and esoteric: “The unity of nature, the ideal possibility of a fundamental reduction of the world, the unfastening power of the word, the age-old struggle between a secret and an utterance, the idea that total knowledge can only be discovered all at once, like a lock that opens after a thousand unsuccessful attempts.”

The critical point

Equations were not always as prominent in, nor as identified with, scientific methodology as they are now. At the beginning of modern science, laws such as Galileo’s law that we now express in the form of equations (d = 1/2at2) were expressed through proportionalities (d α t2). One may well wonder about the role played by the increasing centrality of equations in the advance of science.

The equations in Farmelo’s book are all from the 20th century. Which equations would be on the list if it were expanded to include the greatest equations of all time? I invite you to send me your candidates, the reasons why they deserve to be on the list, and what value, if any, you find in discussing their greatness. I shall report on the results in a future column.

  • What is your shortlist of the greatest equations in science, and what makes them great? Send your thoughts to Robert P Crease at the address or e-mail given below, or by fax to +1 631 632 7522.

New challenges for the EU

As membership of the European Union increases from 15 to 25 states this month, research and development has a higher profile than ever before among the member states. The powers that be within the EU have slowly come to realize that the only way that Europe can compete with the US and Japan is to increase its investment in research. Moreover, it seems as if the shortcomings in the EU’s approach to R&D are finally being recognized, although it remains to be seen if this will lead to meaningful change.


The good news for researchers is that the EU is determined to increase spending on R&D from the 2% of gross domestic product (GDP) at present to 3% by 2010 to match current levels in the US (2.8%) and Japan (3%). The bad news is that R&D spending in the EU remained essentially constant between 2001 and 2002, and actually fell in France and the UK. The expansion of the EU will also make the 3% target more difficult to achieve because the GDPs of the accession states are likely to rise rapidly after they join, making it difficult for R&D expenditure to keep up. In terms of basic science, it is generally agreed that the EU matches the US in terms of quantity but not quality or impact.

The 10 new member states are a diverse bunch when it comes to science and research. Some, such as Poland and Hungary, have long track records, especially in physics, but others, to be blunt, do not. Overall, however, the 10 only spend €3bn per year on R&D, compared with €175bn for the 15 existing member states. However, countries such as Greece, Portugal and Spain should note that after two decades in the EU they spend less on R&D as a fraction of GDP than Slovenia and the Czech Republic.

It is good that R&D features so high on the EU agenda, but how can intentions be turned into actions? Much of the responsibility lies with industry, because the bulk of the gap between the EU and its major rivals is due to the historically low levels of investment in R&D by European companies (and high levels of expenditure on defence R&D in the US). Moreover, firms in Europe employ less than half the number of researchers that their US competitors do.

The European Commission also needs to change its approach to R&D. It is disappointing that the current Sixth Framework Programme, which will spend €17.5bn on R&D between now and 2006, is attracting many of the same criticisms as its predecessors – everything is too complicated, success rates are too low and too much of the money goes to industry. There are suggestions that the budget of the next Framework might be double that of the current programme, but the money will be wasted unless there are major changes in approach.

And what of the much discussed European Research Council (ERC) for basic research? The EU is currently considering a proposal from an “expert group” chaired by Federico Mayor to establish a European Fund for Research Excellence that would be overseen by such a council (www.ercexpertgroup.org). Many of the ideas put forward by the group – and by other interested parties like the Royal Society in the UK – have merit. To be specific, the ERC needs its own budget (ball-park figures are €1-2bn per year) and real autonomy from the Framework programme. Moreover, existing budgets for R&D should not be reduced to pay for the ERC, and, crucially, only those projects that are truly excellent must be funded.

The EU needs to take these messages on board and accept the fact that ERC research grants are more likely to go to institutes and nations that are already have strong track records. There are enough other EU schemes to support less prosperous regions – including various regulations in the Framework programme – for this not to be a problem.

Accelerating universe will limit technology

It is well known that the amount of information that can be processed and stored in any device is ultimately limited by the laws of quantum mechanics. However, Krauss and Starkman have shown that the nature of the universe itself also places limits on computation because it is not possible to transmit or receive information beyond the so-called global event-horizon in an accelerating universe.

The acceleration of the universe is driven by something that has repulsive rather than attractive gravitational interactions. However, although this so-called “dark energy” is thought to account for around two-thirds of the universe, no one knows what it is made of. Possible explanations for dark energy include a “cosmological constant” or something known as quintessence.

Krauss and Starkman have determined how far an observer could travel in such a universe and still be able to transmit energy back to Earth. They then determined how much energy could be transmitted this way. To calculate the total amount of information that could be processed, they assumed that the universe has a minimum temperature, below which no energy — and therefore no information — can be extracted. Theory predicts that this minimum temperature exists if the universe has a cosmological constant.

The duo calculated that the total number of computer bits that could be processed in the future would be less than 1.35×10120. This means that the effective information available to any observer within the event horizon of an expanding universe will be significantly less than the total so-called Hawking-Beckenstein entropy — the entropy that is associated with a black hole — in the universe. Many cosmologists predict that an accelerating universe will ultimately contain nothing but black holes, which will then eventually disappear themselves.

“It is remarkable that results from cosmology can provide such definite limits on the nature of technology,” Krauss told PhysicsWeb. “In addition, it is also remarkable that simple laws of physics put such robust constraints on life, and technology, even when we don’t know what that technology will be like.”

Krauss expects that the work will generate discussion on the limits of computation, intelligence, consciousness and civilization. “Ultimately arguments of this kind will have an impact on how we view ourselves, and our place in the universe — which is what science is all about.”

Viruses go from strength to strength

The shell or “capsid” of a virus is made of protein and is intended to protect the DNA in the virus when it attacks another cell. The structure of these capsids is well understood, but very little is known about their mechanical properties.

Wuite and co-workers started by placing the sharp tip of an atomic force microscope (AFM) onto the shell of a bacteriophage – a virus that infects bacteria. Next, they slowly increased the force applied by the tip and recorded how the shell deformed. They operated the AFM in a so-called “jumping mode”, which allowed them to carefully control the maximum force exerted on the shell.

From these measurements, Wuite’s team calculated that the shell had a Young’s modulus of 1.8 gigapascals, which is comparable to that of hard plastic. Moreover, they found that the shells could withstand forces of several nanonewtons and could be flattened by up 30% of their original height without cracking. “It came as a surprise to us to find that the bacteriophage capsid was as strong as it was,” Wuite told PhysicsWeb.

“From a medical point of view, this research might reveal new insights into the transport strategies of different viruses and the shell strength might also relate to the time a virus stays infectious outside a host cell,” said Wuite. “From a nanotechnology point of view, viral capsids could be used as nanocontainers which are strong and able to self-assemble. The capsid proteins could also function as building blocks to make other complex structures.”

The team is now investigating other viruses using this technique.

Dark matter remains at large

Dark matter was originally proposed by astronomers to explain why galaxies rotate much faster than can be explained by the amount of visible matter they contain. This mysterious form of matter does not emit or absorb electromagnetic radiation — hence the name “dark” — and can only be detected by its gravitational influence on ordinary matter. Black holes and other objects are known to make up some of the dark matter in our galaxy. However, many cosmologists believe that galaxies also contain exotic particles left over from the big bang. These include so-called weakly interacting massive particles (WIMPS) and other particles not included in the Standard Model of particle physics.

The CDMS II experiment is located in a disused iron mine several hundred metres below ground in Soudan, Minnesota. It is necessary to build dark-matter detectors underground to shield them from cosmic rays and background radiation, which can mimic a dark matter signal. WIMPs are extremely difficult to detect because they rarely interact with ordinary matter. The CDMS II team hopes to detect these rare interactions by measuring the charge and vibration produced by particles as they pass through a tower of germanium-silicon detectors maintained at cryogenic temperatures. It should be possible to identify WIMPs because they release less charge than other particles for the same level of vibration.

The first results from CDMS II show with 90% certainty that the interaction rate of a WIMP with a mass of 60 gigaelectronvolts must be less than about one every 25 days per kilogram of detector. According to the CDMS II team, these measurements are at least four times more sensitive than the best previous result, obtained by the EDELWEISS detector in France. The scientists now hope to improve this sensitivity by a factor of 20 or more.

The extension of the Standard Model favoured by most particle theorists — supersymmetry — predicts that WIMPs should be found in the region of parameter space that CDMS II is entering. “The discovery of WIMPs and thus supersymmetry would simultaneously solve the most important problems in particle physics and cosmology,” says Blas Cabrera of Stanford University, one of the spokespersons for CDMS II.

Physics meets archaeometry in ancient Greece

Pantos and co-workers analysed a “Corinthian-type” battle helmet that dates from the 7th century BC and is currently being exhibited at Manchester Museum. Alistair Jackson, an art historian at the museum, suggested that the helmet was made by beating out a single lump of bronze — a technique that was so efficient that it was still being used in Italy in the 15th century — but he also suspected that the nose-guard of the helmet might date from much later.

To investigate this, Pantos and co-workers subjected the helmet to a variety of techniques — surface X-ray diffraction, X-ray fluorescence and infrared spectroscopy — using the Synchrotron Radiation Source (SRS) at Daresbury. They also used the ISIS neutron source at Rutherford to determine the microstructure of different parts of the object. Neutrons are able to probe the bulk of the object, as well as the surface.

“We have confirmed Alistair Jackson’s suspicions that the nose-guard is indeed a modern prosthesis,” Pantos told PhysicsWeb. According to the team, the nose-guard dates from the 19th century and was probably added by the person who found the helmet. It is made from a copper-zinc alloy (brass) while the helmet itself is made of copper and tin (bronze).

Moreover, information about the orientation of crystallites in the bronze provided by neutron texture analysis supports Jackson’s theory that the helmet had been hammered out from a single piece of alloy.

“We believe that this high-profile case will now encourage other investigators and museum curators to use our approach,” said Pantos. “We have the potential at SRS and ISIS to probe whole statues, whether made of bronze or even marble.”

Nanotube transistors speed up

The feature sizes in conventional microelectronic circuits are getting smaller and smaller and look set to reach the limit imposed by the fundamental properties of silicon in a decade or so. The semiconducting properties of carbon nanotubes – rolled up sheets of graphite just nanometres in diameter – make them a promising alternative to silicon, and nanotubes have already been used to fabricate a variety of electronic components, including diodes and field-effect transistors.

Conventional transistors have three terminals: the source, drain and gate electrodes. The gate controls the electron density in the central region of the transistor, which is usually made of a semiconducting material. If the electron density is high, current flows from the source to the drain. However, current does not flow if the electron density is low. This property allows the transistor to operate as a switch.

Burke and colleagues made their transistor by sandwiching a semiconducting single-walled nanotube between source and drain electrodes made of gold (see figure). When they varied the gate voltage in the device, they found that the circuit operated at 2.6 gigahertz (2.6 x 109 Hertz). This means that current can be switched on and off in about 0.1 nanoseconds, making it the fastest nanotube transistor made to date.

At present, the device only works at 4 kelvin but Burke is confident that it can be made to operate at room temperature. Moreover, he believes that the transistor could be made to switch at even higher frequencies. “I estimate that the theoretical speed limit for these transistors should be terahertz (1012 Hertz),” he said. “This is about 1000 times faster than modern computer speeds.”

Carbon-50 makes its debut

The most common fullerene is carbon-60 – also known as buckminsterfullerene or “buckyball”. This molecule, which contains 60 carbon atoms arranged in a spherical structure made up of pentagons and hexagons, was first created in 1985. Since then larger fullerenes containing between 70 and 500 carbon atoms have also been produced.

All the fullerenes made so far obey the isolated pentagon rule (IPR): this rule states that the most stable molecules are those in which every pentagon is surrounded by five hexagons. However, it is not possible to satisfy this rule in a molecule with fewer than 60 carbon atoms. This means that so-called non-IPR fullerenes should have unusual properties, but it also makes them structurally unstable and difficult to synthesise. Until now, fullerenes with fewer than 60 carbon atoms have only ever been made in the gas phase.

Zheng and colleagues succeeded in stabilising and capturing solid-state carbon-50 molecules using a graphite arc-discharge method. They added 0.013 atmospheres of carbon tetrachloride vapour to 0.395 atmospheres of helium in a sealed stainless steel vessel and then applied an electric field of 24 Volts. After purifying around 90 grams of soot that contained carbon-50 chloride (C50Cl10), they obtained about 2 milligrams of C50Cl10 that was 99.5% pure.

“The C50Cl10 looks like a spacecraft or a spinning planet with 10 reactive carbon-chlorine arms ready for further chemical functionalization,” team member Su-Yuan Xie told PhysicsWeb (see figure). Like derivatives of carbon-60 and 70, Xie says that carbon-50 could easily react with a variety of organic groups to form new compounds with interesting chemical and physical properties. Moreover, the technique could also be extended to synthesise other small fullerenes, such as carbon-54 and carbon-56.

Making new minerals on the Moon

Although the Moon has not been hit by a large meteorite for about 3.8 billion years, it has been continually bombarded since then by much smaller interplanetary rocks called micrometeorites. These tiny objects, which can be just tens of microns across, have been responsible for altering the lunar surface over time.

Micrometeorites travel at speeds of up to 100 000 kilometres per hour, and the heat generated when they crash into the Moon can flash-melt the silicate rock particles present on the lunar surface. Splashes of this molten material can then vaporize and dissociate into their constituent elements before coating other fragments of rock in a process is known as “space weathering” (see figure). This process is virtually unknown on Earth.

Lawrence Taylor and Mahesh Anand from the University of Tennessee, together with co-workers at the Vernadsky Institute of Geochemistry in Moscow and the Carnegie Institution of Washington, analysed a piece of rock from the Moon that had been found in Oman in 2000. They expected to find grains of pure iron in their sample, but instead found that the rock contained hapkeite – a mineral made up iron and silicon in a ration of two-to-one – and other iron-silicon phases.

The discovery was completely unexpected, says Taylor. At first he thought that the surface of the sample was tarnished as a result of oxidation. “I asked Mahesh Anand to perform electron microprobe analyses on these grains to see how much nickel and cobalt they contained,” he told PhysicsWeb. “Little did I know that the grains were iron silicides – the first report of such minerals in lunar samples.”

According to Taylor, the minerals were formed when elemental iron and silicon re-combined from the vapour state. He believes they provide direct evidence for the space-weathering process, and show how important it is for creating the soil on the Moon. The team named the mineral hapkeite after Bruce Hapke of the University of Pittsburgh, who predicted some 30 years ago that vapour-deposited coatings would be found on lunar grains.

Physics and fame

ben-Avraham and colleagues started by compiling a list of 449 condensed-matter physicists who have published papers on the cond-mat pre-print server. Then they determined how “famous” each scientist was by counting the number of Google hits their name produced in a search. To calculate “merit” they counted the number of papers that he or she had posted in cond-mat since it began in 1991. They found that fame is linearly proportional to merit.

This is completely different to what is found for people who enjoy what the Clarkson physicists describe as “true fame”, such as sports stars and actors. Fame for these people increases exponentially with merit. Moreover, fame for the truly famous follows a power-law distribution, whereas it falls off more rapidly for scientists.

“The simplest explanation for this is that scientists cite their colleagues in web pages, in relation to their published work,” ben-Avraham told PhysicsWeb. “Thus, more work published results in an equivalent increase in citations in web pages. Essentially, the fame of scientists is limited to within their peer group, hence the title of our paper “How famous is a scientist? – Famous to those who know us.”

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