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Home » Page 1739

Pythagoras

Pythagoras’s theorem changed the life of the British philosopher Thomas Hobbes (1588- 1679). Until he was 40, Hobbes was a talented scholar exhibiting modest originality. Versed in the humanities, he was dissatisfied with his erudition, and had little exposure to the exciting new breakthroughs achieved by Galileo, Kepler and other scientists who were then revolutionizing the scholarly world.

One day, in a library, Hobbes saw a display copy of Euclid’s Elements opened to Book I Proposition 47, Pythagoras’s theorem. He was so astounded by what he read that he used a profanity that his first biographer, John Aubrey, refused to spell out: ” ‘By G__’ Hobbes swore, ‘this is impossible!’.” He read on, intrigued. The demonstration referred him to other propositions, and he was soon convinced that the startling theorem was true.

Hobbes was transformed. He began obsessively drawing figures and writing calculations on bed sheets and even on his thigh. His approach to scholarship changed. He began to chastise philosophers of the day for their lack of rigour and for being unduly impressed by their forebearers. Hobbes compared other philosophers unfavourably with mathematicians, who proceeded slowly but surely from “low and humble principles” that everyone understood.

In books such as Leviathan, Hobbes reconstructed political philosophy by establishing clear definitions of terms, then working out implications in an orderly fashion. Pythagoras’s theorem had taught him a new way to reason and to present persuasively its fruits.

Before Pythagoras

Pythagoras’s theorem is important for its content as well as for its proof. But the fact that lines of specific lengths (3, 4 and 5 units, say) create a right-angled triangle was empirically discovered in different lands long before Pythagoras. Another empirical discovery was the rule for calculating the length of the long side of a right triangle (c) knowing the lengths of the others (a and b), namely c2 = a2 + b2.

Indeed, a Babylonian tablet from about 1800 BC shows that this rule was known in ancient Iraq more than 1000 years before Pythagoras, who lived in the sixth century BC. Ancient Indian texts accompanying the Sutras, from between 100 and 500 BC but clearly passing on information of much earlier times, also show a knowledge of this rule. An early Chinese work suggests that scholars there used the calculation at about the same time as Pythagoras, if not before.

But what we do not find in these works are proofs – demonstrations of the general validity of a result based on first principles and without regard for practical application. Proof was itself a concept that had to be discovered. In Euclid’s Elements we find the first attempt to present a more or less complete body of knowledge explicitly via proofs.

Euclid does not mention Pythagoras, who lived some 200 years previously, in connection with Proposition 47. We credit it to Pythagoras on the authority of several Greek and Latin authors, including Plutarch and Cicero, who wrote half a millennium after Pythagoras. These authors seem to be relying, in turn, on a single source – a certain Apollodorus – about whom next to nothing is known. Apollodorus does not even show how Pythagoras originally proved the theorem.

Pythagoras’s theorem is unique for the peculiar way in which it has become a challenge to devise new proofs for it. These proofs are not necessarily any better; most rely on the same axioms but follow different paths to the result. Leonardo da Vinci, Christiaan Huygens and Gottfried Leibniz contributed new proofs. So did US Congressman James Garfield in 1876, before he became the 20th US president.

Indeed, more than a dozen collections of proofs of Pythagoras’s theorem have appeared. In 1894 the American Mathematical Monthly began publishing proofs, but stopped after about 100. That did not prevent one reader – a teacher from Ohio called Elisha S Loomis – from publishing a book with 230 proofs in 1927; its second edition in 1940 contained 370. The Guinness Book of World Records website, under “Most proofs of Pythagoras’s theorem”, names someone who, it is claimed, has discovered 520 proofs.

The appeal of the theorem

One may wonder what there is to gain by proving a theorem over and over again in different ways. The answer lies in our desire not merely to discover, but to view a discovery from as many angles as possible. But what is it that is so fascinating about Pythagoras’s theorem in particular? First, the theorem is important. It helps to describe the space around us and is essential not only in construction but – suitably adapted – in equations of thermodynamics and general relativity. Second, it is simple. The Hindu mathematician Bhaskara was so enamoured of the visual simplicity of one proof that he redid it as a simple diagram – and instead of an explanation wrote a single word of instruction: “See”.

Third, it makes the visceral thrill of discovery easily accessible. In an autobiographical essay, Einstein wrote of the “wonder” and “indescribable impression” left by his first encounter with Euclidean plane geometry as a child, when he proved Pythagoras’s theorem for himself based on the similarity of triangles. “[F]or anyone who experiences [these feelings] for the first time,” Einstein wrote, “it is marvellous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking.”

The critical point

Small wonder that Pythagoras’s theorem became a model of what a proof is and does. In Plato’s dialogue Meno, for instance, Socrates coaxes a slave boy (ignorant of geometry) to prove a simplified version of the theorem: that the area of the square formed on the diagonal connecting the corners of another square is twice the area of the first square. Socrates leads the boy to see the inadequacy of the obvious answers, provoking bewilderment and curiosity. Then he helps the boy to recast the problem within a larger, richer context where the path to the solution is clear. Socrates does this exercise not to educate the slave boy, but to illustrate to his owner what learning is all about.

For Hobbes and countless others, Pythagoras’s theorem was far more than a means to compute the length of hypotenuses. It shows something more, the idea of proof itself. It provides what philosophers call categorical intuition; it reveals more than a bare content but a structure of reasoning itself. It is a proof that demonstrates Proof.

Shelf life: Dava Sobel

What are the three best popular-science books?

Three of my top picks are Silent Spring by Rachel Carson (1962) because it changed the world for the better; The Cosmic Connection by Carl Sagan (1973) because it was as movingly written as it was factually informative; and Naturalist by E O Wilson (1994) because the author shared his childhood fascination for his subject in a manner likely to inspire young readers to pursue careers in science.

What science books are you currently reading?

I am currently serving out the second year of my second term as a judge for the Los Angeles Times book prize in science and technology, which means I am trying to read every science book published in 2005, including Big Bang by Simon Singh and A Sense of the Mysterious by Alan Lightman.

What else are you reading?

Extremely Loud and Incredibly Close [a novel] by Jonathan Safran Foer.

Which popular-science book have you never read, but feel you ought to have tackled, and why?

Just Six Numbers, In the Matrix and other works by the cosmologist Martin Rees, because I am interested in his research and in the office of the Astronomer Royal, and because he seems to be such an interesting thinker.

What advice do you have for physicists who want to write a popular-science book?

I would advise them to pick one person they know – either friend or family, but not a scientist – to whom they might address what they have to say. I would then write the book with the fervent intention of making that person both understand the material in the story and enjoy the telling of it.

•The Planets was reviewed in Physics World last October, pp42-43 (see “Secret lives of the solar system “). The book was published exactly 10 years after Longitude, which has so far sold over 1.5 million copies in the US and UK

The might of military research

The problem with all books on military technology is that they deal with topics that are subject to secrecy, hype and misinformation. Judging the reliability of the information in such books can therefore be tricky. One good sign is when authors include plenty of high-quality references and can be seen to have drawn on technical reports as their main source of evidence. I was not therefore filled with confidence when I saw that this book’s bibliography opens with The Hitchhikers’ Guide to the Galaxy.

Although Weapons Grade is clearly aimed at military-technology enthusiasts, it does discuss issues of interest to a wider readership. David Hambling’s main thesis is that the military has historically had close links with science and technology, and played a sometimes pivotal role in certain technological developments in the civilian sector. Current military research is therefore where we will find many of the civilian technologies of tomorrow.

Hambling begins by outlining a number of historical case studies where military science and technology have led to civilian spin-offs. He cites familiar examples such as rocketry, aviation and nuclear power, but also includes less commonly discussed military research involving computers and lasers.

In his discussion of aviation, he traces the development of aircraft from airships and propeller-driven planes to turbojets, pulse jets and turbofans. He looks at how military research – especially in Germany, the US and the UK before, during and after the Second World War – led to rapid developments in aircraft technology that eventually changed the face of everyday travel.

Hambling also examines some areas where military research did not lead to civilian spin-offs, despite early indications that they might. Examples here include atomic aircraft, rocket planes and atomic ships.

The author then looks at where military research might take us. He starts by outlining major research programmes mainly in the US, which cover everything from airborne robots and hypersonic bombers to nano-sized explosives and “e-bombs”, which could potentially render electronic equipment useless from a distance. There are also projects on artificial intelligence, studies of millimetre-wave vision that could let soldiers see through solid objects, and plans to design “less lethal” weapons, such as microwave-based armaments.

For each area, Hambling outlines a few possible future civilian spin-offs, such as space tourism, “mood enhancement” using magnetic fields, and surveillance and crowd-control technologies for the police. While Hambling makes a concerted effort to avoid the common exaggerations to which many authors in this area are prone, the book nevertheless has some significant shortcomings.

First, the book’s inadequate references make it hard for the reader to critically assess the current state of the technology, although the author does acknowledge the difficulty in getting reliable information about military research. Second, while some of the developments the author examines sound highly plausible, others seem to be on the fringes of science fiction. His unnecessary discussion of UFOs and repeated references to science-fiction films do not help us judge which is which.

These problems are compounded when Hambling starts to discuss possible civilian spin-offs, which is perhaps why several of the later chapters only devote a page or two to limited – and sometimes quite speculative – discussions of the possibilities. Given that one of the main aims of the book is to look at the possible benefits to the civilian world of current military research, such limited material leaves the reader feeling short-changed.

To his credit, Hambling does in a couple of cases mention that the negative effects of such spin-offs might outweigh the positive. For example, the work on “less lethal” weapons is likely to lead to technologies that will be procured by undemocratic regimes, possibly making them much harder to dislodge. He also outlines how some weapons – such as e-bombs – may be especially attractive to terrorists. Unfortunately, in areas such as computer surveillance or magnetic brain stimulation, the possible misuses get very scant attention.

Another significant flaw with the book is the paucity of policy analysis. Hambling does briefly discuss a few issues – for example how high levels of secrecy can impede the uptake of military technologies by the civil sector, or how Russia’s preference for simple, reliable military technologies restricted their hi-tech programmes – but any broader analysis is missing.

That is a pity because science policy, economics and global-security issues have a huge impact on military research and how it affects society. A fuller analysis would have revealed the flaw in thinking that today’s military research might lead to lots of civilian spin-offs. After all, many of the technological developments that emerged from the military during and after the Second World War did so because a far higher proportion of research funding was spent on defence than is the case now.

Some of the author’s historical accounts are also problematic, particularly when it comes to the civilian application of military research. Sure, mobile phones have some roots in military communications research, but it has taken huge investment over several decades by the civilian sector to turn this technology into something that is widely used. It is oversimplistic to point to a particular source of research funding a long time in the past and say that society would not have certain benefits today if the money had been spent on something else. Technological innovation is a far more complex process.

I would also like to have seen the author give more attention to major ethical issues. Hambling does admit that some civilian spin-off technologies like nuclear power pose serious security problems, but he fails to acknowledge just how widespread the potential problems are, especially if the military retains significant influence in a particular technological field. He also glosses over ethical issues related to the size of military research budgets.

As many defence economists have pointed out, using military investment in research to gain some future civilian spin-off is an ineffective way of using public money. At a time when we need more research into solving the urgent problems of international poverty and global environment threats, large military budgets seem to raise especially serious ethical issues.

Hambling is clearly very enthusiastic about military technology. Unfortunately, this leads to lengthy descriptions of military research programmes at the expense of discussing the broader issues that are just as important in determining when and whether military research leads to civilian spin-offs and – critically – the extent to which such spin-offs are good or bad for society. Overall, the book is a frustrating read.

Review of the year

1. January: All-silicon laser makes its debut
2. February: Saturn and Titan reveal their secrets
3. March: The passing of a legend
4. April: Negative refraction goes optical
5. May: Particle physicists discover new meson
6. June: Europe beats Japan to fusion prize
7. July: Quantum boost for optical clocks
8. August: Fibres control the speed of light
9. September: Comet reveals its secrets
10. October: New look for Hall effect
11. November: Electrons lose their mass in carbon sheets
12. December: Entanglement reaches new levels

1. January: All-silicon laser makes its debut

The year got off to a good start for solid-state physicists when researchers at Intel announced that they had made the first all-silicon laser. The laser was manufactured using standard semiconductor processing techniques and was heralded as bringing cheap, high-speed optical circuits one step closer. Integrating lasers and electronic devices on the same silicon chip is one of the holy grails of microelectronics. But most existing optical devices — such as the lasers inside DVD players — use exotic and expensive semiconductors like gallium arsenide or indium phosphide.

All-silicon laser makes its debut

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2. February: Saturn and Titan reveal their secrets

February saw the first flood of results from the Cassini–Huygens mission to Saturn. The Cassini mothership had started orbiting Saturn in June 2004 and eight new papers revealed evidence from that mission for new moons and ring structures. Later in the year, the European Space Agency’s Huygens probe, which landed on Saturn’s largest moon Titan in January, revealed fascinating new information about Titan’s nitrogen-rich atmosphere.

Cassini reveals Saturn’s secrets
Saturn’s moon reveals its secrets

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3. March: The passing of a legend

It was a sad month for physics when Hans Bethe — one of the giants of 20th-century physics — died on 6 March. Bethe was awarded the 1967 Nobel Prize for Physics for his theory of nuclear reaction inside stars and was one of the key figures in the Manhattan atomic-bomb project during the Second World War.

The year also saw the death of Joseph Rotblat, who was the only physicist to resign from the Manhattan project and who later worked tirelessly for a world free from nuclear weapons as founder and secretary-general of the Pugwash peace movement. Rotblat and Pugwash shared the 1995 Nobel Peace Prize. Other deaths during the year included Philip Morrison, Jack Kilby, Hermann Bondi and John Bahcall.

Atom bomb designer dies
Bomb builder turned critic passes away
Jack Kilby: 1923–2005
John Bahcall dies
Joseph Rotblat dies
Sir Hermann Bondi: 1919–2005

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4. April: Negative refraction goes optical

Physicists at Purdue University in the US announced in April that they had demonstrated “negative refraction” at optical wavelengths for the first time. The researchers obtained the result in a material consisting of an array of pairs of parallel gold nanorods, and said the structure could lead to optical “superlenses” that reflect no light and operate with sub-wavelength resolution. Negative refraction continued to be a hot topic throughout the year, with other papers describing an optical superlens from a thin layer of silver and the discovery of a new type of negative-index material made from layers of superconducting and ferromagnetic thin films.

Negative refraction goes optical
Superlens breakthrough
Ferromagnets and superconductors make negative-index materials

April also saw a rather over-hyped announcement by researchers at the Brookhaven National Laboratory in the US that they had found strong evidence for a “quark–gluon” plasma — the state of matter that is thought to have existed in the first millionth of a second after the Big Bang. The researchers stopped short of saying they had actually discovered the elusive state of matter, which consists of quarks, antiquarks and gluons. But they did find evidence suggesting the quark-gluon plasma behaves more like a liquid than a gas as was previously thought.

Quark-gluon plasma goes liquid

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5. May: Particle physicists discover new meson

With the Large Hadron Collider still being built at the CERN lab in Geneva, it was a quiet year for much of particle physics. However, in May members of the Belle collaboration at the KEK laboratory in Japan discovered the first “hybrid meson”. The particle, first predicted over 25 years ago, appears to contain a gluon in addition to the quark and antiquark that are usually found in mesons. The new meson, which decays into two well-known particles called the Omega and J/psi, adds to a string of other hadrons with mysterious properties that have turned up at KEK and other laboratories in the last year or so. However, 2005 also saw the final chapter of the “pentaquark” story, with dedicated searches at the Jefferson Laboratory in the US ruling out the existence of the five-quark state that had been tantalizing experimentalists since 2003.

Particle physicists discover new meson

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6. June: Europe beats Japan to fusion prize

After long, drawn-out and frankly rather dull negotiations, it was finally announced at the end of June that France had won the prize to host the €10bn International Thermonuclear Experimental Reactor (ITER). The decision to build the reactor at Cadarache in southern France came after an 18-month battle between the European Union (EU) and Japan. Two of the project’s six partners — Russia and China — had backed the EU bid, while South Korea and the US supported the site in Japan. As a sop, Japan was allowed to pick Kaname Ikeda as boss. It will also be given more than its fair share of industrial contracts when construction begins. ITER will be the next step before a prototype commercial fusion reactor dubbed DEMO is built, and could be ready by 2016

Europe beats Japan to ITER prize

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7. July: Quantum boost for optical clocks

This year saw physicists celebrate the golden jubilee of atomic clocks, which were invented in pioneering experiments at the National Physical Laboratory in the UK in 1955. Atomic clocks rely on microwave transitions in caesium atoms, but a new generation of devices based on much faster optical transitions could be even more accurate. Work on these “optical clocks” moved forward in July when physicists at the National Institute of Standards and Technology (NIST) in the US demonstrated a new form of laser spectroscopy that could lead to more accurate devices. Such clocks could lead to the second being redefined and could also be used to check if the fundamental physical constants are truly constant.

Quantum boost for optical clocks

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8. August: Fibres control the speed of light

Work on “slow light” led to a flurry of papers this year, including the demonstration by physicists in Switzerland that the speed of an optical pulse can be reduced in an optical fibre based on off-the-shelf technology. Over the past decade physicists have had to use exotic media such as ultracold atomic gases and various crystals to make “slow” or “fast” light. Although some of these techniques work at room temperature, they have never before been suitable for use in a fibre-optic network. The result could therefore lead to practical applications in optical delay lines, optical memories and, ultimately, the development of an all-optical router that speed up the flow of data over the Internet.

Later in the year, scientists at IBM built a silicon chip that combines miniature heaters and photonic crystals to control the speed of light pulses. The silicon waveguide circuit reduces the “group velocity” of light by a factor of up to 300.

Fibres control the speed of light
Silicon chip puts the brakes on light

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9. September: Comet reveals its secrets

It was a good month for astronomers when the first results from the Deep Impact mission to the comet Tempel 1 were released. The NASA probe — roughly the size of a washing machine — had crashed into the comet on 4 July after leaving its mothership a few hours earlier. The data showed clouds of dust and ice being ejected from the impact site. The impact was designed to disturb material inside the comet that were then analysed by the instruments on board the flyby spacecraft as well as various ground and space-based observatories.

Comet reveals its secrets

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10. October: New look for Hall effect

October saw physicists in France demonstrate the Hall effect with phonons — vibrations of a crystal lattice — for the first time. The classic Hall effect occurs when an electric current flows through a conductor in a magnetic field. If the current and magnetic field are at right angles to each other, electrons deflect to one side and a Hall voltage builds up in a direction that is at right angles to both the current and the magnetic field. It had been assumed that the Hall effect could not exist for phonons because they have no charge. But a team at the Grenoble High Magnetic Field Laboratory proved otherwise. When a heat current was passed in one direction down a crystal of terbium gallium garnet and a magnetic field was applied at right angles to it, the Hall effect was observed as a temperature difference normal to both directions.

New look for Hall effect

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11. November: Electrons lose their mass in carbon sheets

Exotic behaviour was discovered in two-dimensional sheets of carbon atoms by two teams of physicists from the UK, Russia and the Netherlands in November. The researchers found that the electrons in graphene behave like relativistic particles that have no rest mass and travel at about 106 metres per second. Although this is a factor of 300 slower than the speed of light in vacuum, it is still much faster than the speed of electrons in an ordinary conductor. Both teams also observed a new “half-integer” quantum Hall effect, which is the relativistic analogue to the conventional integer quantum Hall effect that is seen for free electrons in semiconducting systems.

Electrons lose their mass in carbon sheets

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12. December: Entanglement reaches new levels

Work on entanglement went from strength to strength in 2005 as physicists managed to entangle the largest number of particles ever. Two rival teams, one from the National Institute of Standards and Technology in the US and the other at Innsbruck University in Austria, entangled up to eight calcium ions. The results were the latest steps on the long road to large-scale quantum computers. Later in the month, physicists took another big step towards this goal with the creation of an ion trap on a semiconductor chip.

Entanglement reaches new levels
Ions trapped on a chip

And finally: The physics of everything

2005 saw a steady stream of papers on some unconventional topics. These include: fluids that “unmix”, a theory for exploding “dark matter balls”, and how sound can move faster than the speed of light. Other unusual papers described how physics can be used to analyse works of abstract art, explain how spaghetti breaks and how animals find things. A back-pack that generates electricity while you walk and a recipe for making “superstrings” in the lab were among the other less conventional results of the year (see below for a full list).

Devices controlled by thought move closer
How to make a blockbuster
A recipe for making strings in the lab
How animals find things
The physics of pasta
Power walking
Physics goes abstract
Could sound move at the speed of light?
Fluids mix in reverse
Exploding dark-matter balls predicted

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Ferromagnets and superconductors make negative-index materials

The existence of materials with a negative refractive index was first predicted by the Russian physicist Victor Veselago in 1967. He speculated that materials in which the electric permittivity and the magnetic permeability are both less than zero would refract light in the opposite direction compared to conventional materials. The electric permittivity is the ability of a material to store electrical charge and the magnetic permeability is the ability of a material to acquire magnetisation in magnetic fields.

In 2000, John Pendry, a theorist at Imperial College in London, showed that such materials could also behave as perfect lenses and so overcome the “diffraction limit” of the resolution of optical devices. Although negative-refractive-index materials do not occur naturally, several groups have successfully made them recently from metamaterials and photonic crystals.

The new material, made by Andrei Pimenov and colleagues of the University of Augsburg and co-workers at the Polish Academy of Sciences and Northern Illinois University consists of a multilayer stack of ferromagnetic manganese oxide and superconducting copper oxide thin films. With no external applied magnetic field the sample is superconducting and has negative electric permittivity. Then, when an external magnetic field of 3 Tesla is applied, the magnetic permeability becomes negative close to the so-called resonance field. In this way, both conditions for negative refraction are satisfied.

Pimenov and colleagues were able to directly determine the refractive index of the material as a function of applied magnetic field by measuring the field dependence of the transmittance and phase shift of the sample. Moreover, they found that the index of refraction can be tuned between negative and positive values by varying the strength of the applied magnetic field.

The physicists now plan to try and reduce the value of the magnetic field required to achieve negative magnetic permeability. They will also look at replacing the ferromagnetic layers with antiferromagnetic materials, which have “internal magnetism”. This means that negative refraction could be observed in the absence of an external magnetic field, says Pimenov.

Hubble reveals dark-matter details

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. According to the standard model of cosmology, the universe is thought to contain about 5% ordinary matter, 25% dark matter and 70% dark energy — the nature of which is unknown.

Galaxy clusters are the largest gravitationally bound systems in the universe and are composed of three major components: visible galaxies, a hot intra-cluster medium, and dark matter. Jee and co-workers employed a “gravitational lensing” technique to map dark matter from two galaxy clusters — thought to have formed when the universe was much less than half its present age — in the southern sky using the Advanced Camera for Surveys (ACS) instrument on the Hubble Space Telescope.

Astronomical bodies, such as galaxy clusters, behave as lenses because their gravitational fields bend the light emitted by background galaxies towards Earth as they pass throught the clusters. This means astronomers can observe dark matter despite it being invisible. Using the Hubble data eliminates the problem of atmospheric turbulence, which blurs similar images taken by ground-based based telescopes.

Using mass reconstruction techniques, Jee and colleagues show how “clumps” of dark matter surround the cluster galaxies (see figure). Although these clumps have been seen before, Jee and colleagues were able to image them in much more detail and discovered additional “satellite” clusters within the clusters.

According to the researchers, the results support the theory that dark matter and visible matter should be found in the same places because gravity pulls them both together. They also agree with current theories on how dark matter particles do not collide, unlike ordinary matter particles, but simply pass through each other. “If dark matter particles were collisional, we would observe a much smoother distribution of dark matter without any small-scale clumpy structures,” explains Jee.

Fluids mix in reverse

David Pine of New York University and colleagues at the Haverford College, the California Institute of Technology and the Israel Institute of Technology studied the motion of tiny polymer beads suspended in a viscous fluid trapped between two concentric cylinders held 2.5 millimetres apart. When the team rotated the inner cylinder in one direction and then back again, they found that the beads returned to their starting positions. But the behaviour is only seen if the solution is relatively dilute and the beads are stirred for a short time. At higher concentration and longer times, mixing becomes irreversible.

According to the researchers, the observed behaviour can be explained by collisions between individual beads. Mixing can be reversed if the particles do not collide with each other, which is the case at low concentrations. But as the solution becomes more concentrated — and more collisions occur — the process becomes irreversible.

“The irreversibility of these particles may be explained by the extreme sensitivity of their trajectories to imperceptibly small changes of the particle positions,” explains Pine. Such perturbations might arise from almost anything – from small imperfections in the particles or by small external forces – and are magnified exponentially because of the motion of other particles suspended in the liquid, he says. Physical systems that exhibit such extreme sensitivity to small perturbations are said to be ‘chaotic’, which means that their behaviour cannot be determined in advance.

The US-Israel team says that an irreversible flow could be transformed into a reversible one at a predictable point by reducing the number of particles since this makes collisions between the particles less likely. This could be important for scaling up laboratory experiments to industrial levels, which is difficult simply because of the unpredictable behaviour of the particles involved. Possible applications include mixing of pharmaceutical suspensions and the catalysis of petrochemicals in fluid beds. The work could also help in understanding particle migration during ceramic processing and in the culture of blood-making cells.

Ions trapped on a chip

Classical computers store and process information as bits with one of two values: “0” or “1”. But a quantum computer would exploit the ability of quantum particles to be in superpositions of two or more states at the same time. These “entangled” states, in principle, allow a quantum computer to outperform a classical computer for certain tasks.

One of the most exciting candidates for such a quantum bit or “qubit” is a trapped ion, whose internal energy states can be manipulated using a laser. The basic requirements for such a qubit have already been met, and very recently two groups set a new record by entangling up to eight calcium ions in a single trap. But a real quantum computer will require such trapping and manipulation to take place for millions of atoms. “This will require a departure from the typical method of fabricating ion traps ‘by hand’ and involve integrated approaches like the one reported by us,” says Chris Monroe of the University of Michigan.

In the lab
In the lab

Monroe and colleagues at the University of Maryland have now made an ion trap from four alternating layers of aluminium-gallium-arsenide and gallium-arsenide grown on a substrate using molecular-beam epitaxy. The team created a hole through the chip and fashioned a set of cantilevered electrodes over it using techniques routinely employed in fabricating microelectromchanical systems (MEMS). They then mounted the chip on a socket inside a vacuum, and introduced a gas of cadmium-111 atoms into the hole using a pulsed laser.

By carefully controlling parameters such as electrode voltages and the laser wavelength, Monroe and colleagues were able to produce a single cadmium ion in the trap and manipulate its quantum state. “Such ion traps look to be useful for scaling up the trapped ion quantum computer, as we can fabricate hundreds of thousands of electrodes using MEMS lithographic technology,” says the team.

DNA pyramids make their debut

DNA — the “building block of life” — consists of two linear strands wound into a double helix with one of four different “bases” attached to every sugar group along the strands. DNA is an attractive engineering material because strands with complementary base sequences recognise and bind to each other, enabling complex molecular structures to be made by self-assembly.

Previous attempts at making DNA nanostructures, in the shapes of cubes or octahedra, required many steps and did not produce much material. The new method, invented by Andrew Turberfield and Russell Goodman of the University of Oxford, overcomes this problem.

It creates nanoscale tetrahedra of DNA that self-assemble in a single step in just seconds with a yield of up to 95%. The tetrahedra are made from four short strands of synthetic DNA, each of which runs around one face. Each edge is formed where neighbouring strands have complementary base pairs.

The new method is simple and involves heating DNA strands in a salt solution to just below boiling point. When the strands are then rapidly cooled, they bond together to form a tetrahedron. The team can then link different tetrahedra together using single strands of DNA.

“Tetrahedra are used extensively in architecture and engineering because their structure is simple but very strong, making them ideal for use in DNA nanostructures,” says Turberfield. “These atomically precise nanostructures are ideal building blocks for nanofabrication and can be produced cheaply in large quantities – all you have to do is mix the components together.”

Indeed, colleagues at the Vrije Universiteit in Amsterdam showed that the tetrahedra can withstand forces of up to 100 picoNewtons by compressing them with the tip of an atomic force microscope. These compression tests also allowed the Anglo-Dutch team to measure the elastic properties of DNA for the first time.

The team now plans to make 3D scaffolds for molecular devices such as electronic circuits using linked tetrahedra. The DNA pyramids could also act as containers for individual protein molecules and so be used to deliver drugs.

“We have designed a family of DNA tetrahedra which are structurally stable and can be manufactured in a quick and simple step,” adds Turberfield. “Ultimately we hope to use them as building blocks for nanofabrication, to act as templates for much more complex DNA nanostructures.”

Perseus arm is closer than thought

The Milky Way is believed to contain either two or four spiral arms that swirl out from the galaxy’s centre. Previous measurements of the distance from the Sun to the Perseus arm were made using two methods. The first relies on measuring the apparent luminosity of massive young stars. The other “kinematic” method involves comparing the rotation speed the Sun with that of an object inside the Perseus arm, and then combining the difference between the two values with a model for the rotation of the Milky Way itself. However, these two methods produced very different results.

Now, Mark Reid of the Harvard-Smithsonian Center for Astrophysics in Massachusetts and colleagues in China and Germany have measured the distance from the Sun to a star-forming cluster in the Perseus arm called W3OH. Their value of about 5.86 x 1016 kilometres agrees with results obtained by luminosity techniques but is 10 times more accurate. The distance is also much shorter than that obtained by kinematic measurements, which produce values of around 1.32 x 1017 kilometres (about 13,962 light years).

The team made their measurement using the Very Long Baseline Array (VLBA) – an array of ten radio telescopes spanning the Earth from Hawaii to New England to the Virgin Islands – and a technique called “triangulation”. This method, which is similar to that used by ground-based surveyors, uses the Earth’s orbit around the Sun as one leg of a triangle. By then measuring the change in angle of W3OH from its average position over a year, the researchers can calculate the distance to W3OH by simple trigonometry.

The results also show that stars in the Perseus arm have unexpectedly large motions and move in ways that are beyond the simple rotation of the Milky Way around its centre. “Studies such as ours are the first steps to accurately mapping the Milky Way,” says Reid. “We are now embarking on a large project to measure the distances to a dozen regions of star formation spread across several spiral arms in our galaxy. In addition, we hope to use our data to better understand how spiral arms form.”

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