The USS Enterprise drops out of warp and slips into a parking orbit around an uncharted alien planet. The Captain orders a scan for lifesigns and, within seconds, he is told exactly what lifeforms are present, including the pre-industrial humanoids on the southern continent. How feasible is this, really?
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Summer is rubbish for two fundamental reasons. One: wasps set about stinging everything with two legs without doing anything useful with their time like making honey or pollinating stuff. Two: girlfriends always want to go off on picnics. Avoiding having to eat al fresco was the sole reason our ancestors stopped messing about in trees and found some good caves instead. Picnics are inherently stressful; the apple juice invariably leaks into the bag, the Sports section blows away, and it is always always impossible to find a spot good enough to settle down on. The problem is that as soon as you approach that idyllic lush area of grass you spied from afar it starts looking nasty and patchy. The grass really does seem to always be greener on the other side, or at least further away.
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The word “stealth” is often associated with high-tech bombers built to be invisible to enemy radar. This technology works through the aircraft’s surface being specially designed and having a covering of radar-absorbent skin that ensures minimal radio waves are reflected back to the enemy radar transmitter.
There is another kind of stealth, however, that does not rely on hiding the presence of an object, but on masking the fact that it is moving. If the pursuer approaches along a particular trajectory it appears to remain perfectly stationary from the point of view of the target. The pursuer can use this “motion camouflage” to rush right up to the target before it is perceived as a threat. This technique could be used by missiles to remain undetected for as long as possible, and even appears to have been discovered by nature. There is good evidence that hoverflies and dragonflies have evolved this strategy to fly without being detected.
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The world around us is full of relationships, rhythms, correlations, patterns. And mathematics underlies all of these, and can be used to predict future outcomes. Our brains have evolved to survive in this world: to analyse the information it receives through our senses and spot patterns in the complexity around us. In fact, it’s thought that the mathematical structure embedded in the rhythm and melody of music is what our brains latch on to, and that this is why we enjoy listening to it. It is perhaps not surprising then that there is a great deal of overlap between mathematics and the art that our brain finds so pleasing to look at.
This article is a whistle-stop tour of some of the types of art with a strong mathematical component, or conversely where a mathematical visualisation has an astonishing beauty.
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We’ve all experienced it. It’s Monday morning, you’ve slept through your alarm and are now in a hopeless rush to get in on time. The toast comes out of the toaster, you give it a quick sweep of butter, or in these more health-conscious times, margarine, and pick it up to take over to your newspaper on the kitchen table. And then it happens. Whether it simply slips out of your fingers, or it burns slightly and you subconsciously release it, the toast begins to drop towards the filthy floor. You watch in dismay as the toast falls, neatly performing a half-turn and landing flat on the floor, butter-side down in the grime. You don’t even know why you tentatively hoped for the toast to land otherwise – the Universe seems out to get you as far as free falling toast is concerned.
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Cuttlefish have an extraordinary, and almost instantaneous, control over their appearance. They can produce hundreds of distinct patterns, which they use for camouflaging, courting mates or startling predators. One dynamic pattern, where thick black and white bands flow rapidly over the skin of the cuttlefish as it near its prey, is somewhat of a mystery. Why, just as the cuttlefish approaches an unsuspecting target, should it switch from camouflage to a highly conspicuous display?
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The BA (British Association for the Advancement of Science) is a nation-wide organisation dedicated to connecting science with people and promoting openness about science in society. It organises Science Week (11-20 March 2005) and an annual Festival of Science, which was hosted this year by the University of Exeter. Mathematics was well represented at this year’s Festival, with an exhibition of mathematical art, the launch of the National Cypher Challenge, and a morning of fascinating lectures on the Clay Institute Millennium Problems. The three ‘Million Dollar Maths’ problems covered were P vs. NP, the Riemann Hypothesis, and the Poincaré Conjecture.
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This article has been reprinted in the February 2005 issue of Mathematics Today (vol.47, no.1), the magazine of The Institute of Mathematics and Its Applications (IMA). Download pdf
The sleek animal glides effortlessly above the ground. The skin of her bullet-shaped body can smoothly change colour and pattern to match the surroundings, providing an almost perfect camouflage. The animal is on the hunt, and scans ahead with her acute vision. She spies her prey in the murky distance and banks gently towards it. As she approaches, the hunter slows and then hovers, the skirt-like fin along her flanks rippling gently to provide lift. The design on her back morphs into an all-together different pattern, this one a vivid, dynamic display. The whole animal appears to be pulsating as thick black and white stripes race across her surface, from the base of her body to the tip of her tentacles. The predator carefully edges nearer to its quarry and then suddenly jets forward, her tentacles exploding outwards to envelop the hapless prey in their clutches. It is quickly dragged in towards the beak, which crushes through the prey’s exoskeleton and gulps down its flesh. After this burst of ferocity the hunter switches back into perfect camouflage, and slips away unseen.
The USS Enterprise drops out of warp and slips into a parking orbit around an uncharted alien planet. The good Captain orders a scan for lifesigns, and within seconds he is being informed exactly what lifeforms are present, including the preindustrial tribes of humanoids on the southern continent. How feasible is this really? Well, unfortunately for Star Trek fans, identifying a species from orbit will perhaps forever remain in the realm of science fiction. For astrobiologists, however, revealing the presence of life on a remote planet is becoming possible even now, on 21st century Earth. Within a decade there will be telescopes capable of detecting the chemical fingerprints of life on planets nearly 50 light years away. And within out lifetimes there may even be telescopes able to image the oceans and continents of alien worlds.
Alan Turing is considered to be one of the most brilliant mathematicians of the last century. He helped crack the German Enigma code during the Second World War and laid the foundations for the digital computer. His only foray into mathematical biology produced a paper so insightful that it is still regularly cited today, over 50 years since it was published. In it he described how a set of ‘reaction-diffusion equations’ explain how the wonderful diversity of animal patterns may be generated.
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This article has been reprinted in Muse, the YouthAgency magazine. The agency is run by the National Association for Gifted Children and aims to inspire able students to cultivate their abilities. pdf copy of the reprint.
As we saw in the last edition of +plus, mathematical techniques have been applied very successfully to analysing certain types of games. The two examples that we looked at were the simple subtraction game Nim, and the much more complex case of chess endgames. The next step is to see how computers, which are no more than automated maths machines, are being programmed to actually play chess themselves. It is theoretically possible to play chess perfectly, but neither humans nor machines will probably ever accomplish this. Computers have, however, already practically achieved perfection in draughts, and soon may be said to have ‘solved’ the game. Continue reading
Mathematicians love games. Not only can they have fun while looking like they’re busy working, but even the simplest games can demand clever tactics and strategies to win. These are the perfect kinds of problems for solving with maths.
One branch of mathematics, called Combinatorial Game Theory, was developed around 30 years ago specifically to deal with the analysis of games. It prescribes a way of breaking games down into smaller parts that are easier to examine, and then using a special kind of algebra to add up the values of the individual subgames. And if there’s one thing that mathematicians are good at it’s counting.
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