快猫短视频

SAME old decorations, same old films on telly, same old arguments with
relatives. So how about doing something different this Christmas鈥攍ike
inventing a new branch of science? If you thought you needed a particle
accelerator or a telescope orbiting the Earth to make significant scientific
discoveries, think again. Some of the biggest discoveries have come not from
pondering great questions about the cosmos, but from musing about a trivial
little puzzle that most people don鈥檛 give a monkey鈥檚 about. And let鈥檚 face it:
if you can鈥檛 think trivially at Christmas, when can you?

There is, however, a huge barrier between you and your yuletide breakthrough:
the astonishingly pervasive idea that only Big Questions can possibly lead to
Big Answers. Despite the lessons of history, stern warnings against wasting
one鈥檚 time on 鈥渢rivial鈥� problems have been handed down to would-be scientists
for years. Even the late Nobel prizewinning immunologist Sir Peter Medawar, in
his otherwise genial little book Advice to a Young 快猫短视频, declared:
鈥淚t can be said with complete confidence that any scientist of any age who wants
to make important discoveries must study important problems.鈥�

Percentage players of the game of science know that you can boost your
chances of getting a Nobel prize by joining a world-class institution, and
signing up for the search for the Key to the Universe, or the Cure for Cancer.
But there鈥檚 no getting around the fact that many Nobels have come about by
kicking around an itsy-bitsy problem just for the hell of it.

The most famous case of the ho-hum leading to the humongous is, of course,
the fall of the apple that led Isaac Newton to the law of gravitation. Newton
himself cheerfully recounted the story about how, as a student at home with his
mum, he twigged that the same force that pulled the apple to the ground might
also pull the Moon to the Earth. The rest, as they say, is a totally astonishing
scientific revolution the like of which had never been seen before.

Fame from games

Yet even now, historians take a snooty view of this demonstration of the
power of trivial problems. 鈥淭he story vulgarises universal gravitation by
treating it as a bright idea,鈥� sniffs Richard Westfall of Indiana University in
Bloomington, professor of the history of science and doyen of Newtonian
biographers.

According to Westfall, 鈥淎 bright idea cannot shape a scientific tradition.鈥�
Oh really? How about a disreputable pastime, then? In 1654, Chevalier Gombaud
Antoine de M茅r茅鈥攂on viveur, courtier and inveterate
gambler鈥攁sked for a bit of advice from his erudite friend, the French
philosopher and mathematician Blaise Pascal. De M茅r茅 had been
putting bets on dice, and wasn鈥檛 sure about what odds to put on the various
combinations that could turn up. There were some rules of thumb doing the rounds
at the time, but de M茅r茅 didn鈥檛 entirely trust them. Did Pascal
have any bright ideas about how to work them out?

Yes he did, and he also knew a man who had more: the lawyer and mathematician
Pierre de Fermat. Between them, Pascal and Fermat set about laying the
foundations of the theory of probability.

As so often with mathematical trivia, it turned out that de
M茅r茅鈥檚 simple questions were anything but. One of the hardest
turned out to be an interrupted game of dice. Two players try to reach a certain
number of points, but before either of them succeeds, their mates drag them off
to the pub. How should the prize money be split? To solve such questions, Pascal
and Fermat had to work out the probabilities of getting the different scores
that could exist when the game was interrupted. And to do that, they had to
devise systematic ways of counting all the different ways in which events can
occur, which led them to invent what is now called combinatorics.

From such lowly origins, probability theory went on to underpin vast tracts
of science, from quantum theory to the use of statistical inference to draw
conclusions from experiments. Combinatorics became the foundation of statistical
mechanics and solid-state physics. Now biochemists are using the method, in a
technique called 鈥渃ombinatorial chemistry鈥�, to combine millions of different
compounds to find new drugs.

All of which makes it just as well that Pascal didn鈥檛 simply tell his friend
to stop gambling and get a life. But then, back in Pascal鈥檚 time, the best
brains seemed positively to relish the idea of tackling what looked like silly
questions. And they don鈥檛 come much sillier than this one, first raised by
Johann Bernoulli in 1696: into what shape should you bend a piece of wire so
that a bead on the wire will slide down from one point to any other in the least
possible time?

An immediate response might be 鈥渨ho cares?鈥�, quickly followed by 鈥渙h, it鈥檚
obviously a straight line鈥�. Obvious, perhaps, but wrong except when the two
points are in a vertical line. If they鈥檙e in any other direction, you鈥檙e not
making the most of the vertical pull of gravity to maximise the bead鈥檚 speed,
and thus minimise its descent time.

Egged on by Bernoulli, some of Europe鈥檚 greatest mathematicians, including
Newton and Gottfried Leibniz, had a shot at solving the problem, which turned
out to be decidedly nontrivial. For a start, it demanded the application of the
new-fangled technique of calculus to integrate the ever-changing speed of the
bead down the length of the wire. But Bernoulli鈥檚 question had introduced a new
twist: the result of the integration鈥攖he time taken by the bead鈥攈ad
to be as small as possible.

Being a genius, Newton cracked the problem after a hard day鈥檚 work at the
Royal Mint. The wire must be bent into a curve known as a cycloid, the shape
traced by a single point on the rim of a bicycle wheel as it turns. No one much
cares about the answer anymore, but every modern theoretical physicist has to
know how to solve such problems. That鈥檚 because Bernoulli鈥檚 silly question sowed
the seeds for the development of one of the most important techniques in modern
physics: the calculus of variations.

Piffling problems

The laws of motion, magnetism, electricity and even Schr枚dinger鈥檚 famous
wave equation in quantum mechanics all flow directly from solutions to integrals
just like that thrown up by Bernoulli鈥檚 bead game. The key equations for probing
the mysteries of nature in each of these huge fields of physics are found using
the calculus of variations.

These techniques were given their modern form by the 18th-century Swiss
mathematician Leonhard Euler,who also happens to be the all-time champion
at turning the piffling into the profound. A classic example emerged in 1736,
when Euler published the solution to a problem of mind-boggling parochialism: Is
it possible to take a trip round Konigsberg that crosses each of the town鈥檚
seven bridges once and once only?

Euler鈥檚 12-page analysis was bad news for the burghers of
Konigsberg鈥攖hey could not complete their tour of the town without crossing
one of the bridges twice. The good news for the rest of us, however, is that
Euler extended his research to cover the general case of any number of bridges
spanning any number of rivers. In the process, he helped create the foundations
for two hugely important areas of applied mathematics: graph theory and
topology.

Graph theory is the study of networks of points connected by lines. All of
which sounds recherch茅, until one realises that problems ranging from
designing microprocessor circuits to sending company reps out to sell carpets
all involve some type of network. So-called Eulerian circuits, trips around
networks that visit each point just once, and their more technical relatives,
are today at the heart of attempts to solve hard-nosed business problems such as
finding the cheapest route for international telephone calls.

Topology, which the Bridges of Konigsberg problem also helped create, is
roughly speaking the mathematical study of shapes. Long regarded as pretty
useless even by mathematicians, topology is now casting light on truly Big
Problems, ranging from how enzymes extract genetic information from tightly
coiled DNA to the physicists鈥� quest for a Theory of Everything that unifies all
subatomic particles and forces. Which seems an awful long way from taking a
stroll over a few bridges.

Euler didn鈥檛 live to see these amazing consequences of his work, but trivial
problems don鈥檛 always take centuries to reveal their true depth. In 1921, the
physicist Chandrasekhara Raman was sailing back to India from a physics
conference when he started musing about why the sea was blue. Of course,
everyone knows why the sea is blue. As the great Lord Rayleigh had explained
years earlier, it is simply a reflection of the blue of the sky. In that case,
thought Raman, looking at the sea through a polariser, which cuts out the light
reflected from the surface of the sea, should reveal the sea鈥檚 true colours. But
Raman found that the sea still looked blue, rather denting Rayleigh鈥檚 simple
explanation.

Casting around for other possibilities, Raman hit on the idea that water
molecules scatter light to reveal the blue part of the spectrum, while the other
colours tend to go straight through the water. Experiments back in India
confirmed his idea鈥攚hich won him the 1930 Nobel Prize for Physics, and led
to Raman spectroscopy, widely used today for the chemical analysis of liquids
and solids.

Just seeing a plate spinning through the air in the cafeteria of Cornell
University in Ithaca, New York, was enough to set the American physicist Richard
Feynman on the path to a Nobel prize. Intrigued by its rapid wobbling, Feynman
worked out that as long as the wobbles are small, they occur at twice the rate
of spin of the plate. Delighted by his discovery, Feynman ran off to tell his
friend and colleague Hans Bethe, who thought it was all rather, well, trivial.
Yet it inspired Feynman to investigate the spin of the electron, and from there
to work on quantum electrodynamics, for which he won the 1965 Nobel Prize for
Physics.

Even today, trivial questions show no signs of losing their ability to lead
to major breakthroughs. The implications of physicist Lewis Fry Richardson鈥檚
simple question 鈥淗ow long is the coastline of Britain?鈥� are still being worked
out 70 years after he first raised it.

Richardson had noticed that different textbooks gave radically different
answers to this apparently straightforward question. Investigating further, he
found that the length of a coastline depends on the scale of the map, which is
pretty obvious. The more detailed the map, the more wiggles it shows, and the
longer the coastline. But Richardson found there was a relationship between the
scale and the length of the coastline such that a single number could capture
the otherwise ineffable concept of the 鈥渞oughness鈥� of the coastline.

Quirky fractals

Richardson鈥檚 work is now recognised as a pioneering study of fractals,
objects with fractional rather than whole-number dimensions. Fractals have
since become a major research topic in mathematics, while their quirky
properties are applied in fields as diverse as data compression, analysis of
brain scans and studies of gold-bearing rocks.

So, suitably inspired by past examples of the value of trivia, how should you
set about conjuring up your own scientific revolution this Christmas? History
suggests that just about anything鈥攁part from watching telly鈥攃ould do
the trick. Playing poker inspired mathematician John von Neumann to invent game
theory, the mathematics of competitive strategies now widely used by economists
and animal behaviourists.

Alternatively, doing a bit of DIY might provide the spark. When Euler was
asked to design a fountain for Frederick II the Great of Prussia, he worked out
the basic principles for putting on a decent display鈥攁nd invented fluid
dynamics.

Or you could just spend the whole day in bed. The French polymath Ren茅
Descartes is said to have being doing just this when he spotted a fly and
started musing about how to describe the insect鈥檚 position numerically. The
result: the invention of Cartesian coordinates, one of the greatest ideas in the
history of mathematics.

The lesson seems clear. Despite what scientific snobs might have us believe,
nature doesn鈥檛 seem to know what trivial means. Everything from the birth of the
Universe to the wobbles of a spinning plate is another manifestation of nature鈥檚
laws. It鈥檚 just that sometimes we can鈥檛 tell the difference between the cosmic
and the quotidian until that revelatory trip to the cafeteria, say, or the
cruise on the Med.

So finish your Christmas dinner, lie down on the sofa鈥攁nd think
trivial.