September 25, 1994, Sunday

SECTION: Pg. 21

LENGTH: 692 words

HEADLINE: Scope: Off-the-wall ideas win through HOBBY-HORSES Great discoveries can arise from the most useless preoccupations

BYLINE: By ADRIAN BERRY

BODY:
GOVERNMENTS often insist that the research which they fund must be "useful" - without realising that some of the greatest advances in technology have come from people exploring ideas that seemed to have no practical value whatever. An example is the 19th century studies - by unknown researchers - into the vibrations of violin strings, which indirectly led Heinrich Hertz to discover radio waves. Another is that geometric curiosity known as the Mobius strip, a looped strap with a single twist, which enabled conveyer belts to be used twice as long before wearing out. The German chemist Friedrich von Kekule had a dream in 1865 that must have seemed unlikely to be of any use to anyone. He imagined a snake which was trying to swallow its own tail. Luckily he remembered the dream when he awoke, and the experience inspired him to visualise the highly complicated structure of the benzene molecule. For 23 centuries, since Euclid proved the infinity of prime numbers (numbers like two, three, five and seven, which can be divided only by themselves and one), primes were no more than a fascinating curiosity. One of the biggest - and apparently most useless - challenges still facing mathematicians is to find a proof of Christian Goldbach's "conjecture" of 1742 that every even number greater than two is the sum of two primes. (For example eight is the sum of the primes three and five.) Although verified for every even number up to 100,000, a proof that it is true for all even numbers greater than two has yet to be found. Then, in the 1970s, a James Bond-like use was found for prime numbers with the invention of the highly practical RSA cipher which depends for its security on the sheer difficulty of finding two primes that have been multiplied to make a larger number. (The coming of the "quantum computer", however, may destroy the RSA - see Scope, June 12 this year.) The 17th-century French mathematician, Blaise Pascal, was approached by an irritated gambler anxious to know why he always lost when betting on the appearance of certain combinations in the fall of dice. Pascal, after consulting Pierre de Fermat (of "last theorem" fame), responded by inventing the "Pascal triangle", an inverted pyramid of numbers in which each was the sum of the two above it. This enabled the gambler to understand the odds on dice throws and win some money, and it also created modern probability theory on which most statistical science is based. While no outcome is absolutely certain, it can be seen, with the aid of the triangle, that, for example, if a coin is tossed a sufficiently large number of times, the odds on getting roughly an equal number of heads and tails rises towards infinity. The subject of triangles brings up one of the most extraordinary pieces of "useless" research yet to appear. This is a design for wallpaper by Prof Charles Radin, a mathematician at the University of Texas at Austin which poses challenges that are both mathematical and psychological. "It appears to be infinitely complicated and unpredictable, until you are told how it works," said his colleague Prof Ian Stewart, of the University of Warwick at Coventry, author of an article in this week's New Scientist, which reproduces the diagram. "Then, suddenly, you see it as being simple." Most wallpaper patterns, however complicated, involve repetition, but Radin's seems truly random and chaotic. On a big enough scale, it would be the ultimately difficult jigsaw puzzle - and yet its creation was simple. It starts, at the centre, with a single right-angled triangle. Four more identical triangles surround it. Each of these four is surrounded by four more such triangles, and so on. The result is that the triangles point in all directions with an infinite number of orientations. But what can it do? "I don't know for sure," Radin said, "but I have a strong suspicion that it resembles the crystalline structure of some material. A physical chemist might at once see some meaning to it that I cannot." If Radin is right, he will be, like Kekule, the second great scientist to have had a crazy-seeming dream that turned into something useful.