### Of Fibonacci and Da Vinci Code

(Reuters)London, July 19 : If the numbers 13-3-2-21-1-1-8-5 ring a bell, it might be because you have been reading The Da Vinci Code.

Dan Brown's bestseller uses that series -- a mixed-up version of a sequence of numbers brought to the Western world by a 13th century Italian mathematician -- as a clue to a secret Swiss bank account.

As Brown's book, and the film, turned into a global entertainment phenomenon, the work of Leonardo Fibonacci of Pisa was given a new lease of life for millions of people.

Not so for Elizabeth Miller. The technical analyst had long been a devotee of the Fibonacci sequence which mirrors patterns found in art, music, architecture and nature.

Working for Redtower Research out of a farmhouse in Aberdeenshire, Scotland, Miller looks at charts of price movements to work out future financial market trends, and she says Fibonacci numbers help her.

Fibonacci numbers are used by technical analysts to determine price objectives from percentage retracements, or market corrections.

Miller says the Dow Jones stock index collapse from the dotcom boom that peaked in 2000 followed a Fibonacci pattern.

"Fibonacci retracements work on pretty much everything, but they work particularly well with stock markets--

it's natural mathematics."

However, the Fibonacci series' origin lay in something a little more earthy and tangible than stocks and bonds..

Fibonacci published a book in 1202 which calculated the reproductive cycles of pairs of rabbits and showed that they followed a pattern 1-1-2-3-5-8-13-21 -- a sequence where the sum of any two consecutive numbers equals the next highest number.

Each gestation lasts one month and rabbits become fertile after two months, so that the reproductive cycle of one pair of newborn rabbits follows the Fibonacci sequence month-by-month.

If you calculate the ratio of each number in the Fibonacci sequence to the previous one, it evens out over time to the magic ratio of 1.618.

**From music to markets**

The Fibonacci sequence exists in nature -- in the arrangements of seeds on the head of a sunflower, the spirals on a snail's shell and the way that leaves arrange themselves to get the most sunlight.

Mathematicians have found the Fibonacci ratio in the building of the pyramids, in the aesthetically pleasing golden section in paintings, and also in music.

"Some people have found it in Mozart and Beethoven, though we do not know if it's there naturally or by design," said Ron Knott, visiting fellow in Maths and Computing Sciences at the University of Surrey in southeast England.

Interest in Fibonacci numbers as a way of forecasting markets started in the 1930s, with a stock-picking newsletter written by accountant Ralph Nelson Elliott.

Elliott developed his own technical analysis theory based on Fibonacci numbers called Elliott wave analysis -- which says that the market follows a repetitive pattern with each cycle made up of a five-wave rise followed by a three-wave fall.

A market bestseller published in 1978 on Fibonacci and Elliott wave boosted the Italian mathematician's popularity through the 1980s, and the attraction has not waned.

"It's amazingly common to read sentences written by technical analysts involving Fibonacci ratios," said Roy Batchelor, professor of banking and finance at Cass Business School in the City, London's financial district.

**How does it work?**

The main premise of the Fibonacci ratio for financial markets is that any large move is followed by a retracement, or counter-move, which works out as a Fibonacci-based proportion of the original move.

In percentage terms, 61.8 is among the key price retracement targets for modern traders of currencies, stocks and commodities. The ratio is known as the "golden mean" for its universal applications.

Another key retracement is 38.2 percent, which is 1 divided by 1.618 squared.

Technical analysts scour the charts for large moves followed by retracements, and then use Fibonacci to work out when these market corrections may end.

"When there is a big major upmove, the Fibonacci retracement -- 61.8 percent or 38.2 percent -- is a good target, a sign that maybe the latest sell-off is going to finish," said Miller who has been a technical analyst for 14 years.

Use of Fibonacci can become something of a self-fulfilling prophecy for stock markets where there is a crowd mentality and where everyone starts to watch out for the same levels.

Tom Pelc, chief technical strategist at RBS Financial Markets in London, finds Fibonacci patterns in recent moves in London's FTSE stock index.

"If you look at the FTSE's decline in May and June and its bounce since then, it's struggling right now at exactly the 61.8 percent retracement," he said in mid-July.

But some academics are unconvinced.

Batchelor and his colleagues at Cass Business School have looked at whether the ratio of rises to declines in the Dow Jones followed the Fibonacci rule.

"The evidence on this is contrary. People will show you a picture where it happened, not the 100 pictures where it did not happen -- but it's got entertainment value," he said.

In "The Da Vinci Code", characters follow a complex series of clues in a chase around Europe to discover a secret concealed for centuries. Batchelor says analysts are also chasing an elusive, and ultimately illusory, truth.

"Fibonacci is popular for the same reason that 'The Da Vinci Code' is popular. We suffer from the illusion of control, that things have to have an explanation -- people hate randomness."

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