## golden ratio

Continued fractions reveal the good, the bad and the beautiful side of numbers.

The Fibonacci sequence – 0, 1, 1, 2, 3, 5, 8, 13, ... – is one of the most famous pieces of mathematics. We see how these numbers appear in multiplying rabbits and bees, in the turns of sea shells and sunflower seeds, and how it all stemmed from a simple example in one of the most important books in Western mathematics.

A commonly held belief about medieval Europe is that academic pursuits had fallen into a dark age. The majority of scholars were churchmen, and their enquiry often related to some principle of church practice. But is there a value to respecting the tenacity of historic mathematicians?

Dan Brown in his book, The Da Vinci Code, talks about the "divine proportion" as having a "fundamental role in nature". Brown's ideas are not completely without foundation, as the proportion crops up in the mathematics used to describe the formation of natural structures like snail's shells and plants, and even in Alan Turing's work on animal coats. But Dan Brown does not talk about mathematics, he talks about a number. What is so special about this number?

The Plus anniversary year — A word from the editors

Many people find no beauty and pleasure in maths - but, as Lewis Dartnell explains, our brains have evolved to take pleasure in rhythm, structure and pattern. Since these topics are fundamentally mathematical, it should be no surprise that mathematical methods can illuminate our aesthetic sense.
It isn't often that a mathematical equation makes the national press, far less popular radio, or most astonishingly of all, is the subject of a debate in the UK parliament. However, as Chris Budd and Chris Sangwin tell us, in 2003 the good old quadratic equation, which we all learned about in school, reached these dizzy pinnacles of fame.
It was Euclid who first defined the Golden Ratio, and ever since people have been fascinated by its extraordinary properties. Find out if beauty is in the eye of the beholder, and how the Golden Ratio crosses from mathematics to the arts.
During World Mathematical Year 2000 a sequence of posters were displayed month by month in the trains of the London Underground aiming to stimulate, fascinate - even infuriate passengers! Keith Moffatt tells us about three of the posters from the series.
One of the most striking and powerful means of presenting numbers is completely ignored in the mathematics that is taught in schools, and it rarely makes an appearance in university courses. Yet the continued fraction is one of the most revealing representations of many numbers, sometimes containing extraordinary patterns and symmetries. John D. Barrow explains.