A picture, the old adage goes, is worth 1,000 words. But in science a diagram can describe things that transcend the written word. A single image can convey the simple underlying pattern hidden by words or equations, says Marcus du Sautoy.

Draw the right picture and you can literally transform the way we see the world. But a diagram is more than just a physical representation of what we see with our eyes.

The power of a diagram is to crystallise a new way of seeing the world.

Often it requires throwing away information, focusing on what is essential. Other times it changes a scientific idea into a visual language providing a new map where the mathematics of geometry takes over and helps us to navigate the science at hand.

Copernicus certainly understood the power of a good picture. In his great opus De Revolutionibus Orbium Coelestium published shortly before his death in 1543, Copernicus takes 405 pages of words, numbers and equations to explain his heliocentric theory.

Find out more Marcus du Sautoy's six-part series on The Beauty of Diagrams started 18 November

It's on BBC Four on Thursdays at 2030 GMT Watch it on the iPlayer

But it is the diagram that he draws at the beginning of the book that captures in a simple image his revolutionary new idea: it is the Sun that is at the centre of the Solar System, not the Earth.

His picture encapsulates some of the essential elements of the best diagrams. The concentric circles are not meant to describe the precise orbits of the planets.

Copernicus knew they weren't circles. The uniform distances between the circles aren't meant to tell you how far the planets are from the sun. Rather this picture conveys the simple yet shocking idea that we aren't at the centre of things. His diagram transformed our view of our place in the universe

But some diagrams do more than just crystallise the essential underlying structure of a complex system. A diagram has the power to create a whole new visual language to navigate a scientific idea.

Florence Nightingale was the first to use the idea of a pie chart to represent data Marcus du Sautoy

Newton's optics diagrams for example transform light into geometry.

By representing light as lines Newton is able to use mathematics and geometry to predict the behaviour of light. It was a revolutionary idea. Look at the light that illuminates the world around you. There are no lines. Newton's diagram translates the slippery science of optics into the concrete world of geometry where mathematics becomes the eyes to see what is happening to light.

Sometimes a diagram is the crucial step in making people believe in the impossible.

Diagram of deaths

Mathematicians had been struggling with the idea of the square root of minus one. There seemed to be no number on the number line whose square was negative. Yet experts knew that if such a number existed it would transform their subject.

But where was this number? It was a picture drawn independently by three mathematicians at the beginning of the 19th Century that brought these numbers to life.

They created a two dimensional map of numbers where the numbers we'd known about since the Ancient Greeks ran east-west along the horizontal axis while these new imaginary numbers like the square root of minus one extended vertically in the north-south direction.

The Pioneer plaque diagram 1. The image of a naked man and woman might prove meaningless scribbles to a life form that looks very different to our own. 2. Echoing Copernicus's picture of the sun-centred solar system, Sagan and Drake drew a picture of the planets and included Pluto. 3. A picture of the Pioneer space probe at the end of a line emanating from the third planet of the Solar System tells aliens where the message comes from. 4. The numbers in the Pioneer plaque are binary - a vertical line for 1 and a horizontal line for 0. The numbers on planets indicate relative distance from the Sun. 5. There are further binary numbers. We write numbers in decimal because we have 10 fingers. Aliens probably have an entirely different anatomy and hence number system 6. The star map locates the Sun. The radial lines locate pulsars, stars emitting a regular electromagnetic pulse. A binary number shows the frequency. 7. This is a unit of measurement of time and distance. The circle represents a hydrogen atom. When the orbiting electron flips states it corresponds to a wave with frequency 1,420 MHz and wavelength 21cm. These are the diagram's units of measurement. 8. The image of the Pioneer space probe gives aliens an idea of the height of humans relative to the probe. 9. The height of the woman is in binary: 1,000 units of length. The unit of length is 21cm given by the wavelength of the hydrogen atom as it flips states. As 1,000 is eight in binary, the woman's height is 8x21cm=168cm.

Called the Argand diagram after one of its creators, this picture helped mathematicians to believe in these new numbers. Not only that, the diagram was a potent tool in manipulating these new numbers since the geometry of the diagram reflected the underlying algebra of the numbers they depicted.

One of the most powerful uses of diagrams though has been in visualising data. Given that we live in an age that generates huge reams of numerical information, finding ways to make sense of all these numbers is essential.

Transcend language

One of the first to use the visual world to navigate numbers was Florence Nightingale.

Although better known for her contributions to nursing, her greatest achievements were mathematical. She was the first to use the idea of a pie chart to represent data.

Image caption Nightingale's diagrams were designed to highlight deaths in the Crimea

Nightingale had discovered that the majority of deaths in the Crimea were due to poor sanitation rather than casualties in battle. She wanted to persuade government of the need for better hygiene in hospitals.

She realised though that just looking at the numbers was unlikely to impress ministers. But once those numbers were translated into a picture - her Diagram of the Causes of Mortality in the Army in the East - the message could not be ignored. A good diagram, Nightingale discovered, is certainly worth 1,000 numbers.

One of the strengths of all these diagrams is that they transcend language. They can be read and understood by people across the globe.

Which is why when we launched our first space craft out of our Solar System in 1972, scientists recognised that a diagram was probably our best bet at communicating to any intelligent life out there in space.

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Frank Drake and Carl Sagan created in some sense the ultimate diagram (see above), an engraving that was attached to the Pioneer space probe which would communicate in visual language who we are and where we come from.

It's unlikely that anyone or thing has received our first message to outer space but when they do, it is the clever use of diagrams that has the best chance of saying hello.

Marcus du Sautoy is the Simonyi Professor for the Public Understanding of Science at the University of Oxford. His new series, The Beauty of Diagrams, is on BBC Four at 2030 on Thursdays and also available on iPlayer.