I was captivated when I came across the work of a young geometer called John Martineau while he was studying at my School of Traditional Arts some years ago. He decided to make a close study of how the orbits of the planets relate to each other and how the patterns that can be drawn from them fit so precisely with things made down here on Earth. He found many rather beautiful relationships but, as an example, consider the one that exists between the orbits of Venus, Mercury and the Earth.

The five-pointed star

Ancient astronomers long ago discovered that Venus makes an intriguing journey across the sky. It takes eight years -- or thirteen "Venusian years" (note the relationship) -- for it to return to exactly the same position it is found in today, and I mean exactly. It is such an accurate cycle that many cultures down the ages and in very different parts of the world have all noticed this and used Venus as a way of regulating their calendars. For example, there is a vast pre-historic site at Newgrange in Northern Ireland which, for hundreds of years, was thought to be some sort of prehistoric burial mound. More recently studies have shown that, whatever else it might have been used for, Newgrange is also a very precisely aligned "Venus trap." Once every eight years, and only for a very short period on that one day, the light cast by Venus when it appears at sunrise as the Morning Star passes down the long entrance tunnel and hits the wall at the back of the inner chamber that lies at the very centre of the mound. It does this so regularly that the researchers who carried out their study claim its accuracy was only slightly improved with the invention of the atomic clock.

As ancient astronomers charted Venus's progress through its eight-year cycle, they discovered that it describes a swirling rose-like pattern. The illustration was made by John Martineau as he tried to verify how ancient cultures devised the symbols that are still so familiar today. The Earth is at the very centre of the picture. There are moments when the line comes closer to Earth and then moves away again, creating a circle of five petal-like shapes. If we were to join the tips of that pattern together, as ancient astronomers clearly did, then what is revealed is a shape familiar the world over, the endless line that forms the five-pointed star. It is a shape that contains some breathtaking secrets.

The orbits of the planets are not perfectly circular, but it is possible to refine their elliptical shapes without altering their length so that they become perfect circles. Such a circle is called the "mean orbit" of the planet. John Martineau found that putting scaled drawings of the mean orbits of the Earth and Mercury together on a piece of paper reveals an extraordinary correspondence between them. In the image the Earth's orbit is the bigger circle that contains the five-pointed star. The smaller circle is the mean orbit of Mercury, which sits within the orbit of the Earth in such a proportion that it fits exactly over the pentagon at the heart of the five-pointed star. If that were not itself astonishing enough, the same thing happens if a scaled drawing of the actual physical body of the Earth is overlaid with a scaled image of the actual physical body of Mercury. Mercury, once again, sits inside the circle of the Earth's circumference in exactly the same proportion. The pentagon shape at the heart of the five-pointed star is once again enclosed by Mercury's circumference.

This may, of course, all be a coincidence, but such is their precision it does begin to challenge the popular notion that we live in an accidental universe, especially when the same things happen at levels of the material world that we cannot see without the aid of a microscope. For example, we are now all familiar with the double-helix shape of the DNA molecule. Deoxyribonucleic acid is present in nearly all forms of life and transmits all of our genetic information from one generation to the next. The less familiar image of the molecule is the view taken from the top of the double helix. When we look down on the molecule through a microscope the image is not dissimilar to the one of Venus's journeys across the night sky. It is a swirl of patterns with ten protruding petals. If every other petal is connected by a series of straight lines, once again what emerges is the same five-pointed star. This five-pointed star, found in so many petal arrangements on flowers, appears constantly in the patterns and architectural designs in Islamic buildings as it does in Christian structures and it also underpins the structure of some of the most familiar man-made objects we know.

Take the image of a Stradivarius violin, for example, and place it in a circle that already contains a five-pointed star and the impact is just as breathtaking. All of the key proportions of the violin fit the geometry of the star perfectly. And notice that the base of the violin is also the product of those two overlapping circles that create the mandorla shape with which we began this demonstration. For the ancients the two overlapping circles also represented the Sun and the Moon. The Sun, of course, is much bigger than the Moon, although from Earth this does not always appear so. Every so often we are still drawn to marvel at a total eclipse of the Sun when the Moon, seen from Earth, is exactly the same size as the Sun. So, even in the sky, the grammar of harmony is at play. This is all pretty remarkable evidence that there is a mysterious unity about the patterns found throughout the whole of creation. From the smallest of molecules to the biggest of the planetary "particles" revolving around the Sun, everything depends for its stability upon an incredibly simple, very elegant geometric patterning -- the grammar of harmony.