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What do the branching of leaves on a stem, the structure of quasicrystals, the architecture of Le Corbusier, and the music of Debussy all have in common? These phenomena share a close association with a single, extraordinary number, known as the 'Golden Ratio'. This number, which is equal to half the sum of 1 plus the square root of 5, was first defined by the systematizer of Greek geometry, Euclid of Alexandria, around 300 B.C.

Euclid's definition was geometrical. Take any line segment and divide it into two parts, in such a way that the longer part is in the same proportion to the shorter part as the entire line segment to the longer part. In this case, the ratio in question is equal to the Golden Ratio.

The Golden Ratio is often represented by the Greek letter phi f . Euclid's definition of the Golden Ratio gives us a quadratic equation that we can solve using the Formula to give an algebraic expression for the value of phi.

The quadratic equation also implies a special relationship between the value of phi and the value of its inverse.

Like pi (the ratio of the circumference of a circle to its diameter), the Golden Ratio is an irrational number, one that cannot be expressed as a ratio of two whole numbers, such as 3/2, or 23/41. What makes the Golden Ratio so interesting is its propensity to pop up where least expected. A rectangle whose ratio of length to width is equal to the Golden Ratio is known as a golden rectangle.

Golden rectangles have the property that, if a square is sliced away from them the rectangle remaining is a smaller golden rectangle, as shown above. This process can be continued indefinitely. We can draw a logarithmic spiral within the original golden rectangle, that passes through a vertex of each of the smaller golden rectangles. This is the shape of the cross section of the Nautilus shell shown above.

© Salvador Dali, Gala-Salvador Dali Foundation, DACS, London 2006
© ADAGP, Paris and DACS, London 2005

Over the past two centuries, many authors have claimed that the golden rectangle is the most aesthetically pleasing of all rectangles. Although numerous psychological experiments have given ambiguous results, the Golden Ratio has been used in many works of art. For example, the shape of Salvador Dali's 'The Sacrament of the Last Supper', shown above, is a golden rectangle. Furthermore, a huge dodecahedron, a regular polyhedron with twelve pentagonal faces, one for each disciple, is seen floating above the table. The dimensions of regular figures with five-fold symmetry, such as pentagons, dodecahedra and icosahedra are intimately related to the Golden Ratio. For instance, the vertices of an icosahedron are situated at the centres of the faces of a dodecahedron, but they are also the same vertices as three perpendicular golden rectangles, as shown in the animation above.

If a regular pentagon has edges of length one, the length of the edges of the inscribed pentagram will be phi. The diagram above shows how the Golden Ratio makes its appearance in the pentagon and by extension all other figures with fivefold symmetry.

Step 1

One edge of a pentagon is coloured red and one edge of the inscribed pentagram is coloured black. The ratio of the lengths of these lines is R.

Step 2

The edge of the pentagram has been divided into two parts. The red part is the same length as the edge of the pentagon. The remainder is black.

Step 3

The black remainder is equal in length to the edge of a smaller pentagon. The red line is the edge of a pentagram inscribed in this smaller pentagon. So the ratio of the length of these lines must also be R. Therefore the lengths of the edges of a pentagon and the inscribed pentagram stand in the same relationship as that given in Euclid's definition of the Golden Ratio.

To read more about the Golden Ratio see the book: 'The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty', by Mario Livio (October 2002, Headline).       

The illustrations on these pages have been adapted from the CD-ROMs: 'Art and Mathematics' and 'Life, the Universe and Mathematics' by Nick Mee, published by Virtual Image.