Science Spin November 2009

Beauty is in the 'Phi' of the beholder

They say if you could bottle beauty, you would make a fortune. Well, mathematicians - with the help of nature - figured out the secret of beauty hundreds of years ago and many of them died in poverty. So much for what 'they' say! You see beauty can be reduced down to a number. It even has a name. It is called phi.

So, what exactly is phi? Phi is an irrational number that approximates to 1.6180339887. It can also be expressed as a mathematical constant. Phi has many aliases; throughout history it has been known as the golden section, golden mean, golden number, golden ratio, golden cut, the divine proportion.

Important in mathematics, art and geometry, phi can be defined as 'the proportion arising from the division of a straight line into two segments so that the ratio of the whole line to the larger part is exactly the same as the ratio of the larger part to the smaller part'. Quite a mouthful.

INFLUENCE

As you will see this simple line division has had a massive influence on mathematics, philosophy, geometry, art and architecture to name but a few, and botany, physiology and cosmology to name a few more. Some of the greatest thinkers of all time have been fascinated by this number. From the ancient Greeks, Euclid and Pythagoras, to Italian mathematician Leonardo of Pisa, and Renaissance astronomer Johannes Kepler, the mystery of phi has perplexed, captivated and inspired many of our greatest artists, philosophers and scientists from as early as the 3rd century.

For instance, Pythagoras and his followers believed so strongly in the power of the number, that they felt if they worshipped it, it would reveal to them the hand of God. Leonardo da Vinci paid homage to this number by incorporating the 'divine proportion' into his iconic painting the Vitruvian Man. As science guru Professor Mario Livio put it, 'the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics'. Read on and you'll see why.

FIBONACCI

Now, one of the simplest methods of deriving phi is to use what is arguably the most famous mathematical sequence of all time - the Fibonacci sequence. For those of you not familiar with Leonardo Fibonacci's famous series, here's a crash course introduction. Beginning with the numbers 0 and 1, each subsequent number in the series is found by summing the two numbers preceding it. For instance: 0,1,1,2,3,5,8,13,21,34,55,89,144...... You will find that the ratio of each successive pair of numbers in the series is approximate to phi e.g. 5 divided by 3 is 1.666...; and 8 divided by 5 is 1.6. After the 40th term in the series the ratio is accurate to 15 decimal places.

Phi can also be derived by solving the quadratic equation which comes from the geometric solution of a golden cut: n² - n - 1 = 0. The positive solution to this equation is the square root of 5 plus 1 divided by 2:

(√5 + 1) / 2 = 1.6180339... = Phi.

BEAUTY

The real beauty of this number lies in the fact that it is ubiquitous; it is everywhere; it is in everything. As I will exemplify, it not only crops up in G.H. Hardy's metaphysical notion of a mathematical reality that 'lies outside us', but reveals itself in our physical world too. The real beauty of phi lies in the fact that this magical number is within the grasp of anyone who cares enough to look for it. Let's look for it.

We'll take some accessible everyday examples to prove the point - and what could be more accessible than the human body? It is here that you will find approximations of phi in the most unlikely places. If you divide the length of your forearm (elbow to wrist) by the length of your hand, you'll get phi. The distance between your shoulder line and the top of your head, divided by the length of your head will give you phi.

Divide your height by the distance between your navel and the bottom of your foot and again, you'll come up with phi. And that's just the human body. Examples are evident in nature too. Divide the length of a dolphin's dorsal fin by the height of its dorsal fin and you will get an approximation of the golden ratio. Divide the distance between the two upper markings on the wings of a Death's Head moth by the distance between the two lower markings and you will see they conform to phi. Look at the nautilus sea shell. You will find that each new chamber is proportioned by phi relative to the preceding chamber.

It couldn't all just be a coincidence, could it? Well, evidence from the worlds of business, astronomy and geometry suggest that the answer to this question is no. In the stock market, for example, phi-based analysis software is used to identify key turning points in the timing of highs and lows and price resistance points. Look at the planet Saturn's magnificent icy rings and you will see that they show a division of the length by the width of the rings relative to phi.

And in geometry, the five platonic solids; the tetrahedron, hexahedron, octahedron, icosahedron and dodecahedron are all prime examples of beauty in mathematics. Many of their proportions also conform to phi. This list of examples is almost endless. The seeming randomness of so much of that which surrounds us is pulled together by this fascinating number.

COMPARISONS

Phi is also one of the quirkiest numbers. It's the only number known to man which, when diminished by unity equals its reciprocal. No other number to our knowledge possesses this quality. Find another number that does and you will be on a par with the person who solved de Fermat's last theorem.

So how does phi hold its own in relation to the more established and proven mathematical constants? Take for instance pi (π). This Greek letter expresses one of the most important mathematical constants in existence. Like our friend phi, pi is irrational, meaning it cannot be represented as a terminating or repeating decimal.

Therefore it cannot be expressed as a fraction, although the estimation is commonly seen in mathematics text books. Equal to approximately 3.14159, pi expresses the ratio of any circle's circumference to its diameter. Many formulas from mathematics, physics and engineering are derived from pi.

The constant e is another mathematical heavyweight. The irrational number e (2.71828.....) is commonly defined as the base of the natural logarithm. e is a mathematical cornerstone in that its use is necessary to calculate exponential growth. And the exponential function, e, is the only function which is also its own derivative; the true loner of the mathematical world.

When put head-to-head with these and other heavyweights, such as Planck's constant or the gravitational constant, phi could never be described as an imposter. Two quantities are in the ratio of phi if the ratio between the sum of those two quantities and the larger quantity is the same as the ratio between the larger quantity and the smaller quantity. This definition gives us some small idea of the importance of phi in relation to a whole world of mathematical possibilities.

So now, back to the main purpose of this essay - beauty in mathematics and the beauty of this divine ratio. How do we define beauty? Well here I believe the words of philosopher Thomas Aquinas to be as good as any a definition; 'Beauty is that which pleases in contemplation'. In other words, when we discover hidden patterns or truths through the use of our mental faculties, a true beauty is uncovered.

So contemplate this; you will find that the top models in the world have a phi-face; their head forms a golden rectangle with their eyes at its midpoint while their mouth and nose are each placed at golden sections of the distance between their eyes and the bottom of their chin. George Clooney, twice voted Sexiest Man Alive, has a face that conforms to this proportion.

Perhaps Thomas Aquinas liked to contemplate the sunflowers in his garden? But did he realise that sunflower seeds originate from a central point and move radially outwards as new seeds are formed? In fact, it was only 15 years ago that it was proven that the optimal angular displacement for newly-formed seeds is a phi-defined fraction of a circle (0.618034*360, about 222.5 degrees).

The apparent opposing spirals of seeds observed in sunflowers are an optical illusion due to the fact that the ratio of the successive Fibonacci members approximates phi. If you count the apparent number of arms in these spirals, they'll always equal two adjacent Fibonacci numbers whose ratios to the succeeding numbers are slightly above and below phi. Yet another aesthetically pleasing example of the golden ratio.

ART

Without mathematics there is no art'. These famous words of Italian mathematician Luca Pacioli are apt when one considers some of the most beautiful works of art ever created. And the reason? Phi has been used as the template for so many of these works. Take for instance, Michelangelo's David. You will discover the sculpture's proportions conform to phi from the location of the navel with respect to height and the placement of the joints in the fingers.

Another example is Leonardo da Vinci's unfinished canvas Saint Jerome. Look closely and you will see that a 'golden rectangle' fits so perfectly around the saint himself that some scholars believe the artist deliberately painted the figure in proportion to the golden ratio. Knowing da Vinci's fondness for geometry and the divine proportion, this may well be likely. A final, more modern representative of the golden number in art can be found in The Sacrament of the Last Supper by 20th Century Catalan artist Salvador Dali.

You will find that the painting itself fits neatly inside a 'golden rectangle', and the golden proportion was used in the positioning of the figures. A section of an enormous dodecahedron even floats above the table, almost in explicit worship of this golden ratio. The polyhedron in the painting consists of 12 regular pentagons - all of which have golden connections. There is much substantial evidence here to suggest that these famous artists, armed with the understanding that conforming to this golden ratio would make their work more aesthetically pleasing, intentionally included it in their masterpieces.

From the magnificent, structural extravagance of the Parthenon to the most perfectly designed human features, the inclusion of phi cannot be ignored. As we have seen, this wondrous number is everywhere. But it is more than just a number. It permits us to truly appreciate beauty beyond simply a mere aesthetic level.

Perhaps beauty is more than just a metaphysical ideal. Perhaps we can explain beauty on a scientific level. Maybe phi is the key to unlocking the secret of beauty. Could it be that there is a format for aesthetically pleasing objects? If so, the golden ratio could very well be that format. It is no coincidence that many great artists, men who devoted their lives to the artistic expression of beauty, included this 'divine proportion' in their work.

Surely also, it is no coincidence that this fascinating number is found in the most naturally beautiful of objects. From the spiral of the Nautilus sea-shell to the majestic brilliance of Phidias' Zeus, phi is present in both natural and man-made beauty. This simple number, so much more than just another ordinary decimal, allows us to appreciate the ordinary world in a most extraordinary way.

Perhaps that in itself is beauty.

REFERENCES

(1) The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty by Mario Livio
(2) The Divine Proportion: A Study in Mathematical Beauty by H.E. Huntley
(3) A Mathematician's Apology: G.H. Hardy
(4) http://goldennumber.net/ (14 April)
(5) http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm (15 April)

The RDS McWilliams Young Science Writers' Competition is open to students in all parts of Ireland.