Monday, 10 September 2012

The Mathematical Mechanics of the Pleasing Proportions

                          Most people will remember pi (π) from their schooldays, as in πr2, which is the ratio of the radius to the circumference of a circle. Pi can be variously calculated – a rough approximation is the fraction 22/7, a slightly more accurate form is 3.142. Another ratio, perhaps not as well known, is phi (φ), which is the ratio of one quantity to another, wherein the ratio of the first quantity to the second is in the same ratio as the first quantity to the whole. This can best be seen in this diagram, where a is to b as a+b is to a

Algebraically, this is (1+√5)/2, which gives a rough value of 1.618. This proportion has been given a number of names in addition to phi; it has also been called the Golden Mean, the Golden Ratio, the Golden Section or the Divine Proportion. 

You can easily construct your own Golden Section with a ruler and a pair of compasses. 

Take a line of length AB. 

Draw a vertical line at B equal to half of AB, forming a line BC, then join A to C to produce a right-angled triangle. 

With your compasses, draw an arc from centre C with a radius CB, from B to cross AC at point D. 

Re-set the compasses to radius AD, and from centre A draw another arc from D to cross AB at point E. 

The proportion of AE:EB is the same as AE:AB – the Golden Section or φ. 

Conversely, to increase a line by φ, take your line AB. 

Form a rectangle above AB and divide this rectangle into two exact halves, with a point x halfway between A and B. 

Draw a diagonal line from x to the top right angle above B, giving y. 

Set the compasses with a radius xy and with centre x draw an arc from y to cross the extension of AB at point z. 

That gives you the Golden Section again.
(On the other hand, you could just measure the line and multiply it by 1.618, adding the product to the end of your original line).

The Golden Ratio is not just of interest to mathematicians; artists, architects, biologists, historians, musicians, psychologists and proponents of just about any other discipline have found phi to be of interest. In the arts, phi is particularly interesting because it is so aesthetically pleasing to the human eye. In 1509, the Franciscan friar Luca Pacioli published his three-volume De Divina Proportione in which he explored aspects of the Divine Proportion, together with Vitruvian proportions, and with illustrations by Leonardo da Vinci, Pacioli’s works were enormously influential on Renaissance and later art; the Golden Section continues to be used by artists and designers to this day. However, because the ratio is so satisfying to the eye, many people discover it where it was not deliberately employed, a case in point being the Parthenon of ancient Greece. 

There is no evidence that the Greeks were aware of phi when the Parthenon was built (begun 447 BCE, completed 438 BCE), and Euclid, in his Elements (308 BCE) merely discusses it as an interesting irrational number – the ancients, notably Vitruvius, were more interested in the ratios of whole numbers. This hasn’t stopped people overlaying phi on the Parthenon and finding instances of it, but this is a highly selective procedure, conveniently ignoring instances where phi doesn’t occur. The builders of the Parthenon built to pleasing proportions, certainly, but it was more like the photographer’s Rule of Thirds, where the subject of a picture is placed off-centre, at about a third of the frame, to give a more interesting shot. It just so happens that phi is also very roughly at about a third, which causes some to see it where it hasn’t actually been used. 

It’s not unlike things like leys, the supposed lines linking ancient ‘holy’ sites. It’s fun to get a map and a ruler and make your own – I’ve seen lines that perfectly align petrol stations, branches of Woolworths or telephone boxes across the landscape – showing that if you look hard enough, you’ll eventually discover whatever it was you set out find.

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