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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