Beautiful Science and a Garden full of Sunflowers

Life is mathematics. We are forever trying to add to our income, subtract from our weight, divide our time and avoid multiplication.  Yet mathematics seems to be a 'dirty word' for a great many people, who like to tell you they 'don't do math.'

Not true.  This post is a little excursion into mathematical theory, and posting it today came from watching the birds in our garden at Bondi, harvesting the sunflowers.  Bear with us -- it's really very simple...

Draw a line A to B on a sheet of paper, then put a cross C on it at some point, not the middle, which you find aesthetically pleasing.  Statistically, the place you are most likely to make this mark is at a point called the Golden Section - that's where the long bit relates to the total length in exactly the same way as the short bit relates to the long bit. (go read that again... it does make sense)

In geometric terms CB:AC=AC:AB or, more philosphically, "the lesser is to the greater as the greater is to their sum."

The ratio of these two lengths is called phi. It works out matehmatically at approximately 1.618 and is believed by experts on these things -- which I clearly am not -- to be fundamental not just to the psychology of aesthetics but to Life, Art, the Universe and Everything.

Really. You may not have known this (I only learned about it rather recently) but it seems everyone else has been on to it for ages.  Stone Age Man used it in the design of his stone circles (which are sort of egg shaped, the length of the long half being ab out 1.618 times the length of the short half). Phi is the ratio of line lengths in any pentagram, the ancient symbol of health and magic. In 1509 Pacioli published a dissertation on phi, De Divina Proportione, illustrated by Leonarda da Vinci. Much more recently, the book and movie The da Vinci Code hinged on this very principle.

In the nineteenth century, a German psychologist discovered that a Golden Rectangle (that is one in which phi is the ratio of the width to the height) was the rectangle most of his subjects found 'pleasant to the eye.'  You can see this rectangle in window sizes, playing cards, writing paper, computer screens, book covers. The frontage of the Parthenon fits almost exactly into a Golden Rectangle.

The Greeks would have calculated this (having first calculated calculations) but it's doubtful if Stone Age Man had a slide-rule handy. He must have copied  from nature. Easy to imagine, since it appears again and again in the natural world, and in the mathematics of growth. You just need to be looking.

You may know it as the Fibonacci series.  Leonardo Fibonacci was the twelfth century mathematician who introduced Arabic numerals to Christian Europe.  Among other things, he devised a mathematical puzzle to predict the monthly progeny from a pair of breeding rabbits.  You calculate each number by adding the two previous numbers, and if you hop to it, you'll find it goes 0,1,1,2,3,5,8, 13, 21, 44...  (or, as rabbits count, some, more, many more, lots and lots, and enough to wipe out Australia)

If you divide any number in the series by the number before, it approximates more and more closely to 1.618 as the series progresses. Take our word for it.

The Fibonacci series is common in the natural world. It describes the relative lengths of the bones of your hand from the endmost fingerjoint to the wrist, say 2 cm, 3 cm, 5 cm, and 8 cm.  If you draw a spiral in which the radius increses according to the Fibonacci series, you are drawing the Fibonacci spiral -- a curve tht increases constantly in size without changing its basic shape and is therefore the classic description of growth in nature. You can see this illustrated in the shape of certain seashells, the spiral nebulae of galaxies, in the shape of the cochlea (that organ in your ear that lets you hear)

Or, closer to home and right here in our garden, the face of a Sunflower -- which in fact is home to TWO Fibonacci spirals -- one turning left, one turning right, both interlaced. It's everywhere in nature, and this little video helps illustrate it.

Curiously the presence of this spiral in the ear links up rather neatly with observations about phi in relation to music. If you write the wavelength of sound as a number (Middle C, for example, is 256 cycles per second) then divide this up by various multiples of phi, you come out with what we would recognize as an harmonious progression of notes.

Which implies that our instinctual appreciation of music and art, that which we like to call a 'sense of beauty', is simply an inherited ability to recognize mathematical symmetry in noise, as well as in colour and shape.  It has been likened to recognizing the face of a friend in a crowd of strangers -- there are certain constant patterns which you can appreciate (either analytically -- like the Pythagoreans -- or recognize -- as Stone Age Man may have done -- from the recurring designs in nature, astronomy and the human form.

In other words aesthetics is not a highbrow thing, nor is it useless or effete. It is an essential component of our psychological ability to create things because it enables us to measure what we've made against the most time honoured blueprints. As we keep re-discovering, effective art and design is not about the shock of the new. What works is the shock of the strangely familiar.  All of which argues for the inclusion in our educational system of not only attention to mathematics, but music and art as well.

Thanks for the science in this post goes to an old article by Dr. John Collee. Thanks for the sunflower photo that opens the post goes to our very own Sarah.