John Smithiam #1: Golden Ratio

Having just completed a chemistry exam and feeling especially nerdy, i thought i'd post one of the curious bits of trivia garnered from maths tutorials with JS.

For those of you who don't know, JS is a one-of-a-kind erudite (he taught us that word!) who doesn't just teach maths formulas and proofs, but also the whys and whereins of philosophy, languages, the sciences and mathematical history (yes remarkably maths didn't just turn up in Cambridge text books, fully formed and impossible), peppered with slightly self-indulgent anecdotes on men in Port Macquarie who run entire Italian restaurants by themselves. And then he ushers his students out into the world, more informed citizens than when they came in a few hours ago. Good value for a maths lesson indeed!

So... yes, here goes-


John Smithiam #1: The Golden Ratio

The Golden Ratio (1.61803399 : 1) has fascinated intellectuals of diverse interests for over 2400 years:


"Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in Ancient Greece, through the medieval Italian mathematician Leonardo of Pisa, and the Renaissance astronomer Johannes Kepler to present-day scientific figures such as Oxford physicist Roger Penrose have spent endless hours over this simple ratio and its properties.


But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics."


—Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number



So how did a classroom of yr 11 3-uniters get introduced to its wonders?


We had just completed a question that asked "what dimensions should a can made from a [insert area] sheet of alumimium have, so that it has the maximum volume possible?" and the answer had been 5 x 5 x 5 cm or some other cubish answer.

Question: This got JS pondering "if the most economical shape for a can is in cube-like proportions, then why are most cans more rectangular looking?" - as you do.

Answer: Apparently a few years ago, some experiments were done where people were given a sheet of paper filled with rectangles and squares and told to pick the rectangle which was most aesthetically pleasing. And most people picked the Golden Rectangle (the one in Golden ratio). Hmm..

Research was done. It turns out that Golden rectangles are pretty special because when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, which leads to an approximation of the golden spiral. Lots of golden things!

The neat thing about the golden spiral is that its not just some random abstract concept, but it is present in nature.
- snail shell
- sunflowers
- special crystal structure.

And the golden ratio is found in art (it was purportedly used in da Vinci's Mona Lisa)

http://www.geocities.com/jyce3/ - other weird Golden ratio applications

So yes, that is why Campbell's cans of soup come in Golden-ratio, rectangular height cans; because that's what most pleasing to the human eye. And what's pretty sells well. A sort of round about way of saying it. But still.

-- C.L --