6 + 2 = 8 is not the same as 8 = 6 + 2

6 + 2 = 8 is not the same as 8 = 6 + 2

The azaleas on the small roundabout in front of my home have been delighting me for many weeks as I have walked out of my front door each morning. They are probably past their peak now, but they have been a bright sea of pink against the green and grey backdrop of the mist-covered peak of Ma On Shan, the mountain that overwhelms the view from the front of my home.

I have been enjoying the blooming of the azaleas, but it was only this week that I realised something. Whether it is the azaleas, or any of the other flowers in my spring garden - such as hibiscus, sweet peas, daisies, or whatever - they all have numbers of petals that are Fibonacci numbers. The Fibonacci sequence is perhaps the nearest thing to magic in mathematics: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, and so on. I had never thought about it before, but it is evidence that the Golden Ratio (the ratio of any two sequential Fibonacci numbers, which approximates to 1.618034) does represent a kind of natural law - or God’s Number - depending on your perspective.

Fibonacci numbers and the Golden Ratio crop up everywhere in nature - the number of petals on flowers, the ratio of distances between parts of the human body, the spiral of a sea shell, the pods on a pine cone, the spikes on a pineapple - everywhere - it seems to be the basic mathematical building block of our planet. The well-known rarity of the four-leaf clover (4 not being a Fibonacci number) is the exception that helps proves the point! And this wonder has been happening outside my front door without my really being aware of it.

Mathematicians claim that to be appreciated, proofs needs to be beautiful and elegant. The Golden Ratio may not be a proof of anything - although it may be evidence of something! - but it IS undoubtedly beautiful. We need only to look at the Mona Lisa, which is based largely on multiples of the Golden Ratio, to understand its beauty! As a celebration of the beauty of the Golden Ratio, the image at the end of this blog shows the Golden Ratio, expressed to the first 10,000 places, as interpreted by me this week in Photoshop.

If the widespread occurrence of the Fibonacci sequence in nature is something we can take for granted, other aspects of mathematics seem much less certain. Mathematics is not my specialty, but I was reading this week that back in 1913, two mathematicians (B Russell and AN Whitehead) took 362 printed pages of algebra to prove that 1 + 1 = 2. And yet, despite this thorough claim of certainty, just 20 years later the Austrian mathematician, Kurt Gödel was able to prove mathematically that it is impossible to prove that any formal mathematical system is free from contradiction. He didn’t say that mathematics actually contains any contradictions, only that we cannot be certain that it doesn’t.

In this context of this background, I really appreciated the clarity of a short item written by Edward de Bono on his website back in 2004. Discussing logic and creativity rather than mathematics, he made the point that most people cannot distinguish between 6+2=8 and 8=6+2. As he points out, the difference can be rather important!

The addition of 6 and 2 cannot produce any answer other than 8. On the other hand, 8 can be made up of combinations other than 6 and 2 (5+3, 4+4, 7+1). This is important because people tend to believe that if they have the ‘right’ answer, then there is no need to think further because you can never be more than correct.

Too many people think that having the right answer means you do not have to listen to other answers because they can never be ‘more than right’. De Bono claims that the result is a severe limitation on thinking - and I agree with him!

Quite simply, it is the difference between convergent thinking - which is linear and limiting - on one hand, and divergent thinking, which is creative, liberating, and full of potential, on the other!

Sunday, 25 February 2007