φ

Think of two numbers, a and b. Add them together - you have a+b. Add the two previous numbers together - a+2b. Continue and you get 2a+3b, 3a+5b, 5a+8b, etc.

Notice how if you add the coefficients of a and b together, you get one of the next coefficients. That's obvious, because we're constantly adding the previous numbers together. Also notice how the coefficients correspond to the Fibonacci sequence (1,1,2,3,5,...) What's less obvious is the relationship between these numbers - a:b and b:a+b. Let's for a moment assume these two ratios are equal to each other. Algebraically, this can be represented as:

a/b = b/(a+b) = 1 + b/a

Substituting x in for a/b, we see that x = 1 + 1/x - multiply both sides by x, and you see that x^2 = x + 1. Both of the last two equations show something interesting - that when you square x, you're merely adding 1 to x; and when you subtract 1/x from x, you get 1. So what is x?

x is called the golden ratio, which is usually written as the Greek letter phi - φ - and is equal to (1+√5)/2. This happens to also be equal to sin (54) + cos (36), though why this is the case I'm not sure. And φ is everywhere!

It's famously said that φ can be found everywhere in nature, specifically through the Fibonacci numbers. Possibly the most recognisable example is that of sunflowers, where, paraphrasing this book (page 99), during phyllotaxis, the number of spirals is usually a Fibonacci number, and where the seeds themselves are arranged according to the golden angle (about 137.5°).

Another example is that of the golden rectangle, where the sides are of a ratio of 1:φ and which has been known for thousands of years, perhaps with the Babylonians (but we're not sure and Mario Livio doesn't think so, and he's written a book on φ). Supposedly the rectangle was used as the foundation of the construction of the front of the Parthenon in Athens. Even the Great Pyramids of Giza are supposedly based on φ. We also know about how φ has been applied in art, with painters such as Leonardo Da Vinci taking a keen interest in this number and Salvador Dali using a golden rectangle-esque canvas for his painting The Sacrament of the Last Supper. Mozart seemingly based his sonatas on φ, and the same goes for Bela Bartok's work (supposedly). And for Michael Jackson, Tool, and seemingly countless other musicians.

Going back to Livio, who said "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" (an audacious claim to say the least), it's interesting to see how φ continues to fascinate people and serve as an easy topic for newspapers to write about when they want to mention maths. Adobe have a whole bloody article on φ, for goodness sake, and they have a second one mentioning "golden ratio enthusiasts". And whilst I don't know just how prevalent φ plays a part in all these examples, and whether they were intentional or not, it's clear either way that φ has made a lasting impact on maths. I can't deny that the golden spiral looks brilliant, nor that it has an uncanny resemblance to large swathes of nature and the world, nor that the fact it's got such a simple definition yet is so complex isn't fascinating. It's a golden number, that's for sure.

Researching this post made me realise just how many golden things there are in maths. There's even a silver ratio - 1:1+√2! Maybe I'll write about it in the future.