The Golden Ratio Ah, the Golden Ratio. No other number has ever received so much mystical devotion. Old texts on aesthetics even call this number the “Divine Proportion”, such is its reputation in arts. Also known by the Greek letter φ (phi), this curious irrational number has a closed-form given by: φ = (√5 + 1)/2 = 1.61803398875… Since it is one of the solutions of the polinomial equation x² - x - 1 = 0, this number is considered an algebraic number, as opposed to being trasncendental, like π or e are. From nautilus shells, the human body to spiral galaxies, the Golden Ratio seems to be everywhere in Nature, right? Well, not really. A very large portion of what you have probably heard about this number is just hype, widely propagated myths, extremely far-fetched analysis of data or, putting it mildly, just made-up nonsense. Now, don’t get me wrong here. The Golden Ratio really is a very interesting number with a number of outstanding mathematical properties. This is why it saddens me to see so many people praising it for all the wrong (and wildly innacurate) reasons. For instance, several spirals in nature are logarithmic spirals because they are the same independent of the scale. This sort of thing is bound to show up whenever you have exponential growth in a circular fashion, two phenomena that are extremely common in nature. In the end, logarithmic spirals are really just exponential functions in polar coordinates. However, not all logarithmic spirals are Fibonacci spirals. In fact, what it is known as the Fibonacci or Golden spiral, derived from the famous construction using nested squares and golden rectangles (shown below), is a very gimmicky geometric construction that really shouldn’t be expected to show up in nature at all. Nature doesn’t work with squares and rectangles!
Posted on: Sat, 10 Aug 2013 01:03:27 +0000
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