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Why the golden ratio isn’t so goldenAlthough many people find mathematics difficult to understand or boring, the media often has these occasions where something mathematical really fascinates it and the whole world’s attention is suddenly turned to mathematics, or, at least, that part of it. A topic that has ignited many people’s passion for mathematics and mine too is the golden ratio. What the golden ratio is exactly I will come back to, however, it appears to be really closely linked to nature, which made many people shocked and amused. Unfortunately to all these people, I’m here to show them that this golden ratio is not so golden and that there are many ratios that are also fascinating.To understand what the golden ratio is, we must go back to an Italian mathematician of the 12th century called Leonardo Bonacci or, more famously, Fibonacci. He was responsible for popularising the Hindu-Arabic numerical system in the Western world and was considered to be “the most talented Western mathematician of the Middle Ages”. Amongst his most notable works was the Fibonacci sequence which was actually the solution to a problem that he had posed.Fibonacci was interested in a question about the growth of populations, in particular, he asked the question involving the growth of rabbits based on a set of idealised assumptions that he had made. The solution to this was a sequence of numbers that later was called the Fibonacci sequence. To generate this sequence, start with the numbers 1 and 1. After that, your next term is the sum of the previous two terms. 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5, and so on. If you continue doing this, you end up with the following sequence:1, 1, 2, 3, 5, 8, 13, …Mathematicians, centuries later, had a look at this sequence and wanted to study it further. What they wanted to find out was how this sequence was behaving as you tend towards infinity. They found that the ratio of consecutive Fibonacci numbers, numbers in this sequence, tends towards a whole number. Before I tell you what this number is, let us try to figure it out.If we take the ratio of the first two terms, the answer is (1/1 =) 1. The ratio of the next two terms is (2/1 =) 2. We can continue on like this to get a sequence of ratios that is as follows:1, 2, 1.5, 1.67, 1.6, 1.625, …It seems like the ratios are not blowing up to infinity or shrinking to zero. In fact, it looks as though they might be approaching a number that is somewhere around 1.62. This is the case. We can work out this number by, say, calling it x. If we go out large enough, then a Fibonacci number, say, F would be followed by two more terms. To get from one term to the next, you multiply by this constant ratio, x, so your next two terms would be F times x and F times x squared. But, remember, an interesting property of the Fibonacci sequence is that for any given term, the Fibonacci number is the sum of the previous two terms, which means that:F + x*F = x2*FIf you cancel out the F’s and rearrange the equation you end up with:X2 – x – 1 =0This is a quadratic equation that we are all familiar with from high school or even middle school. The solution to this can be found using the quadratic formula and the answer that you get is (1 + 5)/2, which actually comes out to be 1.618…, which is pretty close to the rough estimate that we had at the beginning of 1.62.But why is any of this important? Why does it matter that the sequence of consecutive terms in some sequence is this number? The reason is that this number is actually a very special number and it is called the golden ratio. It is called this because people believe that it appears all around us in nature. It is sometimes called the “divine proportion”. I’ll give you just a few examples of where it appears in nature.The first example is in your own human body, which I find incredible. You’ll find that most of the parts of your body will follow the Fibonacci numbers. You have one nose, two eyes and ears, three segments on each limb and five fingers on each hand. These numbers, 1, 2, 3 and 5, are one of the first few terms of the Fibonacci sequence, whose ratio tends towards this amazing golden ratio. Still not convinced of the beauty of this sequence and this ratio?You also find the Fibonacci sequence in plants, where the leaves of plants rotate by fractions of a full turn that correspond to the ratio of two successive Fibonacci numbers. Leaves may sometimes turn by a ? of a full turn (1 and 2 are Fibonacci numbers) or 3/5 (3 and 5 are Fibonacci numbers) and so on. In fact, the reason that this happens comes down to how the golden ratio is viewed as the most irrational number. This might seem ridiculous because ‘most’ seems like more of a qualitative measure and you can’t really put a value on how irrational a number is, but the surprising fact is that you can. Why this is the case goes back to how the continued fraction expansion of the golden ratio is a series of 1s, but that’s a different topic. What is important is that because the golden ratio is so irrational, plants, for some unusual reason, are inherently making leaves that turn by that fraction of a circle, so that no two leaves overlap and, thus, they can get the most sunshine. This is because if the leaves had turned by a rational amount or an amount that wasn’t very irrational, eventually they would have come to (or have come very close to) completing a full turn, which would make them in the same position as where they had started and their sunlight would be blocked by a previous leaf.All this seems to show that the media is right and does deserve to get excited about the golden ratio. The reality is that there are other ratios that are just as interesting, if not more interesting, than the golden ratio. One example of this is the silver ratio. You can get the silver ratio in a similar way to how you got the golden ratio. All you do this time is you start with 1 and 1 and then your next term is the sum of twice the previous term and the term before that. So, your next term would be (2*1 + 1=) 3. The one after that would be (2*3 + 1=) 7, and so on. Again, the ratios of consecutive terms of this new sequence approach a constant number, which is the silver ratio. This ratio is equal to 1 + 2, which is about 2.414….This ratio also appears in nature, sometimes also where you were used to finding the golden ratio appearing. Some plants actually produce petals, as well, that turn a fraction of a whole turn equal to the ratio between consecutive terms of this sequence, which tends towards the silver ratio. The silver ratio also has properties of its own. For example, if you have a rectangle whose side lengths are 1 and the silver ratio and you take out the two largest squares from either side, you will get another rectangle whose side lengths to have the silver ratio between them. You can continue to do this, and you will always get a ‘silver rectangle’. So, it turns out that the silver ratio is just as enthralling.Even more jaw-dropping, there is actually a whole family of these types of ratios that you can get starting from 1 and 1 and building a sequence from that. As you change the amount by which you multiply the previous term and sum it with the term before to get the next term, you form new sequences. Depending on this number and the resulting sequence, you will get a corresponding ratio. If this number is one or two, the ratio will be the golden or silver ratio, respectively. If it is bigger than that, 3, say, it will be the bronze ratio or, in general, a metallic ratio. All of these ratios have special properties and find a place in nature. It turns out then that the golden ratio isn’t that special, instead, it is the elegance of the mathematics that gives you these numbers, like the golden ratio and metallic ratios, that is very special. ................
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