Again we see that the two sequences match up, so that the denominator of the approximation cancels out to produce an integer:īeyond 1 + √2 and the golden ratio, we can easily find other numbers with the same property. If we expand the sequence of powers, we can compare this with a sequence of approximations – this time from the continued fraction for √2. My internet surfing landed on a reference to a similar property for √2 + 1 (sometimes known as the silver ratio), whose powers converge towards integers even more quickly: But this isn’t just a peculiar feature of that number. Much has been written – including much nonsense – about the special properties of the golden ratio. But looking at the powers in this way does help to make some sense of what’s happening as the powers approach integers. I would hesitate to say that this “explains” the phenomenon of near-integer powers – there’s no doubt some deeper, underlying reason why the sequence of approximations fits the sequence of powers so neatly. But as we go up the sequence of powers, substituting the nth approximation for the GR in the expression for the nth power, the denominators cancel out each time and we get better and better approximations. This example comes early in the sequence, and 7 is not a particularly good approximation for GR 4. This means that, for example, if we select the 4th approximation and substitute it for the golden ratio in the expression for the 4th power (on the right of the table above), the denominator of the approximation cancels out to produce an integer. Note also that, in general, the nth approximation is F n+1/F n. The first few approximations are grossly inaccurate, but by the time we get to A 5 the approximation is accurate to around 1%, and A 8 is within 0.05%. It produces approximations (or convergents) made up – once again – of Fibonacci numbers: This is one of the simpler continued fractions, being made up entirely of 1s: The final piece of the jigsaw involves the continued fraction for the golden ratio. And more generally, the nth power is F n x GR + F n-1. For example, the 4th power is F 4 x GR + F 3. Note that each power can be expressed as a Fibonacci multiple of the GR plus another Fibonacci number. This is in itself a remarkable property, and is a convenient way of looking at the power series. More generally, the coefficient of the √5 term for the nth power is half of the nth Fibonacci number.īut we can alternatively express the powers in terms of the golden ratio, as on the right-hand side of the table. If we look (left side of table) at the coefficient of the √5 term for say the 4th power, we see it’s 3/2 – or half of F 4. In fact the Fibonacci numbers appear in two ways. Second, if we expand the powers of the golden ratio, Fibonacci numbers appear again: Let’s start by labelling numbers in the Fibonacci sequence as follows:Ĭomparing this with the sequence of powers above, it’s apparent thatĪnd in general we can say that (1/2 + √5/2) n ≈ F n-1 + F n+1. The link between the golden ratio and the Fibonacci numbers, however, provides an intriguing clue.įirst, it’s worth noting that the integers to which the powers above converge are sums of two Fibonaccci numbers. What was more difficult to find was any clear explanation of why the powers approach integers. The effect starts to show up from the 4th power and becomes very obvious thereafter:Īs John D Cook shows in his blog, the values become closer to integers at an exponential rate as the powers increase. While exploring the world of continued fractions, I came across a reference to a seemingly unrelated fact: the powers of the golden ratio (1/2 + √5/2) are remarkably close to integers. It turned out that 13/15 was a remarkably good approximation for √3/2, which was crucial for placing the stars. My interest in continued fractions was sparked by working out how to knit the stars in the EU flag earlier this year.
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