![]() ![]() Lastly although I came across these results concerning Fibonacci powers on my own (see also my previous comment about 5), I daresay they aren't new discoveries. For example 13 takes 7 numbers to get to its square, and 13 occupies position 7. Note also the number of numbers in each sequence, which is equal to the position of the start number in the standard Fibonacci sequence. This can be remedied perhaps with: 0 1 1) (My treatment of 1 seems a bit anomalous here since although it 's a perfect square, I haven't presented it as the result of any addition. In fact the only start numbers we can hit the square with seem to be the Fibonaccis and no others: But then 4 and 6 aren't in the Fibonacci sequence either. The same goes for 6: 6 6 12 18 30 48 doesn't include 36. Try this with 4 however: 4 4 8 12 20, and we don't land on 4 squared. Now let's pretend 5 is the first Fibonacci number instead of the usual 1, but still use the same addition algorithm: 5 5 10 15 25. To find out more read The life and numbers of Fibonacci. The sequence is also closely related to a famous number called the golden ratio. You can find it, for example, in the turns of natural spirals, in plants, and in the family tree of bees. Real rabbits don't breed as Fibonacci hypothesised, but his sequence still appears frequently in nature, as it seems to capture some aspect of growth. And from that we can see that after twelve months there will be pairs of rabbits. Starting with one pair, the sequence we generate is exactly the sequence at the start of this article. Therefore, the total number of pairs of rabbits (adult+baby) in a particular month is the sum of the total pairs of rabbits in the previous two months: Writing for the number of baby pairs in the month, this gives Writing for the number of adult pairs in the month and for the total number of pairs in the month, this givesįibonacci also realised that the number of baby pairs in a given month is the number of adult pairs in the previous month. He realised that the number of adult pairs in a given month is the total number of rabbits (both adults and babies) in the previous month. You might be surprised how often it will turn out to have a golden ratio after all or even a rule of thirds.Fibonacci asked how many rabbits a single pair can produce after a year with this highly unbelievable breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month). Avoid distractions and use lines and curves to connect the elements in your composition. I think it’s much wiser to find balance in a photo. You might end up being more of a mathematician instead of a photographer. It also illustrates that using these rules to the letter is often not very artistic. It’s nice to have all these rules, and I find it important to know about its origin to understand why you should use it or not. ![]() On the other hand, the rule of thirds is independent of the aspect ratio of an image. So, you may think you’re using it, but you're not. But this way, you’re not looking at the lines of a golden ratio anymore, because that has an aspect ratio 1.618:1. This can be 3:2, 4:3, or 5:4, to name a few. Those crop tools project the rules inside the aspect ratio of the image itself. The golden ratio projected over the 5:4 aspect ratio. The number is 1.618 followed by an infinite amount of digits, and the Greek letter for it is Phi. This irrational number is the golden ratio. This ratio is an irrational number, which means it cannot be written down as a natural fraction. A Greek called Euclid, who lived somewhere around 400 BC, found out that the division of a line according to a certain ratio can go on indefinitely. It had nothing to do with composition in paintings. ![]() The golden ratio is an invention of mathematicians. ![]() The Golden Ratio, Fibonacci, and Archimedes Let me explain why by diving into history. At first look, the similarity with the golden ratio does make it seem the two are related. In fact, the origin has absolutely nothing to do with drawing lines at thirds of the image. Sort of.Īlthough this can help to achieve an acceptable composition in some situations, the rule of thirds was never intended that way. Place your subject on one of the lines or at the intersection of the lines, and you’re done. Divide the image in nine equal parts, by drawing two horizontal and two vertical lines at thirds from the edges. The rule of thirds is quite a simple rule. ![]()
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