The Fibonacci sequence, known to all in the film

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The Fibonacci sequence, known to all in the film "The Da Vinci Code" - a series nature magazine of numbers, as described in the form of riddles Italian mathematician nature magazine Leonardo of Pisa, better known by his nickname Fibonacci, in the XIII century. In short the essence of mystery: Someone placed a pair of rabbits in a certain confined space, to see how many pairs of rabbits will be born at the same time throughout nature magazine the year, if the nature nature magazine of rabbits is that every month a pair of rabbits gives birth to another nature magazine couple, and the ability to produce offspring with They appear to reach two months of age. The result is a sequence of 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, where display number comma pairs of rabbits in each of the twelve months. This sequence can be continued indefinitely. Its essence is that each successive number is the sum of the previous two.
In this sequence, there are a number of mathematical features nature magazine that are sure to touch. This sequence is asymptotically (approaching slower and slower) tends to a constant ratio. However, this ratio is irrational, nature magazine then there is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It can not be expressed nature magazine exactly. So the ratio of any term of the sequence to the preceding nature magazine number is around 1,618, a paz is excellent, it is not reaching it. The following similar approaches the number of 0,618, which is inversely proportional to 1,618. If we divide the elements of the sequence by one, we get the number 2,618 and 0,382, which are also inversely proportional. This is the so-called Fibonacci ratios. nature magazine What is all this? So we come to one of the most mysterious phenomena of nature. Fibonacci in fact not discovered anything new, it just reminded the world of the phenomenon called the Golden Section, which is not inferior to the significance of the Pythagorean theorem. All objects around us, we distinguish between including and form. We like some more, some less, some do push glance. Sometimes interest may be dictated by the situation in life, and sometimes the beauty of the observed object. Symmetrical and proportional form, promotes the best visual perception and evokes a sense of beauty and harmony. The whole image is always consists of parts of different sizes in a specific ratio to each other and the whole. Golden ratio - the highest manifestation of the perfection of the whole and its parts in science, art and nature. If a simple example, the Golden Section - this division of the interval into two parts in such a ratio in which most applies to lower their sum (full line) to the most. If we take the entire length of c to 1, then a segment will be equal to 0.618, the length b - 0,382, so only condition is met Golden Mean (0.618 / 0.382 = 1.618, 1/0, 618 = 1,618). The ratio of c to a is equal to 1.618, but with a b 2,618. It's all the same, we are already familiar, the Fibonacci ratios. Of course there is the golden rectangle, the golden triangle, and even a golden cuboid. The proportions of the human body in many ratios are close to the Golden Section. Image: marcus-frings.de But the fun begins when we combine this knowledge. The figure clearly shows the relationship between the Fibonacci sequence and the golden ratio. We start with two squares of the first size. Add on top of the square of the second dimension. Paint on the next square nature magazine with sides equal to the sum of the previous two parties, nature magazine a third the size. By analogy, there is a square of the fifth dimension. And so on until you get bored, as long as the length of each side of the next square was equal to the sum of the lengths of the sides of the two previous ones. We see a series of rectangles, the length of the sides, which are Fibonacci numbers, and, if not strange, they are called Fibonacci rectangles. If we draw a smooth line through the corners of our squares, we get nothing but a spiral of Archimedes, which is always increasing spacing evenly. Sound familiar? Photo: ethanhein on Flickr and not only in clam shell can find the Archimedean spiral, and in many plants and flowers, they're just not as obvious. Aloe Multivalent: Photo: brewbooks nature magazine on Flickr Romanesco Broccoli: Photo: beart.org.uk nature magazine Sunflower: Photo: esdrascalderan on Flickr Pinecone: Photo: mandj98 on Flickr And if you look a little further away, you can see the Fibonacci sequence in the unattainable galaxies. nature magazine And then it's time to acces