دنباله فیبوناچی در طبیعت

پنج كار وي در اين مدت به صورت زير بود:
لئوناردو فيبوناچي در سال 1170 در شهر پيزا در كشور ايتاليا به دنيا آمد. پدرش  جليلمو بوناچي تاجر بود. پدرش وي را در سال 1192 با خود به بوجيا برد.پدر لئو ناردو وي را براي آموزش كار و تجارت به كشور هاي مصر، سيريا ، يونان ،سيسيل و چند كشور ديگر فرستاد . لئوناردو از اين مو قعيت استفاده كرد و در خلال اين سفر ها با فنون محاسباتي در اين كشورها آشنا شد وآنها را فرا گرفت.
حدود سال 1200 بود كه لئوناردو به شهر خودش پيزا بازگشت و پس از آن شروع به كاركردن بر روي سيستم ابداعي خود شد. كار بر روي ليبر آباچي (1228-1202) ، هندسه كاربردي (1220-1221) ، نامه بدون تاريخ به تئودور (1225) ، مجموعه راه حل هاي مس‍‍أله هايي كه فردريك دوم به او داده بود(1225) ، كتاب نظريه اعداد كه در آن به حل معادله درجه دوم اشاره شده بود. و همين كارها باعث شهرت وي در رياضيات شد.
اتفاقاتي كه پس از سال 1228 بر وي اتفاق افتاده است بر كسي روشن نيست و تنها نكته اي كه به آن اشاره شده است،اينست كه به پاس خدماتي كه او انجام داده بود ، از طرف شاه برايش مقرري اي در نظر گرفته شده است.
لئوناردو فيبوناچي چندي پس از سال 1240 و به احتمال زياد در پيزا چشم از جهان فرو بست.
دنباله فيبوناچي و نسبت طلائي:
0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , …
در دنباله فيبوناچي هر درايه حاصل جمع دو درايه قبل از خود مي باشد:      fn=fn-1+fn-2 , f1=1 , f2=1
نكته جالب توجه در مورد اين دنباله اينست كه حاصل تقسيم هر دو عدد متوالي از اين دنباله بسيار به نسبت طلائي نزديك است.
34 / 55 = 0.618                   55 / 34 = 1.618
89 / 55 = 1.618                   55 / 89 = 0.617
377 / 233 = 1.618               233 / 377 = 0.618

فيبوناچي در گياهان:

1- آفتابگردان : يكي از جالبترين مثال ها در مورد فيبوناچي در طبيعت همين گل آفتابگردان است. دانشمندان پس از شمارش تعداد مارپيچ هاي موجود در سر گل آفتابگردان دريافتند كه يك دسته از مارپيچ هاي كوچك از داخل به طرف بيرون در جهت حركت عقربه هاي ساعت و مارپيچ هاي بزرگ در خلاف جهت حركت عقربه هاي ساعت قرار دارند.وجود اين دو نوع مارپيچ موجي زيبا را در سر گل آفتابگردان ايجاد مي كند. همچنين دو رابطه زير را آشكار مي سازد:
الف ) جفت ها هميشه به درايه هاي دنباله فيبوناچي نزديك اند: يك جفت ميتواند 21 و34 و جفت بعدي مي تواند 34 و 55 باشد.
ب ) عدد حاصل از تقسيم اين عددها به نسبت طلائي نزديك است.
فيبوناچي در جانوران:
1- DNA  : مولكول DNA از 2 زنجيره پلي نوكلئوتيدي ساخته شده است. بين بازهاي آلي آدنين و تيمين 2 پيوند هيدروژني و بين بازهاي آلي گوانين و سيتوزين 3 پيوند هيدروژني وجود دارد.
مطلب جالب در مورد دو رشته پلي نوكلئوتيدي سازنده مولكول DNA اينست كه هر يك از اين دو رشته داراي 34 آنگستروم طول و 21 آنگستروم پهنا است.
2- انگشتان : به انگشتان خود نگاه كنيد.
  شما داراي ...
2 دست و هر كدام از آنها
5 انگشت و هر كدام از آنها
3 قسمت مجزا كه توسط
2 بند از هم جدا شده اند. آيا  اين تنها يك انطباق است يا خير؟
به هرحال اگر شما استخوان هاي انگشت دست خود را اندازه گيري كنيد (بهترين حالت براي اندازه گيري حالتي است كه انگشتان را خم مي كنيد.) خواهيد ديد كه نسبت طول بزرگترين استخوان انگشت به طول استخوان انگشت با طول متوسط برابر نسبت طلائي است. و نسبت طول استخوان انگشت با طول متوسط به كوچكترين استخوان انگشت  برابر نسبت طلائي است.
منبع

Image: brewbrooks

Leonardo of Pisa was born around 1170 AD in (of course) Pisa, Italy. While not quite as famous as some other Italian or Ninja Turtle Leonardos, we do have a lot to thank him for. His most notable contribution to your life is probably found on the top row of your keyboard. While traveling through North Africa, Leo discovered that the local number system of 0-9 was far superior than the obscure combination of X’s, V’s and I’s the Romans had invented a millennium earlier to confuse later generations of elementary school students. Leonardo brought this number system to Europe and eventually we invented Sudoku with it.

As if this were not enough, Leonardo of Pisa gave us another interesting, if less known gift of mathematics. If you have never heard of the Fibonacci sequence, don’t worry. To be honest, the sequence sees little publicity these days outside of a Dan Brown novel and the occasionally nerdy conversation which may or may not involve warp core propulsion mechanics. However, the Fibonacci sequence is an amazing bit of numbers that ties nature and mathematics together in surprising ways. From deep sea creatures to flowers to the make-up of your own body, Fibonacci is everywhere.

The Fibonacci sequence starts with the number 1. Each additional number is the sum of the two numbers preceding it. For example 1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8 and so on. At first glance, this series might look like the idle musings of a bored person before Tivo was invented, but it goes much further than that. Pause whatever live broadcast you’re watching and take a look at the following examples.

Nautilus Shell

Image: Ethan Hein

A Fibonacci spiral is formed by starting with a rectangle whose sides measure one number in the Fibonacci sequence by its consecutive number in the sequence. For the purposes of simplicity, let’s use 13 and 8. If, hypothetically, we place a square inside the rectangle that measures 8 by 8 and it is placed all the way to one side of the rectangle, the remaining rectangle will have sides that measure 8 by 5, which happen to be two more of Fibonacci’s numbers. Repeating the process with a 5 by 5 square would yield a 5 by 3 rectangle, and so on. This might be hard to follow, so take a look at the following informative, yet slightly boring illustration. Efforts have been made to improve the illustration with special effects.

Image via wikipedia

The pattern continues all the way down until you either get bored and fire up the Tivo, or you simply run out of numbers to use. If we connect the corners of the squares to form a spiral, what we have is a perfect linear model of a nautilus shell.

Image via wikipedia

The Nautilus is a cephalopod that inhabits the ocean at a depth of about 300 meters and is officially the ugliest animal we have ever seen.

Image: Hans Hillewaert

Spiral Galaxies

Image: TopTechWriter.US

If we take the above spiral and rotate it around the the central axis, we get an almost perfect approximation of a spiral galaxy.

The Golden Ratio

Most of the interesting things we find that relate to the Fibonacci sequence are actually more closely related to a number that is derived from Fibonacci, called the golden ratio. If we take each number of the Fibonacci sequence and divide it by the previous number in the sequence (i.e. 2/1, 3/2, 5/3, 8/5), a pattern quickly emerges. As the numbers increase, the quotient approaches the golden ratio, which is approximately 1.6180339887. Approximately. The golden ratio actually predates Fibonacci and has been breaking the brains of western intellectuals for around 2400 years. Applications for the golden ratio have been found in architecture, economics, music, aesthetics, and, of course, nature.

Sunflower

Image: Esdras Calderan

Evolutionarily speaking, the best way to ensure success is to have as many offspring as possible (ergo the Baldwin brothers). The sunflower naturally evolved a method to pack as many seeds on its flower as space could allow. Amazingly, the sunflower seeds grow adjacently at an angle of 137.5 degrees from each other, which corresponds exactly to the golden ratio. Additionally, the number of lines in the spirals on a Sunflower is almost always a number of the Fibonacci sequence.

Pine Cone

Image: mandj98

Like the sunflower, the pine cone evolved the best way to stuff as many seeds as possible around its core. Also, in what was surely an accident, it evolved into perhaps the best substitute for toilet paper when in a pinch. The golden ratio is the key yet again. As with the sunflower, the number of spirals almost always is a Fibonacci number.

Human body

The golden ratio is found throughout your body, all the way to your DNA.

Image: Jamie Neely

Here’s one you can see for yourself, dear reader, if you’re still with us. If you use your fingernail length as a unit of measure, the bone in the tip of your finger should be about 2 fingernails, followed by the mid portion at 3 fingernails, followed by the base at about 5 fingernails. The final bone goes all the way to about the middle of your palm, which is a length of about 8 fingernails. Again, it’s Fibonacci at work and the ratio of each bone to the next comes very close to the golden ratio.

Continuing with the length of your hand to your arm is, again, the golden ratio.

Fibonacci applies even down to what makes you, you. A DNA strand is exactly 34 by 21 angstroms.

The Fibonacci sequence is truly a wonder. The examples are vast, and go way beyond the scale of this article. The patterns in which a tree grows branches, the way water falls in spiderwebs, even the way your own capillaries are formed can all be linked to Fibonacci. Science is just beginning to understand the implications of this simple sequence and some of the most amazing discoveries may be yet to come.

منابع اصلی:

http://en.wikipedia.org/wiki/Fibonacci_number http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html#irrational

دنباله فیبوناچی در طبیعت

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