Fibonacci Sequence Illustrated by Nature [PICS]
Fri, May 8, 2009
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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.
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May 8th, 2009 at 10:05 pm
Nature illustrating fibonacci? Isn’t that a bit back to front?
May 9th, 2009 at 1:21 am
Cool article, I love the Fibonacci blocks with the tank!
May 10th, 2009 at 5:31 am
hey that’s pretty cool
May 10th, 2009 at 6:05 am
Maybe mathematics is a truly way to explain creations . And we mistakenly turns to religions
May 10th, 2009 at 6:28 am
Just like the movie “Pi” from 1998. Cool.
May 10th, 2009 at 6:52 am
Nature is amazing.
May 10th, 2009 at 7:25 am
Really interesting stuff, apart from the observation that nautilus shells DO NOT grow with a golden ratio spiral http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
May 10th, 2009 at 8:23 am
“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.”
Thankyou for perpetuating the stereotype.
May 10th, 2009 at 8:29 am
That’s amazing
May 10th, 2009 at 9:02 am
i like penisez lololollolllllll
May 10th, 2009 at 11:58 am
Although the Fibonacci sequence and the golden ratio do make important appearances in nature, many of these examples are misleading or downright wrong.
Nautilus shells and galaxy arms form logarithmic spirals, but NOT golden spirals. The construction of a “spiral” using blocks with fibonacci lengths creates an approximation of a golden spiral, but it’s NOT actually a golden spiral. (see http://en.wikipedia.org/wiki/Golden_spiral).
The idea that bones in the human hand have a relationship to the fibonacci sequence is not supported by science: http://www.citeulike.org/user/tfx/article/3328470.
Finally, what is surprisingly omitted is the connection to nature that first caused Fibonacci to discover the sequence: an investigation of ideal population growth of rabbits.
May 10th, 2009 at 1:31 pm
Great article and very well written :)
May 10th, 2009 at 2:39 pm
Have you ever tried to superimpose the nautilus shell and the Fibonacci spiral? They don’t fit! Try it.
May 10th, 2009 at 2:49 pm
… also shows up in the stock market.
May 10th, 2009 at 3:08 pm
a nautilus does not follow fibonacci’s sequence
May 10th, 2009 at 4:00 pm
Woah!
GR8 article!
May 10th, 2009 at 4:27 pm
really nice post,i’ve learned this golden number in math at university…
When we see real examples is more interesting, so, thanks for this post
adeux
May 10th, 2009 at 4:43 pm
“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.” This article is full of grammatical errors, irrelevant asides, and plain, pure obfuscation. Let’s hope the math is better.
May 10th, 2009 at 5:13 pm
Nice pictures. The so-called Fibonacci sequence is actually Indian. Leonardo Fibonacci published it in Liber Abaci but it was known in India before that:
http://en.wikipedia.org/wiki/Fibonacci
http://en.wikipedia.org/wiki/Fibonacci_number
http://www.cs.okstate.edu/~subhashk/GoldenMean.pdf
from the wiki on Fibonacci_number:
“The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), long before it was known in Europe. Developments have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gop?la (c.1135 AD), and Hemachandra (c.1150 AD).[4]“
May 10th, 2009 at 6:08 pm
Thank you; this is a great post!. Fibonacci is one of my inspirations in my digital artwork.
May 10th, 2009 at 8:44 pm
No romanesco broccoli? Come on.
May 10th, 2009 at 11:49 pm
ferns are another great example you could use
May 11th, 2009 at 1:05 am
As Phil Dale alluded to, you’ve got the cart before the horse.
Nature doesn’t follow mathematical sequences, but mathematical sequences are derived from observations of nature.
May 11th, 2009 at 1:15 am
Very interesting–looks like Brad did his research….
May 11th, 2009 at 4:32 am
penis has a fibonnaci sequence on it!
May 11th, 2009 at 7:32 am
We have to forget about fixing fibonacci to this sort of mathematical production, we should recognize it’s nature in the fractal geometry, where this complexity that is manifested through the iteration and reiteration of several algorithms and simple equations such as the one fibonacci happened to name. we should realize that this process of complexity is produced from a single, simple equation, one that enables us mathematically to apply and comprehend how nature itself is configured and how life itself, on this particular planet, is possible.
never the less, let us consider what took for us, universally, to exist.
this sort of geometry is one we can relate mathematically with our experience of what is REAL.
think about it.
May 11th, 2009 at 7:34 am
How many times do we have to look amazing creations like this and think, “Wow, this must have happened by the process of genetic mutations and natural selection over millions of years”?
That is ABSURD.
I one were to find, in the middle of nowhere, 50 fruits in five rows of ten, would they honestly attribute it to chance?
These examples are incredible–but they beg for an explanation that’s worthy of their complexity.
May 11th, 2009 at 11:32 am
good way to explain fibonacci code
May 11th, 2009 at 7:21 pm
FTA Efforts have been made to improve the illustration with special effects.
May 12th, 2009 at 3:37 pm
The Fibonacci Sequence gives me anxiety
May 14th, 2009 at 3:53 pm
It is intriguing how this math proportion can be applied to real life. It’s almost like our creator used a formula. I’ve tested the golden ratio out and found that even our bicep to forearm proportion matches that golden ratio too. Fascinating! They have good in depth stuff of that on wikipedia too: http://en.wikipedia.org/wiki/Golden_ratio
May 17th, 2009 at 6:45 am
The Great Pyramids at Giza also conform to the golden ratio, curvature of the earth.
May 25th, 2009 at 12:32 am
Interestingly enough the “Golden mean” mapped on earth is The Kabba in Mecca. The biggest gathering of humans on earth (they circumnavigate this) and it is also a prayer direction for the worship of God.
May 29th, 2009 at 6:26 pm
Fibonacci is a great trading indicator.. i always use it with combination other indicator and its give more accurate analysis
June 28th, 2009 at 6:11 pm
It’s interesting that you showed a nautilus shell as a Fibonacci
spiral. Measurements of real nautilus shells indicate a ratio
of lengths of successive shell layers of about 1.33 instead
of the Fibonacci ration of 1.618… It suggests a better
model for the nautilus shell is the Padovan spiral that has
a ratio of 1.324718…
http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
July 13th, 2009 at 3:18 am
sick as!! maths really can be cool
October 30th, 2009 at 4:12 pm
Really great content, I learnt a lot from reading this.