Saturday, April 3, 2010

Fibonacci Patterns

Nature is full of patterns based on the Fibonacci sequence of numbers. The way you get the Fibonacci sequence is to add the last two numbers in the sequence: 1,2,3,5,8,13,21,34, etc.


Fibonacci numbers turn up in the Archimedes spiral, the chambered nautilus, and the pattern of overlapping spirals in a sunflower or a Queen Anne’s lace.

In Dinotopia: Journey to Chandara, I did a page of small oil studies showing Fibonacci patterns in pine cones, pineapples, and thistles.



If you count the rows of seeds going one way around, you get 5, 8, or 13, etc. And if you count the rows going the other way around, you get another one of those numbers.


The video "Nature by Numbers" is a beautiful demonstration of the principles. Even if you’re not inclined toward numbers, there’s an unmistakable visual logic behind it.

A few inspired math teachers make the time in their curriculum to teach Fibonacci theory, along with fractals, topology, and tessellation, the right-brain branches of math that most teachers unfortunately have to skip over.

And maybe a math expert can explain in the comments why those Fibonacci numbers turn up in nature so universally.

From BoingBoing.
More at Lines and Colors.
Wikipedia on Fibonacci numbers.

27 comments:

Steve said...

When I was teaching elementary school, we'd spend a few weeks on tessellations and Fibonacci. We had a couple bulletin boards covered with Escher prints, Moorish tiles, and photos of pine cones, sunflowers, etc. Even fourth graders could construct impressive tessellating shapes from paper squares and rectangles (regular polygons) by cutting out a piece and sliding it directly to the opposite side. That cut and taped paper was then used as a tracing template to create an interlocking (tessellating) pattern. The kids applied some of this learning to designing quilt squares that parents/grandparents helped them sew into quilts that we raffled off, earning money to donate to charity and to fund a big overnight trip we took at the end of the year. Kids have a natural love of cool number patterns and making geometric art.

Ian Swain said...

"And maybe a math expert can explain in the comments why those Fibonacci numbers turn up in nature so universally."

I'm not a math expert, but maybe Fibonacci numbers turn up in nature so universally because there is a Creator creating everything.

Rob Rey said...

I'm also not a math expert, but my understanding is that the Fibonacci sequence represents a repetitive growth pattern. Beacuase life must, by nature, grow from it's origin size, a simple set of intructions like DNA can produce growth at a regulated rate which is then able to be repeated with out the complication of a new starting point. It is the same as when it started, just larger, and can grow larger yet.

When it comes to aesthetics, humans and other animals associate golden proportions with being properly formed and physically fit, a better mate. And evolution rolls on. Our concept of beauty evolves to match.

It makes perfect sense that this would happen, no creator necessary.

Rob Rey said...

ps - I just posted a painting about this on my blog last week!

Zeke said...

Wise words Rob, there's truth in the numbers

I'm fascinated by the repeating patterns at ever increasing scale that our reality is built out of. The recognizable patterns of swirling nebula's replicate themselves in eddying tide pools, or the way that sand dunes and mountain ranges share shapes with the sea floor from the constant forces of air and water undulated across them

there was a Greek philosopher (Plato? not sure) who theorized that there existed pure forms of beauty that we remember from our lives in another realm, and any beauty we see in this life is due to a resemblance to those true forms. Something about that view fits well with how nature patterns itself through numbers

draigstudio said...

You know in Illustrator you can actually overlay these sequences over your work to see if you are creating strong compositions.

Tyler J said...

Draigstudio, Corel Painter offers the same overlay feature too. I like your comment Rob. I had a teacher once that said that he thought that humans have a shared idea of beauty and that good designers have found a way to tap into that. Fun to think about.

My Pen Name said...

And maybe a math expert can explain in the comments why those Fibonacci numbers turn up in nature so universally."

I'm not a math expert, but maybe Fibonacci numbers turn up in nature so universally because there is a Creator creating everything.


i was going to make a joke about a it's the selfish gene
but someone beat me to it:
When it comes to aesthetics, humans and other animals associate golden proportions with being properly formed and physically fit, a better mate. And evolution rolls on. Our concept of beauty evolves to match.

It makes perfect sense that this would happen, no creator necessary


so how does 'evolution' explain our finding beauty in a golden spiral galaxy? because it represents a place where there are planets fit for human habitation? Come on. I believe natural selection and evolution can explain much, but not everything and i see people use it as a catch all theory to explain every aspect of human behavior and life, much like freudians did (and still do even though Freud has been proven to be a complete fraud)

and by the way, I am more interested in the universe from the standpoint of a physicist , so why do my sub-atomic particles want to do that in the first place?

Seriously, it seems a lot of people who focus on current biological science become remarkably 'reductionist' and in many ways are still thinking of the universe as a 'clockwork' - that idea has been outdated in physics for over century, time for Richard Dawkins to catch up.

My Pen Name said...

dragstudio:
I wouldn't rely on it. I read a book about the golden ratio a few years ago (i wish i could recall it's name) but its application to art is highly questionable to say the least - for example it was often claimed the parthenon conformed to the golden ratio - and it turns out it does - as does just about any building, as long as you decide where to measure it from.

and what about pictures that are beautiful that don't conform? Are you going to change them so they fit a certain ratio...

Mary Bullock said...

Ah yes, the age old conundrum! Good discussion. The beauty and order that inherently exists in nature - such intricate order - did it come from a Creator or did the orderliness come from chaos? I know which one makes more sense to me.

Rob Rey said...

My Pen Name:
I'm sorry, I did not intend my statement to be a catch all, but only a general principal. There are too many permutations of golden proportions evidenced in nature for me to intelligently address all of them with a single statement.
However, my statement does still apply to humans finding beauty in the spiral of a galaxy or anything else because we have evolved to find that shape beautiful and pleasing, regardless of where we see it.
About the sub-atomic particles, I am not a physicist, but I would keep in mind that energy also grows and reduces. By that I mean, if you are talking about particle smashing experiments, the velocity of object can diminish at a constant rate while rotational energy remains constant causing a viewer to see a smooth spiral curve. I don't know this to be a factual example, like I said I'm not a physicist, but the key idea behind Fibonacci is growth or it's inverse, and can apply to many different aspects of the world around us.

Andrew Wales said...

I can't prove it by science, but I'm inclined to agree with the Psalmist, "The heavens declare the glory of God; and the firmament sheweth his handiwork."

Ashok said...

I think the simplest explanation of why they occur in nature is:

1+1 = 2
1+2 = 3
2+3 = 5....

Fibonacci (an accountant) got the sequence by measuring the breeding success of rabbits over generations. Most of nature starts with 1+1...Hence fibonacci sequences.

The golden proportion is expressed in the recursion formula for fibonacci numbers
Fn = Fn-1 + Fn-2... (i.e. 5 = 3+2). A very clever mathematician named Binet turned this into a non-recursive formula..

http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html

And the root of that recursion turns out to be phi (1 +/- sqrt(5))/2 which is the golden ratio....

Right on about Fibonacci and high school math. I had a math teacher - Lloyd Walker - who turned out some of the best mathematicians in this nation. I wasn't one of them - but his intro to the Fibonacci Society at St Mary's here in the SF Bay Area, and my playing with the math led to a very productive career in computer graphics...

It took one teacher, and one student, and the rest grew from there :-)

Roberto said...

Beautiful video! Thanx for dharing.
In my opinion...We recognize the aesthetics of beauty, and the dissonance of ugly, because we are creatures out-of/embedded-in the Great Mystery. While evolution, mathematics, and the scientific study of the natural world can reveal much about the structures and principals of our unfolding… I find it even more wondrous how much of the dance is dependant on chaos, mutations, random variations, coincidence, accidents, and the like. If you look at a spiral shell, a pine cone, a snow flake, a human face… while they may all follow patterns of growth, and laws of proportion, and archetypical forms, it is their flaws and deviations away from the norm that makes each aspect/expression of creation unique and real. The Great Mystery indeed. -RQ

My Pen Name said...

However, my statement does still apply to humans finding beauty in the spiral of a galaxy or anything else because we have evolved to find that shape beautiful and pleasing, regardless of where we see it.
well this is no more provable than saying God made us that way - its an unprovable way of framing the universe. Not an unworthy way,.. but can you say why I find it beautiful? and what if i disagree as to your explanation would you say your theory trumped my conscious? what if i say if find it beautiful because it reflects God's creation -would you say to me, "no you are wrong, you like it because of how you have evolved", it ...and what if I don't find it beautiful?


About the sub-atomic particles, I am not a physicist, I meant, ok, you're taking the biological perspective and saying 'evolution made us find it beautiful and 'that's that' Well, i am taking physicist perspective- why do my sub-atomic particles which make up your cells want to do that?

mike said...

The reason has to do with irrationality. When choosing angles at which to produce leaves, the plant does best when they don't overlap and shade each other; if the angle is near any rational proportion of the circle, then the leaves will overlap. The "least rational" number, according to this measure, is the golden ratio.

A similar reason explains the occurrence of the golden ratio in astronomy; if two bodies have orbital periods that are nearly rational, then angular momentum is strongly transferred from one to the other and it either gets kicked out of the orbit or pushed towards a more rational period. So we expect to see either orbits in a golden ratio (least rational) or graviationally locked orbits like the moon and mercury (most rational).

Annie said...

Zeke:
Yes it was Plato, I loved studying the early classical philosophers in university. Some of their ideas were way out there but still interesting. And others werent far off the mark.
Plato believed that all knowledge/ideas were already learnt therefore when we see the beauty in something we are just remembering what we already knew before...hmm I've a funny feeling old Plato was eatin some of those Fibonacci mushrooms that were so popular back then...

treplovski said...

The naturalist Ernest Thompson Seton pointed out that the 'eyes' in a peacock's tail plumage followed the Fibonacci sequence as well. It's everywhere!

Roberto said...

So… this Mathematician, a Philosopher, a Physicist, and a Priest walk into this head shop…

http://www.zazzle.com/fibonacci_adoration_society_of_america_button-145783511993410572

F.A.S.A.nating! ;) -RQ

Jeremy Hodges said...

GREAT post. An amazing testament to the Ultimate Artist.

Megan said...

I met Dan Harwell, the author of a book called, "Searching for Design with Fibonacci and Phi." First it goes into the theory and then shows flower arrangements based on those principles.

extremophile said...

Here's an interesting article on Fibonacci & nature:

http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm

extremophile said...

The article I linked goes a bit deeper defending something along the lines of what Rob Rey said. Fibonacci is not really a big deal.

briggsy said...

Now could everybody who has repeated the myth that the Fibonacci sequence/Golden ratio is found in the chambered nautilus please ask themselves if they have ever, ever compared them for themselves? This woman did:

http://shallowsky.com/blog/science/fibonautilus.html

James Gurney said...

Briggsy, thank you. Bravo for debunkers and empirical thinkers. But should I take Akkana's word for it? Where's my ruler?

briggsy said...

Your welcome James. If you want to read more, her link to John Sharp's excellent paper "Spirals and the Golden section" now leads to a page where you can download it as a free full-text pdf. He talks all about the mathematics of spirals, and gets on to the Nautilus on page 79.

http://www.springerlink.com/content/f7j040k4332143n2/

Roberto said...

Maybe we should start referring to these patterns in nature as "FIBonnacci" or how about "FIBBIN'acci"?
Thanks again for the journey, Jim; and thanks to Briggs, Extremo and the others for your vigilance. Does anyone know where I can get a ‘flaws and deviations measuring device’?
Empirically yours –RQ.