Fibonacci Numbers, the Cochlea, and Poetry

The Fibonacci numbers form a sequence defined by this relation (don’t be scared!):

What this means, in English, is that it is a sequence of numbers whose relationship is this: after the first two numbers, each proceeding number is the sum of the previous two numbers. For example 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…..and so on. Quite simple, really.

Fibonacci numbers have an interesting property. When you divide one number in the sequence by the number preceding it, you are left with a number very close to 1.618. This number is called the “golden ratio,” and rectangle whose sides is equal to the golden ratio is known as a “golden rectangle.”

Fibonacci numbers are not purely artifact, they are also found in nature in an uncurling fern, the branching of trees, and leaflets of the pineapple. The Fibonacci sequence also describes the “golden spiral,” which is when a “golden rectangle” is subdivided in smaller and smaller golden rectangles (example below)–the result being a predictable spiral.

One example of a biological structure in the mammalian body which is very close to a “golden spiral” is the cochlea. It is not a perfect golden spiral, and there is individual variation between and within species.

Another close example is the shell of a mollusk, or a nautilus shell (below).

The Fibonacci sequence has been used to structure poetry, ie the syllables per line follow the Fibonacci sequence. As a former student of poetry, I couldn’t help but try my hand.

These poems are based on the Fibonacci sequence and have been published in the Science Creative Quarterly, when I wrote them about 6 months ago.

Shyness

Fish
dive
deeply,
mouths agape,
fins proud and ragged,
filtering the oceans apart
until shimmer-hooked and then flopping in boat bottoms,
when gills heave, gasp, drowning in air; eyes glaze like dropped
marbles, clouded and cracked, but holding.

Ego

She
will
nod as
you pass her
and you both will know
you are young and raw, half-bitten,
spitten in disgust like fruit picked before its season.

—–

Heaven

leaves
jump back
on the trees,
a reverse whirlwind
and an impossible sunset
seeking their origins, the life from whence they came.

October 12, 2007

Um, “you are left with a number very close to 1.618, and this number is fixed after the 13th number in the series.” I’m not sure what you’re trying to say — the Golden Ratio, phi, is the limit of F(n)/F(n-1) and n goes to infinity, and it’s transcendental, so you get new digits every time you pick a bigger n. There’s nothing obviously magical about F(13)/F(12).

I’ve been thinking about phi quite a bit recently … I’m convinced its not an accident that it shows up so often in biological stuff, but deeply necessary. I haven’t figured out how to make this into a clear argument rather than an intuition, though.

2. #2 Shelley Batts
October 12, 2007

Charlie, have you seen the movie ‘Pi’ by Aronofsky? If not, you should, as it is a very interesting treatment of what you just mentioned. Fictional of course, but cool.

The golden ratio (and the ‘magic’ is just a name and of course not supposed to be magical) to which i was referring is not pi (3.14…). Its described here: http://en.wikipedia.org/wiki/Golden_ratio

3. #3 Nate
October 12, 2007

Wonderful connection between the numerical, and the organic. I have never seen a poem using these fundementaly natural princables. Gaston Julia (designer of the “Julia set”) would have had a lot to say about your mesh of the two as he was both a mathematician and a poet.

4. #4 Matt Penfold
October 12, 2007

I have a kind of pavlovian twitch whenever someone mentions the Fibonacci series. In my defence I can lay claim to having studied computer science at university, and there everytime since, whenever I learnt a new programming language guess what one of the first things I had to write a program to calculate!

I am glad to say I no longer bother much with that programming stuff, these days being a database person.

5. #5 Doug
October 12, 2007

Hi Shelly,

I still speculate about why there is such a close connection or even intimate relation of the cochlea to the semicircular canals?

Some of the African cats seem to capture birds capable of flight better by sound than sight.

6. #6 candygirlflies
October 12, 2007

Hi, Shelly,

Dropping by from Kim Klein’s blog, where I saw your lovely comment. You must be a very kind, classy (as well as brilliant) young woman.

Best wishes to you, and good luck in the scholarship contest.

7. #7 darkman
October 12, 2007

my two favorite examples of fibonacci in nature is patterning of seeds on the face of sunflowers and human form as visualized by the da vinci picture ‘vitruvian man’ – which i think is phi-related. if i’m not mistaken there’s a lot of fibonacci/golden-ratio plant patterning (like placement of leaves on flower stalks and veins on leaves).

and just to clarify, phi and pi are different, phi being the greek letter used to denote the golden ratio, and pi for the relationship between radius and circumference of a circle (charlie was in fact talking about the same thing as you, you just might have missed the ‘h’, an easy mistake). Charlie’s also right about the value of phi as ‘n’ reaches its limit at infinity. the value for phi does change after n=13, just by such a small amount is isn’t going to be noticeable using regular scientific calculators (which crop after a certain number of digits post decimal point).

8. #8 darkman
October 12, 2007

nice poetry by the way. i find this post funny only because maybe two posts below this one (lions skull) you claim having no artistic skills whatsoever… writing ain’t drawing, but it is still art.

9. #9 Shelley Batts
October 12, 2007

Got it darkman! Sorry for misunderstanding you Charlie. You know how us blondes can be. ðŸ˜‰

10. #10 Colugo
October 12, 2007

Spontaneous formation of Fibonacci spiral in elastically mismatched bilayer.
http://www.physorg.com/news97227410.html

October 12, 2007

The Peruvian calendar is also based on Fibonacci numbers and the last possible date on their calendar is 12/31/2012….interesting!

12. #12 The Flying Trilobite
October 12, 2007

Beautiful! Very nice Shelley, thanks for sharing those poems. I totally agree with Darkman. There’s more artistic talent there than many of the pieces I saw during Nuit Blanche.

13. #13 Hai~Ren
October 13, 2007

Great post. I discovered the Fibonacci sequence through xkcd:

http://xkcd.com/289/

14. #14 LeV
October 13, 2007

hi shelly, i discovered your blog in the finalists blogging scholarship 2007. i’m running a german blog on linguistics and poetics and had to carry the great idea of fibonacci-poem from here around the globe. thanks and much luck for the competition from germany.

15. #15 RickD
October 13, 2007

Hi Shelley,

Got sent here after hearing you are up for a scholarship. I’m going to nitpick because I’m a mathematician, but the ratio of consecutive Fibonacci numbers is not exactly constant, not even after the 13th. It may be identical up to however many digits you are looking at, but it isn’t really constant.

The golden ratio is a very interesting number, appearing, as you point out, all over the place in nature.

16. #16 Marilyn Terell
October 13, 2007

Does anyone else notice the similarity between the design of the cochlea and the Flying Spaghetti Monster?
http://www.venganza.org/

17. #17 Damien
October 13, 2007
18. #18 trane
October 13, 2007

Thanks for posting your poems, and the explanation of Fibonacci numbers. There is a good rythm to the poems.

October 14, 2007

Sweetness, I’m a mathematician, remember?

I’ll spank you next time you come through Denver.

20. #20 John P. Baumlin
October 14, 2007

Shelley,
What a great post. I began to notice something oddly regular about the arrangement of scales on a pine cone when I began doing a bunch of drawings of them years and years ago, and lo, their swirls are based on the Fibonacci series. Most have 5 spirals going one way, 8 going the other, or 8 and 13, like the Monterey pine cone I picked up in Big Sur.
And I agree with whoever said that your poems are as artful as any drawing.

21. #21 Josh
October 16, 2007

Sometimes getting funding means spiraling out of the realm of sanity. Yuk Yuk

22. #22 Matthew Havens
October 26, 2010

The soul does not see it’s own image in another, due to the distortions of physicality and psychology. I am exponetially shocked and exited.I didn’t expect to see a cute, intellectual women writing this. Anyway,I was contemplating the effects of fibonacci series as one of the many intelligent geometrics and mathematics existing in nature,such as cymatics. I got off the subject and was considering the quantum mechanics aspects of math and vibrations, and thought some how of hearing vibrations, and thought about the spiral choclea. I concluded that there might a golden spiral that might be indicative of fib sequence. Interesting to ponder, and it’s shocking to see someone like myself who realizes the interconectedness of all the arts,philosophy,mathematics,science,art,religion(spirituality)history, politics,etc.. and to see the way they are all intelligently intertwined. The fibonacci numbers are everywhere, I trade stocks for a living, and there are fib numbers in all markets.. I use them all the time to find support and resistand retracement prices. I saw the movie pi years ago… it’s just like that… it’s something that, in studying these topics will take over your life if your not careful,but if it does it is exponentially worth it, that’s the beauty of it . It’s funny you mentioned that old black and white pi movie… wish I could of seen it at Sundance. If you ever want to discuss the arts in depth, it would be of my utmost pleasure. Keep up the gift.

23. #23 Bill English
July 8, 2011

I would like a chart of how the harmonics line up in the cochlear canal. I theorize the octave-apart tones, might well form a perfect golden spiral, even if it curves in the opposite direction. In fact, I’d expect that. The hairs give extra stimulation to the nerves, where “resonance” occurs for any particular tone. The tone is determined by position of stimulus, and not by frequency. I’d like to “map” the human cochlear the best way, for experimental sensory substitution tactile array delivery systems, for the deaf and some types of hearing loss. I have a 2D array already, with octave apart tones aligned vertically. However, I am now leaning toward an actual spiral of electrodes as a possible better alternative.