# plants and numbers

May 7, 2007 9:03 PM Subscribe

The Mathematical Lives of Plants "Scientists have puzzled over this pattern of plant growth for hundreds of years. Why would plants prefer the golden angle to any other? And how can plants possibly "know" anything about Fibonacci numbers?"

I'ma tell you guys a secret: math is based on nature, not the other way 'round.

posted by Citizen Premier at 9:30 PM on May 7, 2007 [3 favorites]

posted by Citizen Premier at 9:30 PM on May 7, 2007 [3 favorites]

IANAM, but I thought it had something to do with the intersection of the rates of linear growth and geometric growth?

When X = Phi, X+1=X^2. I always figured it had something to do with mimicking geometric expansion by linear means.

posted by Richard Daly at 9:32 PM on May 7, 2007

When X = Phi, X+1=X^2. I always figured it had something to do with mimicking geometric expansion by linear means.

posted by Richard Daly at 9:32 PM on May 7, 2007

Don't miss the really cool video mentioned in the article.

posted by dhruva at 9:36 PM on May 7, 2007

posted by dhruva at 9:36 PM on May 7, 2007

Or reconciling the geometric growth of volume with the linear growth of surface area, maybe?

posted by Richard Daly at 9:38 PM on May 7, 2007

posted by Richard Daly at 9:38 PM on May 7, 2007

*I'ma tell you guys a secret: math is based on nature, not the other way 'round.*

Yes -- this is exactly why I think Roger Penrose is not as smart as people keep telling me he is.

posted by voltairemodern at 9:53 PM on May 7, 2007

Wonderful article.

Here's another site that nicely illustrates the Fibonacci sequence in plants.

Always loved the graceful geometry in Nautilus shells. "

posted by nickyskye at 10:01 PM on May 7, 2007

Here's another site that nicely illustrates the Fibonacci sequence in plants.

Always loved the graceful geometry in Nautilus shells. "

*The Nautilus shell is one of the known shapes that represent the golden mean number. The golden mean number is also known as PHI - 1.6180339... The PHI is a number without an arithmetic solution, the digit simply continues for an eternity without repeating itself. The uniqueness of the golden mean is that it can be found in all living forms such as the human skeleton, the shell and the sunflower seed order. Appleton called this value - "The key for the universe physics".*" wow.posted by nickyskye at 10:01 PM on May 7, 2007

So, plants are better at math than me. I give up. I'm buying velcro shoes tomorrow.

posted by stavrogin at 10:02 PM on May 7, 2007

posted by stavrogin at 10:02 PM on May 7, 2007

I'm of the notion that if you look hard/long enough for the appearance of any mathematical pattern and you will find it somewhere. That doesn't mean that the mathematical pattern had any governance whatsoever over the formation of the context in which it was found.

It's like a horoscope, or having your fortune told. Vague, general answers are seen as specific to the individual because the individual is looking to match those generalities to their own experiences. Somewhat akin to "suggestion".

It's an interesting phenomenon, yes, but no more interesting than all the other occurrences in which the pattern in question was not found.

posted by C.Batt at 10:43 PM on May 7, 2007

It's like a horoscope, or having your fortune told. Vague, general answers are seen as specific to the individual because the individual is looking to match those generalities to their own experiences. Somewhat akin to "suggestion".

It's an interesting phenomenon, yes, but no more interesting than all the other occurrences in which the pattern in question was not found.

posted by C.Batt at 10:43 PM on May 7, 2007

Uh, C.Batt, the Fibonacci sequence isn't exactly "vague."

posted by Citizen Premier at 10:47 PM on May 7, 2007 [1 favorite]

posted by Citizen Premier at 10:47 PM on May 7, 2007 [1 favorite]

apparently this week is all math and uboats.

up next week: join us on MeFi for trig and triremes.

posted by dreamsign at 11:31 PM on May 7, 2007 [1 favorite]

up next week: join us on MeFi for trig and triremes.

posted by dreamsign at 11:31 PM on May 7, 2007 [1 favorite]

*Uh, C.Batt, the Fibonacci sequence isn't exactly "vague."*

Whoosh...

posted by b1ff at 12:25 AM on May 8, 2007

I don't think anyone had to "reach" to identify the common pattern amongst all known organic spiral patterns...

Sort of like saying bipolar symmetry amongst mammals is an urban myth. I guess we've been discounting the 3 armed cyclopian community for far too long...

posted by yeloson at 12:36 AM on May 8, 2007

Sort of like saying bipolar symmetry amongst mammals is an urban myth. I guess we've been discounting the 3 armed cyclopian community for far too long...

posted by yeloson at 12:36 AM on May 8, 2007

Any post about math and plants deserves props to Aristid Lindenmayer, possibly the first "computational biologist" who invented L-Systems in the 60s and was a pioneer in the field of modeling plant morphology algorithmically (if you can find a copy of "The Algorithmic Beauty of Plants" by Mr. L, it's worth it).

posted by distant figures at 1:12 AM on May 8, 2007

posted by distant figures at 1:12 AM on May 8, 2007

It seems like this article is a little premature; lots of observation but to real conclusions. Not that this type of research isn’t necessary, it just seems a bit obvious. But I always like to read about the golden angle and golden ratio. I guess I can add pine cones and the yellow part of a daisy to my list of things that are “mysterious but better left alone”

posted by BostonJake at 1:31 AM on May 8, 2007

posted by BostonJake at 1:31 AM on May 8, 2007

**distant figures**writes

*"(if you can find a copy of 'The Algorithmic Beauty of Plants' by Mr. L, it's worth it)"*

Good recommendation. Free PDF.

posted by orthogonality at 1:45 AM on May 8, 2007

this type of pop-science is mildly offensive to actual learning

Emergence versus self organization (pdf) this paper helped me clear up my daydreams on the topic recently, i a pretty good introduction to alot of topics.

also, there is a book called 'the computational beauty of nature' by flake which will likely appeal to anyone interested in this sort of thing, but doesnt quite know what they're talking about. plenty of preview material on the site.

posted by sponge at 3:49 AM on May 8, 2007 [1 favorite]

Emergence versus self organization (pdf) this paper helped me clear up my daydreams on the topic recently, i a pretty good introduction to alot of topics.

also, there is a book called 'the computational beauty of nature' by flake which will likely appeal to anyone interested in this sort of thing, but doesnt quite know what they're talking about. plenty of preview material on the site.

posted by sponge at 3:49 AM on May 8, 2007 [1 favorite]

But but...isnt the answer "God"? Cause, you know, all of nature seems to follow laws that we discovered! How can plants know laws?!?!

In other words, God must exist cause the Earth is exactly right for us to live.

posted by Dantien at 5:55 AM on May 8, 2007

In other words, God must exist cause the Earth is exactly right for us to live.

posted by Dantien at 5:55 AM on May 8, 2007

*Sort of like saying bipolar symmetry amongst mammals is an urban myth. I guess we've been discounting the 3 armed cyclopian community for far too long...*

Er, you mean

*bilateral*symmetry.

And anyway, obviously things are going to be built with some kind of mathematical structure. I don't see why this is that surprising or interesting.

posted by delmoi at 5:56 AM on May 8, 2007

Among his collected works, in the few, short years before mathematician Alan Turing was driven to suicide, he published "The Chemical Basis of Morphogenesis", theorizing how a standing wave-like distribution of "cannibal" and "missionary" chemicals might explain how plants and animals develop their shape and pigmentation.

Blogger Jonathan Swinton focuses on this more obscure aspect of Turing's research, and reviews some of his posthumous and unpublished efforts — including one of the earliest known examples of digital computation applied to the field of biology.

posted by Blazecock Pileon at 6:29 AM on May 8, 2007

Blogger Jonathan Swinton focuses on this more obscure aspect of Turing's research, and reviews some of his posthumous and unpublished efforts — including one of the earliest known examples of digital computation applied to the field of biology.

posted by Blazecock Pileon at 6:29 AM on May 8, 2007

**delmoi**writes

*"And anyway, obviously things are going to be built with some kind of mathematical structure. I don't see why this is that surprising or interesting."*

I think it's because the mathematical structure is not nearly as simple as doubling, tripling, or the like.

If I tell my colleague "Nice shoes", it is obvious that he will respond with something. However, it might be interesting or surprising if he responded by putting the shoes on his head. I'm not saying that the Fibonacci series appearance in nature is as completely outlandish as that, just that it's the

*type*of mathematical structure that is interesting, not the mere fact that there is a mathematical structure in the first place.

posted by Bugbread at 6:58 AM on May 8, 2007

I graded several papers this semester on exactly this topic. I ended up learning quite a bit in the process of determining the reliability of the students' sources (normally I wouldn't bother, but I was reading conflicting claims in quotes from different sources.)

The occurrence of mathematical structures in nature is hardly surprising. Nobody bats an eye when biological structures turn out to be spherical or circular. However, some people spend a lot of time trying to find number theoretic parallels (prime numbers, Fibonacci numbers, perfect numbers, etc.), and the ratio of bunk to kernel-of-truth seems to be pretty high.

The occurrence of Fibonacci numbers in spiral shapes (sunflowers, pineapples, etc.) seems to be one of the few places where the intersection of number theory and biology seems to make any sense. The cited explanation often hinged on the nature of the golden ratio as a "very" irrational number. ("Very irrational" here meaning that it requires large denominators to approximate rationally. This can be made precise, if you're interested. Look into continued fractional approximations for more.)

It's nice to see an alternative explanation here.

The often-quoted fact that flower petals always seem to have a number of petals that are

Other occurrences of the golden ratio in nature are similarly forced. I've been told (though I don't recall where, so take this with a grain of salt (on the other hand, all I'm asking people to do is take

The other occurrences of Fibonacci numbers in the human body (2 hands, 3 knuckles on each finger, 5 fingers on each hand, etc.) strike me as ridiculously forced, and I don't think anyone treats those seriously.

posted by ErWenn at 7:35 AM on May 8, 2007 [1 favorite]

The occurrence of mathematical structures in nature is hardly surprising. Nobody bats an eye when biological structures turn out to be spherical or circular. However, some people spend a lot of time trying to find number theoretic parallels (prime numbers, Fibonacci numbers, perfect numbers, etc.), and the ratio of bunk to kernel-of-truth seems to be pretty high.

The occurrence of Fibonacci numbers in spiral shapes (sunflowers, pineapples, etc.) seems to be one of the few places where the intersection of number theory and biology seems to make any sense. The cited explanation often hinged on the nature of the golden ratio as a "very" irrational number. ("Very irrational" here meaning that it requires large denominators to approximate rationally. This can be made precise, if you're interested. Look into continued fractional approximations for more.)

It's nice to see an alternative explanation here.

The often-quoted fact that flower petals always seem to have a number of petals that are

*one*of the Fibonacci numbers seems to be completely untrue. Here's an article backing me up here.Other occurrences of the golden ratio in nature are similarly forced. I've been told (though I don't recall where, so take this with a grain of salt (on the other hand, all I'm asking people to do is take

*other*people's claims with a grain of salt)) that the famous pictures of conch shells lined up with the golden spiral required the examination of a lot of different shells until the perfect one was found, and that other non-golden spirals are comparably common in nature.The other occurrences of Fibonacci numbers in the human body (2 hands, 3 knuckles on each finger, 5 fingers on each hand, etc.) strike me as ridiculously forced, and I don't think anyone treats those seriously.

posted by ErWenn at 7:35 AM on May 8, 2007 [1 favorite]

The reason why people find this sort of thing interesting is isn't because it is a plant doing complex math, but because it is a plant using an irrational number. The reason why we find this to be complex is because we don't use irrational numbers much whereas plants (and other organisms) do. The complexity we see is just an artifact of our world view of rational numbers being normal while irrational numbers (which are actually just as simple and easy to use if you think about it...) are odd.

posted by pwb503 at 9:56 AM on May 8, 2007

posted by pwb503 at 9:56 AM on May 8, 2007

I find the Golden Ratio mysterious and compelling, and I do expect it to continue to be on display at the growing points of science and mathematics, just as it on the growing tips of plants.

If you want to read an exposition of the significance of the GR from the sophisticated standpoint of continued fractions (as suggested by ErWenn), try an essay by John D. Barrow, of which this is the penultimate paragraph:

I think cacti could be trying to tell us something about the ubiquity and deep importance of Fibonacci spirals in plant development as a whole: the spines of cacti are remnants of what were leaves in their ancestors, and as you can see from one of the illustrations in dhruva's link, these spines often form Fibonacci spirals which are themselves arranged in Fibonacci spirals; one way of thinking about cacti is to imagine them as more usual plants which have been inflated into spheres and then shrunk down, and their leaves turned into thorns. That their spines/leaves are in compound Fibonacci spirals may imply that this is true, at some level, of plants like elm trees, as well.

posted by jamjam at 12:26 PM on May 8, 2007

If you want to read an exposition of the significance of the GR from the sophisticated standpoint of continued fractions (as suggested by ErWenn), try an essay by John D. Barrow, of which this is the penultimate paragraph:

*Continued fractions are also prominent in other chaotic orbit problems. Numbers whose cfes end in an infinite string of 1s, like the golden mean, are called noble numbers. The golden mean is the "noblest" of all because all of its quotients are 1s. As we have said earlier, this reflects the fact that it is most poorly approximated by a rational number. Consequently, these numbers characterise the frequencies of undulating motions which are least susceptible to being perturbed into chaotic instability. Typically, a system which can oscillate in two ways, like a star that is orbiting around a galaxy and also wobbling up and down through the plane of the galaxy, will have two frequencies determining those different oscillations. If the ratio of those frequencies is a rational fraction then the motion will ultimately be periodic, but if it is an irrational number then the motion will be non-periodic, exploring all the possibilities compatible with the conservation of its energy and angular momentum. If we perturb a system that has a rational frequency ratio, then it can easily be shifted into a chaotic situation with irrational frequencies. The golden ratio is the most stable because it is farthest away from one of these irrational ratios. In fact, the stability of our solar system over long periods of time is contingent upon certain frequency ratios lying very close to noble numbers.*I think cacti could be trying to tell us something about the ubiquity and deep importance of Fibonacci spirals in plant development as a whole: the spines of cacti are remnants of what were leaves in their ancestors, and as you can see from one of the illustrations in dhruva's link, these spines often form Fibonacci spirals which are themselves arranged in Fibonacci spirals; one way of thinking about cacti is to imagine them as more usual plants which have been inflated into spheres and then shrunk down, and their leaves turned into thorns. That their spines/leaves are in compound Fibonacci spirals may imply that this is true, at some level, of plants like elm trees, as well.

posted by jamjam at 12:26 PM on May 8, 2007

I'm pretty sure this exists as one of the massive footnotes within Wolfram's

posted by adipocere at 3:40 PM on May 8, 2007

*A New Kind of Science*, either on paper, or in his head.posted by adipocere at 3:40 PM on May 8, 2007

how funny you posted this, dhruva...i was just going to post this bit of breaking news...apparently they've figured out how to grow these structures in a lab. the trick seems to involve 'elastically mismatched bi-layer structures' whose resulting stress patterns give rise to these spirals. seems to me like the air hardened cells on the outside of a bud would be 'elastically mismatched' with the sweet, juicy stem cells just inside, no?

posted by sexyrobot at 9:53 AM on May 9, 2007 [1 favorite]

posted by sexyrobot at 9:53 AM on May 9, 2007 [1 favorite]

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posted by parudox at 9:23 PM on May 7, 2007