Apparently the current convention for describing the compound eye of the trilobite is to count the number of lenses by rows and columns.
At (approximately) the turn of the century an earlier researcher used a method based on the fact that the lenses are arranged in fibonacci spirals.
I’m sorry that I can’t (at this point) be more exact than this but the aspect I’m interested in is as follows:
As trilobites evolved from one species to another did the fibonacci numbers for each species increase sequentially with respect to the earlier species?
In those instances where trilobites evolved without eyes did the fibonacci sequence of the preceeding species decrease until the eyes had disappeared?
I think you are looking for something that is not there.
As I understand it, when fibonacci numbers show up in spirals, the organism does not code for specific pairs of fibonacci numbers. In pinecones and sunflowers, for example, there is a most common pair of fibonacci numbers for a particular type of plant, but even the same organism will have other pairs showing up. What happens is new nodes are created in the center at a specific angle (I forget what the angle is) and as they grow, they are forced outward. This arrangement “wants” to fall into fibonacci spirals and once the pair of numbers is “chosen” for a particular pinecone, the rest of the nodes just fall into place. A pine tree that usually chooses 8 and 13 may have pinecones with 5 and 8 or 13 and 21.
I don’t know the answer, but I see no reason to expect a pattern at all. I would bet the answer to both questions is no.
Thank you for replying. Though you suggest that I may be looking for something that isn’t there it is what is doing the “wanting” and “choosing” that I am interested in.
Thank you for replying. Though you suggest that I may be looking for something that isn’t there it is what is doing the “wanting” and “choosing” that I am interested in.
I was just using a bit of anthropomorphism. What I meant was, of the possible arrangements of the nodes, those with Fibonacci spirals are much more likely than other arrangements and once a particular pair of Fibonacci numbers occurs in a particular pinecone, the rest of the nodes continue the pattern. I did not mean to imply that there is anything with desires or willful actions.
I might say that water “wants” to flow downhill. I ascribe no desires to water.
The quotation marks were intended to indicate that I was not using the word in its usual sense.
I did realize what you meant and I was actually replying in the same manner.
I’m going to have to read up on why fibonacci spirals are more likely than any other. I seem to recall that it may be due to efficiency which sounds logical except for the fact that organisms aren’t logical, it could be a homeostatic function.
Check out Conway and Guy’s The Book of Numbers, pages 111 through 124. They spend an inordinate amount of the pages of the book on fibonacci numbers and possible reasons for their occurrence. The section from page 113 to 117 is called Phyllotaxis (“The botanical name for leaf arrangement”), and the next section (pages 118 to 124) is called Say Bud, Where Do You Think You’re Going?
If you look at a pinecone, you will notice that there are a number of spirals going in one direction and a different number of spirals going in the opposite direction. The two numbers are almost always successive numbers in the Fibonacci series. The center of a sunflower exhibits the same phenomenon. Apparently the lenses in a trilobite eye also fall into Fibonacci spirals. Successive pairs of numbers in the Fibonacci series occur very often in nature.
The Fibonacci series is defined by the recursive rule that every number (except the first two) is the sum of the previous two numbers. If you take zero and one as the first two Fibonacci numbers, and just follow the rule you get:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
In the Penguin Book of Curious and Interesting Geometry (a wonderful collection of oddities that shows you the Universe is far more Connected than you ever thought) there’s a disappointingly brief discussion of this issue of Fibonacci curves showing up in Phyllotaxis, and the suggestion is made that some other mathematical formulation fits it better than logarithimic spirals. This is another of those cases where I’ve evidently missed some crucial paper, because I’ve always wanted to know more since I read that.
Good post, Dr. Matrix, probably puts choosybeggar’s mind at ease.
choosybeggar, if you start with other numbers like 1,0 or 1,1 or 1,2 you get the same sequence essentially. But 2,1 gives you the sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, … which even has its own name–the Lucas sequence. They don’t have the same name, but they are related.
If you want to avoid doing all the adding to find the nth fibonnaci number, you can use this formula:
nth fibonnaci number = (1/sqrt(5)) times the
difference of the nth powers of ((1+sqrt(5)/2)
and ((1-sqrt(5)/2)
That (1+sqrt(5))/2 is the golden ratio, and the ratio of successive fibonnaci numbers gets closer and closer to it. Rectangles whose sides had that ratio were once thought by ancient Greeks to be more beautiful than others. Nowadays, it would be known as the quaint ratio.
There is a large amount of information at this site (more than 200 pages if it was printed), so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature.
Having studied the damned things for three years in college many years ago, here’s my 0.02c
a)
One species of Trilobite didn’t evolve into another. Evolution and speciation have been covered here many, many times before. The best summary is that it’s a complex and random business, and any orderly progression is extrememly unlikely.
b) Victorian Taxonomy was obsessed with finding patterns, even ones that were not there. You can describe a trilobite’s eyes in terms of spirals, but as Dr. Matrix points out, there will not necessarily be any relationship between the results and any evolutionary relationship. It’s only one step away from saying that one dalmatian is more evolved than another because it has more spots. It’s beautiful mathematics, but poor biology.
For more detailed information, you might check out the best trilobite site on the net
Thank you for the links and info. Someone has suggested “Gould S.J., and N. Eldredge. 1977: Punctuated equilibria” because it includes a lot of info on trilobites so I’ll be checking that out too.
Does saying “One species of Trilobite didn’t evolve into another” mean there is no connection between any of the species?.
I didn’t mean “as one species evolved into another” literally, ie in a linear fashion.
