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Natural Science Forum / Biology / Paleontology / January 2006Wanted: 'Obscure' 1889 Paper on Trilobite Eyes & The Fibonacci Series
Can anyone help me find the 'obscure' paper by JM Clarke that Niles Eldredge refers to on page 64 of "Time Frames"? A scan of the page is available here: http://members.aol.com/jorolat/trilobite.html In fact I would be grateful for any information regarding the appearance of the fibonacci series in the lens arrangement of trilobite eyes! NB apparently this is not the paper (although I haven't read it myself yet): CLARKE, J. M. 1889. The structure and development of the visual area in the trilobite Phacops rana Green. Journal of Morphology, 2:253270.  SignatureJohn Latter Model of an Internal Evolutionary Mechanism (based on an extension to homeostasis) linking Adaptive (Stationary-Phase) Mutations to the Baldwin Effect. http://members.aol.com/jorolat/TEM.html 'Where Darwin meets Lamarck?' Discussion Egroup http://groups.yahoo.com/group/evomech Hi John, > Can anyone help me find the 'obscure' paper by JM Clarke that Niles > Eldredge refers to on page 64 of "Time Frames"? A scan of the page is [quoted text clipped - 5 lines] > appearance of the fibonacci series in the lens arrangement of > trilobite eyes! The individual lenses were arranged in intersecting logarithmic spirals, clearly evident in holochroal eyes. Logarithmic spirals can also be discerned in the arrangement of lenses in schizochroal eyes, but it's less obvious, due to the smaller number of lenses. Several illustrations of both types appear in chapter 3 of R. Levi-Setti, "Trilobites" 2nd ed. U. of Chicago Press, 1993. Logarithmic spirals are directly linked to the Fibonacci series and to the golder ratio, and intersecting logarithmic spirals show the Fibonacci series quite clearly. Here's one example: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#seeds If there are logarithmic spirals, there is a Fibonacci series. > > NB apparently this is not the paper (although I haven't read it myself > yet): > > CLARKE, J. M. 1889. The structure and development of the visual area > in the trilobite Phacops rana Green. Journal of Morphology, 2:253270. Best Regards, John. >Hi John, > [quoted text clipped - 24 lines] >Best Regards, >John. Thankyou for the info John. I'll try at my local library for the book - I wish I had paid more attention to maths at school! Can I just make sure I've understood you correctly?: >If there are logarithmic spirals, there is a Fibonacci series. >Logarithmic spirals can also be >discerned in the arrangement of lenses in schizochroal eyes (albeit difficult to discern) Therefore the fibonacci series can be found in (all?) schizochroal eyes - or can there be logarithmic spirals without a connection to fibonacci? SignatureJohn Latter Model of an Internal Evolutionary Mechanism (based on an extension to homeostasis) linking Adaptive (Stationary-Phase) Mutations to the Baldwin Effect. http://members.aol.com/jorolat/TEM.html 'Where Darwin meets Lamarck?' Discussion Egroup http://groups.yahoo.com/group/evomech Hi John, >>Hi John, >> [quoted text clipped - 27 lines] > Thankyou for the info John. I'll try at my local library for the book > - I wish I had paid more attention to maths at school! FWIW, Levi-Setti's book is excellent, and not expensive. It's not exactly a grad-school level book, of course, but is aimed at the educated or enthusiastic lay-person. The illustrations, especially of trilobite eyes, are marvellous. > Can I just make sure I've understood you correctly?: > >>If there are logarithmic spirals, there is a Fibonacci series. The Fibonacci series and the golden ratio can be extracted directly from a logarithmic spiral. With intersecting logarithmic spirals (going in opposite directions), the Fibonacci numbers pop out even more obviously. Check the link I gave earlier - it's got nice examples of the effect in the structure of a flower head and of a pine cone. The same logarithmic spirals occur in many other growth patterns. >>Logarithmic spirals can also be >>discerned in the arrangement of lenses in schizochroal eyes (albeit difficult to discern) > Therefore the fibonacci series can be found in (all?) schizochroal > eyes - or can there be logarithmic spirals without a connection to > fibonacci? Intersecting logarithmic spirals are the only arrangement I have heard of for the lens arrangement in schizochroal eyes with an adequately large number (hundreds) of lenses. However, some schizochroal eyes had only some dozens of lenses, so other simple patterns could be argued in those cases; parsimony might still point to the logarithmic spiral form. I must also point out that I'm by no means an expert on trilobite eyes. Best Regards, John. P.S. A mathematician might insist on referring to a constant-angle spiral on the surface of a sphere as a loxodrome. A loxodrome can be projected onto a logarithmic spiral, so biologists use the latter term as it's more likely to be understood. http://mathworld.wolfram.com/Loxodrome.html >Hi John, > [quoted text clipped - 34 lines] >educated or enthusiastic lay-person. The illustrations, especially of >trilobite eyes, are marvellous. I really do want the 1889 Clarke paper even though - as its no doubt obvious! - I know very little about trilobites. I've seen other references to Levi-Setti and if its directed towards an 'enthusiastic lay- person' then some background info will do no harm at all. >> Can I just make sure I've understood you correctly?: >> [quoted text clipped - 6 lines] >the structure of a flower head and of a pine cone. The same logarithmic >spirals occur in many other growth patterns. I did look at the link before & have bookmarked it - very interesting stuff! >>>Logarithmic spirals can also be >>>discerned in the arrangement of lenses in schizochroal eyes (albeit difficult to discern) [quoted text clipped - 10 lines] > >I must also point out that I'm by no means an expert on trilobite eyes. But you know a lot more than me & I'm very appreciative of your help! >Best Regards, >John. [quoted text clipped - 4 lines] >as it's more likely to be understood. >http://mathworld.wolfram.com/Loxodrome.html Fortunately I don't think I'll have to get into the maths very much - at least, not at this stage - at the moment I just want to see what Clarke had to say and (providing its relevant) then just cite him. Thanks again :) SignatureJohn Latter Model of an Internal Evolutionary Mechanism (based on an extension to homeostasis) linking Adaptive (Stationary-Phase) Mutations to the Baldwin Effect. http://members.aol.com/jorolat/TEM.html 'Where Darwin meets Lamarck?' 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