Cook – The curves of life

p 95 So much is wrong about this, but it’s still interesting – this self assurance that the mystery had been unveiled:

A few years ago Mr. A. H. Church, a botanist at Oxford, came forward with a new mathematical conception, founded on the fresh mathematical data afforded by the transition of the mor- phological standpoint from that of adult construction to that of growing systems. As a starting-point for such a conception Mr. Church very properly propounded a certain ideal condition. Just as, in the consideration of the Newtonian laws of motion, the purely abstract and mathematical conception of ” uniform motion ” precedes that of varying motion, so the growth of a mass of protoplasm may be conceived as a ” uniform growth,” expansion taking place around a hypothetical central point From this Mr. Church deduces the hypothesis that logarithmic spiral curves (see Fig. 97, Chap. IV.), with the straight line and circle as limiting cases, are the sole curves of uniform growth- expansion. Spiral phyllotaxis must, in future, therefore be based on a logarithmic spiral on a plane surface, and not on Bonnet’s old idea of a helix winding on a cylinder, which is merely the expression of the attainment of uniform volume by members in a special series, and which, if carried on to a plane as a spiral with equal screw-thread, would become a spiral of Archimedes. Of course, when applied to the living and admittedly irregularly growing plant, the conception of a logarithmic spiral would be as difficult to prove by actual measurement as was Bonnet’s theo- retical helix. But now the standpoint is changed, it becomes possible not only to deal with uniformly growing systems, to deduce mathematically their various properties, to watch them (as it were) grow on paper, but also (as soon as the properties of uniform growth are ascertained) to study the possibility of varying rates of growth, and to state the growth of an irregular body in mathematical terms as precise as those which indicate the erratic orbit of a comet.

p 101 Possibly more nonsense has been written in the past about these spiral constructions than in any other branch of botany. Hence anyone who starts a new way of looking at them is usually regarded as a crank by scientific botanists.

p 102 The proof of the logarithmic spiral hypothesis, as put forward by Mr. Church as the only one possible, is extremely involved, but it is the best yet obtained so far as it goes.

 

 

Our email discussions about the direction/admin of the book

Dear All

I like very much the thread proposed by Stéphane – “tension between a “perfect” mathematical object,

and its actual construction step by step in nature”.  There is a quote that I like to use from Theodore Cook’s “The Curves of Life”.  It goes:

“Newton arrived at his theory of the movements of the celestial bodies in our solar system by postulating perfect movement and by calculating from that the apparently erratic orbits of the planets.  In just the same way may it not be possible to postulate perfect growth and from that to calculate and define the apparently erratic growths and forms of living things?”


This idea fits perfectly with the phenomenon of phyllotaxis and the topics of cell division and packing addressed later in the book.

I agree also with the split for the chapters.  I added my name to the Church chapter if we decide to include one.  Also, it goes without saying the Nancy will be involved directly in ALL of them!

Best wishes,

Jacques

_________________________________
Jacques DumaisProfessor
Facultad de Ingeniería y Ciencias
Universidad Adolfo Ibáñez
Avda. Padre Hurtado 750
Viña del Mar, Chile

From: ste do [ste.doua@gmail.com]
Sent: Tuesday, October 02, 2012 1:25 PM
To: Christophe Gole
Cc: Nancy Pick; Jacques Dumais
Subject: Re: How Irving Adler Became Interested in Phyllotaxis
I am not sure to follow exactly the distinctions and separations
between the groups just proposed by Chris (never thought about
the separation in the biological group).
But i deeply agree that there is a fondamental difference between two
approaches, one centered on a “perfect” lattice, with all the consequences
(the problem of selection, the relation with “magical” golden number, etc…),
and the other based on “dynamics”, looking at how the network can
construct itself step by step with all the possible imperfections…
May be a general underlying theme (as i remember it) of the book
can be this tension between a “perfect” mathematical object,
and its actual construction step by step in nature.
Or the surprising fact that even if nothing is perfect in reality,
in the case of plants, it can come very close to a dreamed
ideal perfection … (i am thinking in particular to the “fractal”
plant shapes, as in the romanesco brocoli)
In other words it is in the tension between fixed ideal perfection,
which characterizes mathematical objects, and growing reality…
and how the “near perfection” that can be observed may not be
due to some overall mystical reason, but to realistic empirical
reasons (packing my little spheres) that, for some strong
underlying constraint, end up into something surprisingly perfect…
Best,
Steph
2012/10/2 Christophe Gole <cgole@smith.edu>

Sounds good Nancy, about setting up a skype meeting…

By the way, I just got the $2000 check and the signed contract from PUP yesterday. You too Nancy? Expect the same soon, dear oversea-ers. Stephane, did you manage to talk to the photographer?
I just updated the file BookPlanning/ChapterWorkShare.xls, trying to – somewhat arbitrarily – split the work when 2 of us had volunteered to be editor for a chapter. I put .2  to other person who had volunteered, to keep track. Of course it should be as collaborative as possible, but it’s good to have, for each chapter, one person who keeps the eye on the ball? Let me know if you agree on my choices…
As I was writing this now too long email, I wondered: should we have a different medium for these discussions? I could easily set up  a blog, in which we could have these discussions, with tags to keep track of threads, and  documents, pictures attached. Not harder than email, except you need to log in…
Let me know!
We need to brain storm a bit on the larger structure: it is important that there be some running themes, and stories that tie the chapters together. Here is one possibility of threads, for the phyllotaxis part:
Two modeling threads:
– the lattice centered – quasi static theory
– the incremental accretion of disks (or other geometric shapes) with local analysis of transition (zickzacklinie, fronts)
Lattice centered-quasi-static theory:  the bravais introduced lattices, and the possibility of the constant divergence angle. This started a thread of research in phyllotaxis, in which lattices (and the van iterson diagram) and thus constant divergence angles played a central role: bravais, van iterson, adler, the lausanne school, stephane and Yves, atela-gole-hotton, levitov. The theory is mathematically tight but to my present sense, its quasi static approach does not lend itself well to connection with plant data: in short, convergence to lattice fixed points is much slower than the plant time scale, the divergence angles are more often than not NOT constant in plants, and lattice-centered models are not well adapted to concretely explain transitions.
Incremental accretion of disks
I would like to posit that Schwendener might be the first to have looked at disk packing see figure attached. This was right in the wake of Hofmeister (whose job he eventually took).
Inline image 1
Interestingly, van Iterson, as re-discovered by stephane, continued that thread, giving a pictorial explanation of Fibonacci transition, using zickzacklinie – in the same paper where he developed his famous diagram. So he was a point of intersection of the 2 threads. That makes him doubly important, to my taste.
As for Adler’s criticism of Schwendener, it is about the lattice centered piece of his work, not the disk accretion…
Our recent work takes this thread back on…
The advantage to this point of view is, whereas the overall universe of patterns is much larger than that of lattices in the van iterson diagram, and thus much messier, this approach is much more adapted to the time scales of plants, with all their transitions. It might also be more pedagogical.
I am also perceiving two threads on the biology side. They are much less marked, especially the first one:
 
Two biological threads
– The vasculature centered view of phyllotaxis structures
– the meristem centered view.
If you accept the fact that, in many (most? Jacques, I know you had objections 🙂 “vasculated” plants, orthostichies correlate with vasculature, then you get a functionally based topological data of plant structure (namely: given any node what is the node “above”?  How many nodes in between these two?). If you think about it, this was essentially shimper and braun’s approach. It yielded the phyllotaxis quotients (e.g. 8/3) description of phyllotaxis. There is a very interesting article by Okabe (Vascular phyllotaxis transition and an evolutionary mechanism of phyllotaxis), with many references (e.g. Kirchoff) which tries to tie vasculature with phyllotaxis. I think he overreaches in his claim that vasculature optimization is the evolutionary pressure that drives phyllotaxis – although it’s what Bejan (the charlatan?) also posits. Despite the overreach the vasculature-phyllotaxis view, it seems to me, has been too neglected by modelers.
– The meristem view started with Hofmeister, culminated with the auxin etc. models.
I hope this opens the discussion!
Best,
Chris
On Mon, Oct 1, 2012 at 1:26 PM, Nancy Pick <nepick@gmail.com> wrote:

Hello again. Thanks to all for your feedback — I’m relieved that Adler took the time to write that document, and that his daughter knew where to find it among his papers.

Chris, I think a Skype meeting with Vickie would be great. Should I ask her how soon this might be possible? I know it’s hard to get much done while classes are in session, but as long as we keep things moving gently forward…..
n.

On Sun, Sep 30, 2012 at 5:40 PM, Christophe Gole <cgole@smith.edu> wrote:

Hi all,

That’s an interesting document… Will help us a lot if we’re to include anything about him. Which we probably should at this point. What a fluke of fate!

Maybe we could do a skype meeting with Vickie and designer? we can share screens, so they could show us what they can do, and we could show them what we’d like?
I’m sorry i’ve not been very present lately… lots on my plate…
C

On Sun, Sep 30, 2012 at 4:02 PM, Nancy Pick <nepick@gmail.com> wrote:

Dear Phyllofriends —

In lieu of an interview with Irving Adler (which would have taken place today, if he were still alive), his daughter sent us this — seven pages describing how he became interested in phyllotaxis. Perfect!
Chris, will you have any time this fall when we could meet with Vickie to talk design?  Let me know.
all my best, as ever —

Nancy

———- Forwarded message ———-
From: Peggy Ann Adler <bxzooo@netzero.net>
Date: Sun, Sep 30, 2012 at 3:13 PM
Subject: How Dad Became Interested in Phyllotaxis
To: Nancy Pick <nepick@gmail.com>

Hi Nancy,
Lovely chatting with you a short while ago!  Well here it is.  All seven pages.  Hope this will answer your questions as to how Dad became interested in Phyllotaxis.  Looking forward to meeting you next April 27th.
P.

Myth creating in math

This well written article (http://nautil.us/issue/0/the-story-of-nautilus/math-as-myth) describes, it seems to me, the tight rope we want to walk in our book: revealing the mathematical beauty in nature without misleading the reader with explanations that are too simple to be true. It could be used for a chapter on the myth of the golden mean as an esthetic organizing principle of nature or art, together with Marguerite Neveux’s “Le nombre d’or: radiographie d’un mythe

As a preview, here is a good presentation of Neveux’s book (augmented by “la divine proportion”) on amazon.fr:
“Il a ses adorateurs, ses exégètes, ses livres cultes. Il aurait traversé les siècles, transmis de bouche de pythagoricien à oreille d’initié. On le retrouve, dit-on, dans les toiles de Seurat, dans l’architecture gothique, sur la façade du Parthénon, et jusqu’au cour de la Grande Pyramide. Le nombre d’or, symbole même des pouvoirs occultes du nombre, semble relever d’un mythe immémorial.
Le physicien Herbert E. Huntley, dans La Divine Proportion, dévoile les propriétés mathématiques certes exceptionnelles et fascinantes de ce nombre, mais non sans céder quelque peu aux fantasmes qu’il a suscités. Et c’est une historienne d’art, Marguerite Neveux, qui procède pour la première fois, dans sa Radiographie d’un mythe, à la réévaluation critique et salutaire du rôle d’un nombre qui, hors mathématiques, est sans doute trop doré pour être honnête. Cette nouvelle édition comprend un chapitre inédit qui fait le point sur les avatars culturels récents de cette ” divine proportion ” jusqu’au Da Vinci Code. ”

Here is an incomplete list of articles, books that give the golden mean, fibonacci too much importance:

Or even numbers, as Galileo would have (Dialogo sopra i due massimi sistemi del mondo):

That the Pythagoreans held the science of numbers in high esteem, and that Plato himself admired the human understanding and believed it to partake of divinity simply because it understood the nature of numbers, I know very well; nor am I far from being of the same opinion. But that these mysteries which caused Pythagoras and his sect to have such veneration for the science of numbers are the follies that abound in the sayings and writings of the vulgar, I do not believe at all. Rather I know that, in order to prevent the things they admired from being exposed to the slander and scorn of the common people, the Pythagoreans condemned as sacrilegious the publication of the most hidden properties of numbers or of the incommensurable and irrational quantities which they investigated. They taught that anyone who had revealed them was tormented in the other world. Therefore I believe that some one of them, just to satisfy the common sort and free himself from their inquisitiveness, gave it out that the mysteries of numbers were those trifles which later spread among the vulgar.

 

Turing

Archive of papers, including unpublished
http://www.turingarchive.org/
i
ncludes pictures:
http://www.turingarchive.org/browse.php/K/3
(We should at least include one picture of contour plot – drawn over computer numerical output of concentration of morphogens.)

Here are books about Turing’s work, containing in particular chapters about phyllotaxis:
Alan Turing: His Work and Impact  (see p 834, Swinton’s article)
Alan Turing: Life and Legacy of a Great Thinker

 

Kepler on the golden mean

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold, the second we may name a precious jewel.
–Johannes Kepler

Bonnet: An early definition of Phyllotaxy

… together with an erroneous (? that would actually fit the 6th law of thermodynamic of Bejan) function for it.

LA Transpiration qui s’opère par les Feuilles exigeoit aufïi que l Air circulât librement autour d elles & qu elles fè recouvriflènt le moins qu il étoit poffible

L ART avec lequel la Nature a pourvu au libre exercice de ces deux Fonftions eft un de ces Faits qui font tous les jours fous les yeux qu on avoit mênle vu en partie mais dont on n avoit point encore la caufe finale Il confifte dans une telle diftribution des Feuilles fur les Tiges & fur les Branches que celles qui fè fuivent immédiatement ne fe recouvrent pas parce ce qu elles font pofées fur différentes Lignes