After 3 hours with google, a method for generating these spiral quadrilaterals was not found.
They cold be measured from plants, or traced on the Fibonacci script?

I did locate professor Scott Hotton's gif, (and several mathematically esoteric papers with hyperbolic geometry.)
http://scotton.freeshell.org/phyllo/lattice_5_8.gif
The central "beginnings" and expansion factor are unclear.

Perhaps a rotated scaled Fibonacci curve??

As time permits, I may try some curve fitting in MoI.
Or read some of the papers, including finding the "center point" of "pinecone" curves.
It is an interesting homework assignment:-)

- Brian

This site seems promising:
http://www.math.smith.edu/~phyllo/About/Lattices/SpiralLattices.html

Bon courage!

Yikes! :-)

"Each spiral lattices can be seen as a discrete subgroup of the complex multiplicative group, with the generator Geid. Indeed, in complex notation, points of the lattice can be written as Gkeikd. These form a group isomorphic to the integers (the isomorphism is k-> Gkeikd). Parastichies correspond to cosets of the subgroups 8Z and 13Z in the example given here."

- Brian

The three applets on the Smith site can be run and viewed, on Windows 7, by loading the current Java program, and modifying it by adding to the Exception Site List.
After installing Java, Under Configure Java/Security/Edit Site List, add http://www.math.smith.edu to the exception site list.
Java apparently has some security risks...

The cylindrical phillotaxis applet uses mouse picks, or a "2D Mouse slider," for the parameter space. Some voronoi patterns resemble a pineapple pattern.

Can MoI do a 2D slider, e.g. with Mouse or Mouseover?

The spiral applet uses a hyperbolic parameter disk with mouse picks, or "hyperbolic mouse slider."
There are an "infinite" number of possible phillotaxis results depending upon Growth rate and Divergence angle.
Using nearest neighbor, and next nearest neighbor, the parastichy curves are created, which show the 4 curve-sided quadrilaterals.
The display seems to be a bit cramped.

It should be possible to do some MoI script(s).

- Brian

Also located some Croatian script...The code is in English, comments translated with Google Translate. Needs more study...

Doing a 2D mouse slider in a NodeEditor window, along with a javascript program, with phyllotaxis in the MoI window might work?

- Brian

Hi Brian,

> Can MoI do a 2D slider, e.g. with Mouse or Mouseover?

There isn't any prepackaged control for doing that, but it should be possible to implement a custom one, you would want to put an onmousemove="" handler on an HTML element to do that: http://www.w3schools.com/jsref/event_onmousemove.asp

- Michael

Thank you Michael.
- Brian

Hello Brian and Pilou,
Thanks for sharing the video - great stuff, but it took a while till I understood it.
So like Brian I can't withstand and so I made also some experiments. I started with a expansion factor k is for the main curves given by the construction of the golden spiral. For a 1/4 turn the golden number = r = exp(k*PI/2) -> k=4*log((sqrt(5)+1)/2)/(2*PI) Ok - that's not new and already known by WIKI and the rest of the world. But it's not necessarily.
But I didn't find that:
If you use this k_1 for the main spirals e.g. 13 and have 8 counterclockwise ones, you can use a k_2=k_1/golden number. I think that's the best choice, because you get a minimal distorted element. What do you think about?



Have a nice day
Karsten

Hello Brian,

I'm working on a loxodrome node. I think that makes it possible to go to the 3rd dimension. For undistorted elements the intersections between clockwise/counterclockwise can be calculated by the logarithmic spiral function with a angles n*2PI/next fibonacci e.g. 8S/13S/(21). It seems that everything in this configuration is fibonacci!?!



Have a nice day
Karsten

[offtopic]
Hi Karsten
It seems like you using linux+wine to run MoI. The fonts looks awful. :)
Try to edit FontSmoothingGamma setting in ~/.wine/user.reg (set value from 400 to 600)

In my opinion the best values is:
"FontSmoothing"="2"
"FontSmoothingGamma"=dword:00000512
"FontSmoothingOrientation"=dword:00000001
"FontSmoothingType"=dword:00000002

Hello Max,

I've tested the values and it works perfect:-)
Thank you very much and have a nice day

Karsten

Hi Karsten,
You have an interesting project.

Based upon my limited understanding:

From the Smith site, of 650 plant species, R. Jean found that "about 92% of them have Fibonacci phyllotaxis." (F0 = 1 and F1 = 1).
Some plants follow one particular Lukas sequence with F0 = 1 and F1 = 4. (There are other types of Lucas sequences.)
There is a formula for the n'th Fibonacci number. (Haven't seen a formula for the n'th term of the above type of Lukas sequence.)
As time allows, I'm still studying for a phyllotaxis script. The Java applets seem very broad and very repetitive and inscrutably "simple," so I'll just
start doing some code based upon the given formulas, and Growth rate and divergence angles. Plus more reading and study:-)
So you might try a Lukas sequence. Or numbers which are different, the Lucas numbers?
So a loxodrome is not a Seifert spiral.

- Brian

Hello Brian,

the loxodrome node isn't really a loxodrome creator. It's more a stereographic projection node. It takes various curves and maps it to a sphere. In the example I used logarithmic spirals -> loxodrome.

Have a nice day
Karsten

Out of subject :)
a very cool book! (exists in English and French also! ;)
http://www.bilder-der-mathematik.de/

Hi Pilou,

does the book have also reference links for software, in which you can create those mathematical shapes, or source code included?

Regards
Stefan

In my French version
Nothing about render programs but you have several links about the subject show!
Each function is detailed.
http://www.editions-belin.com/ewb_pages/f/fiche-article-surprenantes-images-des-mathematiques-16456.php
Press book's image under "Feuilletez un extrait" (on the top left page then Choose a page then Slider Zoom)

Thanks for the info Pilou, much appreciated!

Regards
Stefan