Astonishing!

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 From:  Frenchy Pilou (PILOU)
8265.1 
Beauty of the complexity of the simplicity!

EDITED: 14 Jan by PILOU

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 From:  TpwUK
8265.2 In reply to 8265.1 
The wonderfully complex world of Mother Nature - I enjoyed this post, thanks for sharing

Martin
(TpwUK)
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 From:  bemfarmer
8265.3 In reply to 8265.1 
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

EDITED: 15 Jan by BEMFARMER

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 From:  Frenchy Pilou (PILOU)
8265.4 In reply to 8265.3 
Bon courage!
---
Pilou
Is beautiful that please without concept!
My Gallery
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 From:  bemfarmer
8265.5 In reply to 8265.4 
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
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 From:  bemfarmer
8265.6 
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...

EDITED: 17 Jan by BEMFARMER

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 From:  bemfarmer
8265.7 
Doing a 2D mouse slider in a NodeEditor window, along with a javascript program, with phyllotaxis in the MoI window might work?

- Brian
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 From:  Michael Gibson
8265.8 In reply to 8265.6 
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
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 From:  bemfarmer
8265.9 In reply to 8265.8 
Thank you Michael.
- Brian
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 From:  Karsten (KMRQUS)
8265.10 In reply to 8265.3 
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

EDITED: 20 Jan by KMRQUS

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 From:  Karsten (KMRQUS)
8265.11 
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
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 From:  Max Smirnov (SMIRNOV)
8265.12 In reply to 8265.11 
[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
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 From:  Karsten (KMRQUS)
8265.13 In reply to 8265.12 
Hello Max,

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

Karsten
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 From:  bemfarmer
8265.14 In reply to 8265.11 
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

EDITED: 22 Jan by BEMFARMER

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 From:  Karsten (KMRQUS)
8265.15 In reply to 8265.14 
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
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 From:  Frenchy Pilou (PILOU)
8265.16 
Out of subject :)
a very cool book! (exists in English and French also! ;)
http://www.bilder-der-mathematik.de/
---
Pilou
Is beautiful that please without concept!
My Gallery
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 From:  futagoza (STEFAN)
8265.17 In reply to 8265.16 
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
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 From:  Frenchy Pilou (PILOU)
8265.18 In reply to 8265.17 
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)

EDITED: 23 Jan by PILOU

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 From:  futagoza (STEFAN)
8265.19 
Thanks for the info Pilou, much appreciated!

Regards
Stefan
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