How do you construct the pattern of a dragonfly wing using the golden ratio?

[kaleidoscope]

You can see it here starting at 2:30


https://www.youtube.com/watch?v=http://www.youtube.com/watch?v=kkGeOWYOFoA

My question is, how do you determine the position of the points used to draw the triangles and circles.

 

[redbowlover]

i may be mistaken but i don't think the position of the points matters that much. if you start with any (nice) array of points you can connect them to create the pattern...

for example, draw a point and then draw 5 points around in. connect each of the outside points to the center and each point to its neighbor. then around each of the outside points draw 2 more points. connect each of these points to the new center and to the neighbors.

you'll start to see a pattern of 5-sided polygons. every triple of points that are connected to each other will lie in a circle. (this is true because any three points that do not all lie on one line will determine a unique circle.) but the circles aren't necessary to complete the pattern. they are pretty for the video but not practical.

the other lines are drawn perpendicular to the existing lines of the triangles through the midpoints.

so you see, the original placement of the points isn't really important, as long as you don't allow for "bad" arrays... like all the points are on the same line. you could draw all the point equidistant from each other...this would give a perfect pattern not likely to appear in nature.

 

[uoflcard]

i may be mistaken but i don't think the position of the points matters that much. if you start with any (nice) array of points you can connect them to create the pattern...

for example, draw a point and then draw 5 points around in. connect each of the outside points to the center and each point to its neighbor. then around each of the outside points draw 2 more points. connect each of these points to the new center and to the neighbors.

you'll start to see a pattern of 5-sided polygons. every triple of points that are connected to each other will lie in a circle. (this is true because any three points that do not all lie on one line will determine a unique circle.) but the circles aren't necessary to complete the pattern. they are pretty for the video but not practical.

the other lines are drawn perpendicular to the existing lines of the triangles through the midpoints.

so you see, the original placement of the points isn't really important, as long as you don't allow for "bad" arrays... like all the points are on the same line. you could draw all the point equidistant from each other...this would give a perfect pattern not likely to appear in nature.

I have to disagree with this post, extremely. It exhibits 20th century ideas of our perception of clumsiness in nature that has actually turned out to be genius. "Genomes are filled with junk DNA", "cells are blobs of jelly", "blood clots like driftwood clogs a stream", etc.. All of these beliefs, and countless others, have turned out to be just as laughable as flat Earth theories. The dragonfly wing, alone, is studied by dozens of engineering departments around the world trying to study and hopefully apply the engineering design principles that are exhibited by it, from micro air vehicles (MAV's) (1) to structural engineering (2).

(1) http://shyylab.engin.umich.edu/files/papers/Reno_'08_Shyy_et_al.pdf

(2) http://biomimetic-architecture.com/2010/10/23/maria-mingallon-and-the-architecture-of-the-dragonfly-wing/

To believe that dumb 'ol nature haphazardly cobbles together polygons to make dragonfly wings is naive. I do not know if dragonflies use points on a Fibonacci spiral and this geometric algorithm to construct their wing area patterns, but I would be extremely cautious when making this statement after everything else we have woefully under-estimated in biology since Charles Darwin "eliminated the need for intelligence" to explain the designs of life.

Also, below is a website that explains why "perfect" (as you describe it) ratios (round numbers, integers, symmetry) are horrible for things like seed/leaf/petal placement in flowers, and why precisely the Fibonacci sequence (a quite irrational number, not something that nature would stumble upon like the simple ratios of crystals) accomplishes perfection. The same principle can be applied everywhere else in nature that this sequence is found, which is seemingly everywhere.

http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html