Dragon Curve Fractal Using Golden Ratio

EebamXela

I've been fooling around in MS Excel trying to reconstruct this fractal:



I haven't had any issues here making it. I totally understand the algorithm for generating the left turn/right turn ordering. What I really want to know is how this version is generated:



Original image: http://en.wikipedia.org/wiki/File:Phi_glito.png
The sides of the base triangle are equal to:
1.000000000
0.742742945
0.551667082

Is this fractal generated using the same algorithm as the above one? I can't seem to find any explanations anywhere to confirm. I tried using the same algorithm and steps to recreate it in excel but all i get is a fractal that KINDA looks like it, but it's obviously not:



I don't have any code to share because i'm not very good with code. I figured once i nail down how to actually construct the thing manually i'd try coding it.

Please someone tell me what i'm doing wrong. Thanks.

haruspex

When I discovered the Dragon curve back in about 1970 (ok, I can't prove it, but I did), I generated it as shown in your first picture: Draw some figure (short line segment to start with) from point A to point B, take a copy of the figure and rotate it 90 degrees about B, and make the copy of the point A your new point B. So necessarily the points A, B, A' form a 45 degree right triangle. The second picture above appears to be the same but with a magnification of the copy. The question is, how was the magnification selected?
The original construction has this fascinating trick of meshing seamlessly with itself, never overwriting any lines. The magnifying variation doesn't mesh in the same way, but seems to be chosen just so that it touches itself at many points.

TGlad

The clue is in the triangle behind the picture. For every line (e.g. the horizontal line in the triangle), replace it with the other two lines in the triangle. The trick is to note that there are two orientations for each of these two new lines, each is a 180 degree rotation of the other. Hence, if you make both lines use the closest orientation to the horizontal line, you get variants of Levy C curve, if you make both lines 'upside down' then you get variants of the Von Koch curve, and if you make the two lines one of each then you get variants of dragon curve fractals.

Michael Redei

I think you're just too impatient. What you've produced looks execellent, not just "kinda" like the original from Wikipedia. All you need is more steps to get the straight lines "bumpier".

I'd love to see how you did this in Excel. All I've ever used that for is tables, and never graphs.

Michael Redei

According to the caption for the image in Wikipedia, their fractal was not constructed in the way you describe (with a Lindenmeyer system) but rather with an IFS.

EebamXela

Can anyone help me figure out the algorithm that was used for this golden dragon fractal?