T-square fractal

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bwisialo
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Re: T-square fractal

Post by bwisialo » Mon Mar 14, 2016 7:17 am

Time to geek-out. :ugeek:

I noticed that the first square block I posted above had a pattern that would be symmetrical if part of the pattern was flipped 180. It's below. It doesn't have a name, but I found a construction of it (below as well) in this Gallery of Sierpinski-like Fractals. I've thought for a while that the fractal we've created are fundamentally similar to one another. I believe they are all fractals of the same type: iterative function system (IFS) fractals. The subset of IFS fractals in the linked gallery are all generated by mapping a square onto a square. The results include the Sierpinski triangle with a 90 degree angle; the T-square, the ice fractal; and the fractal below, which I think is the one numbered 138. The page notes that "there are 8 ways to map the large image square into each small square," and it uses a numbering scheme to indicate the mapping scheme. I believe there are 256 fractal images on the page (though some are duplicates): 8^3 (512) divided by two because half are mirror images of the others, which he excluded. It seems like a quite a leap, but it may in theory be possible to create all of these fractals in Cell Lab, because it should be possible to map squares in the 8 ways. Many cases may be just ridiculously complex to generate.

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bwisialo
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Re: T-square fractal

Post by bwisialo » Mon Mar 14, 2016 9:18 am

This one is, I think, number 316.

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bwisialo
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Re: T-square fractal

Post by bwisialo » Mon Mar 14, 2016 4:22 pm

By playing around with the T-square genome, I was able to produce quite a few of the fractal in the gallery linked above. They can be produced they are shown in the gallery: covering 3/4 of a square. Some I produced as rectangles. Here they are with their number in the gallery.

118
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124 -- T-square
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166
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181
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183
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316 -- in the post above, but here covering 3/4 of a square
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338
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342
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383
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amor fati
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bwisialo
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Re: T-square fractal

Post by bwisialo » Mon Mar 14, 2016 5:12 pm

Here are some others that, except for first one, apparently don't follow the Sierpinski mapping scheme, but they're cool.

This one is in the gallery -- 475
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Update: Actually, after posted this I realized it's the 3/4 version of one in the previous post -- 181
Update 2: No, I was right the first time: the pattern in the first post is 181. This one is different, and I'm pretty sure it's not in the gallery.
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And some lattices that came out when I tinkered with the Sierpinski carpet genome.

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amor fati
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CandyYAHFT
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Re: T-square fractal

Post by CandyYAHFT » Tue Mar 15, 2016 12:06 am

well boys, you definitely bring this to a whole new level
PEACE & LOVE,
CANDY
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bwisialo
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Re: T-square fractal

Post by bwisialo » Tue Mar 15, 2016 12:16 am

It's pretty crazy! It's like hitting upon one single thing -- like making the pattern symmetrical in that first, weird block square I posted -- and a whole new world opens up in all sorts of directions directions around it, just by playing and exploring. It was the same thing when I produced that first lattice. Like a door was unlocked, because all of sudden there were all sorts of other patterns in the near vicinity.
amor fati
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Alast
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Re: T-square fractal

Post by Alast » Tue Mar 15, 2016 7:06 am

You created yourself a canvas. Now you're painting on it ;-)
Perfection hasnt reached me yet, but its trying hard!
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bwisialo
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Re: T-square fractal

Post by bwisialo » Tue Mar 15, 2016 6:17 pm

It got crazier! :) This one I generated on purpose, starting from a new genome, to try to get the right angle in the corner. The genome is so simple and can be altered to generate so many different patterns, it's just mind-blowing. It's 4 modes, all splitting at right angles. In the genome linked below, they're all 90. Use a 1024 or 4096 cell plate. Start rotating angles 180, ticking and unticking mirroring, and look at what you get. I'm posting one below -- second image -- because it's super cool! It's 318 in that Sierpinski gallery.

Genome: Fractal Generator

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Alast
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Re: T-square fractal

Post by Alast » Tue Mar 15, 2016 6:25 pm

Guess I need to start canvasing ;)
Perfection hasnt reached me yet, but its trying hard!
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bwisialo
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Re: T-square fractal

Post by bwisialo » Tue Mar 15, 2016 11:07 pm

Here's another generator that produces additional results via 6 modes -- with mode 1 and 2 mostly but not entirely duplicated into 5 and 6. I could make the starting point simpler, and I haven't figured out the "rules" for manipulating it to get fractals in the gallery -- only certain sort of changes work out. But that could take some time. In this genome, when there are 4 cells, you'll have two 1's (top) and 5 (bottom left). 1 and 5 produce the same pattern, but I think you need the six modes to create the pattern. I think you can see where this is headed. ;)

Fractal Generator 2 -- 6 mode

Here's the starting pattern:
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