T-square fractal

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bwisialo
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Re: T-square fractal

Post by bwisialo » Tue Mar 15, 2016 11:13 pm

I'm calling it now -- every fractal in that gallery can be generated in Cell Lab! ;)
amor fati
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bwisialo
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Re: T-square fractal

Post by bwisialo » Tue Mar 15, 2016 11:32 pm

There are plenty of cool patterns that aren't in the gallery, of course. And you can do things like deform a pattern by, eg, changing angles by 90 instead of 180 -- still surprised the thing doesn't just rip apart when you do this.

Image
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bwisialo
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 6:20 am

Actually, the pattern I posted with the 6 mode genertor can be produced with the 3 mode one. Clearly I haven't figured this out. ;)
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bwisialo
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 7:02 am

If additional modes generate additional result, I figure that you have to use 7 modes. When you have 4 cells, you'd have three 5's and 4. And you'd use modes 6 and 7 to 1s and 4s. Maybe...
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bwisialo
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 7:44 am

Right, that made more sense. Here's the simple 7 mode generator. I changed the child angles on 1 so that it produces fractal 222 in the gallery. I'm pretty sure that actually does require 7.

https://www.dropbox.com/s/2werrnbkgkgvn ... enome?dl=0
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bwisialo
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 10:03 am

Sratch that as well. That one can be done with 3 modes, using a 90 rotation. I hadn't done much of that because of the peculiar splits.
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 7:34 pm

Here's how the 3-quadrant Sierpinski fractals work, with illustrations of the ways the base pattern can be manipulated -- in effect, the "rules" for how you can use angle rotations and mirroring.

http://ecademy.agnesscott.edu/~lriddle/ ... iation.htm
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bwisialo
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 10:56 pm

I cracked the code -- 2/3 of the fractals in the Sierpinski family are producible! :D Using 4 mode for some, then 7 modes for others. There should be a way to get the final 1/3. May or may not require 10 modes.

Here's a 7-mode fractal. I'll post the generator later. :)

Image
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bwisialo
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Re: T-square fractal

Post by bwisialo » Wed Mar 16, 2016 11:45 pm

https://www.dropbox.com/s/c5wu2j42zvovy ... enome?dl=0

Rules and tips

Use a plate for 1024 or 4096 cells.

When you have 16 cells use Child modes of M 2 and M3 to rotate and reflect the three quadrants with colored cells. (See example at bottom.) Below, 1 is the initial position of the pattern you'll be rotating and reflecting. When you're at 4 cells you'll see the pattern.

Image

2, for example, is 90 degrees, etc. The mirror setting performs #6, and #5 can be done by mirroring combined with rotation. (7 and 8 aren't needed – they produce duplicates of others.) Whatever settings you used in 2 and 3, duplicate them in modes 6 and 7 respectively. That's it!

This website has an immensely useful image gallery of the 232 fractals in the Sierpinski family -- it's a different gallery than the one I posted previously. Click on any fractal you want and it will show you how to orient the pattern in the three quadrants. For example, to create this

Image

do this with the three quadrants using Child mode settings for M2 and M3. Rotate the bottom right quadrant 90 degrees.

Image

Here's the site. Enjoy!

http://ecademy.agnesscott.edu/~lriddle/ ... eryBox.htm
amor fati
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bwisialo
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Re: T-square fractal

Post by bwisialo » Thu Mar 17, 2016 6:25 am

Actually, it appears that the diagonal transformation 7 doesn't work out with a mirror and rotation -- the final pattern isn't right. Everything else seems to work though.
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