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More fun with Fractals

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Posted (edited) · Report post

Fractals is a much bigger subject in mathematics than just the chaos game i proposed to brainden recently. There is also a version of self-similar iterated fractals based off rotation, translation, reflection.

This is an example of one:

post-53485-0-90947100-1375302316_thumb.p

Fractals like these are made by repeating taking a "picture of a picture of a picture sort of like looking at a mirror that is directly across of a mirror. Essentially for each image a picture is taken and either left alone or some how modified (rotation, translation, reflection, etc.)

For example:

post-53485-0-63170800-1375303135_thumb.p

Can you determine how the first image was created (the colorful one)?

Edited by BMAD
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Posted (edited) · Report post

The initial image is in black. For every new image, the three copies of the previous image are in red, blue, and green.

The red copy is the original image flipped across the y=-x diagonal. The blue and green copies are the red image flipped vertically (or simply the original image rotated 90 degrees CCW).

post-54021-0-75840400-1375516022_thumb.p
Edited by gavinksong
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Posted (edited) · Report post

The initial image probably looked sort of like an L rotated 90 degrees clockwise.



It looks like the initial image was rotated 90 degrees counter-clockwise and placed in the third quadrant, a copy was reflected across the x-axis, another copy was translated to the first quadrant, and then the whole process was repeated with the resulting image.
Edited by gavinksong
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Posted (edited) · Report post

post-54021-0-15484900-1375339414_thumb.p
Edited by gavinksong
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Posted · Report post

Excellent solution. There is an even easier (and non trivial solution. The trivial being that the image present is the initial. Can anyone find it?

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Posted · Report post

Excellent solution. There is an even easier (and non trivial solution. The trivial being that the image present is the initial. Can anyone find it?

Okay. Maybe the initial image was moved to the second quadrant, flipped in respect to the y=-x diagonal, a copy was reflected across the x-axis in to the third quadrant, and then another copy was translated to the first quadrant. The initial image still being an L rotated 90 degrees clockwise.

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Posted (edited) · Report post

I just realized that the initial image could just be a pixel.

Everything else is the same as my last answer. To get the next image, the current image is flipped in respect to the y=-x diagonal, two copies are moved to the right of it and below it, and they are both flipped vertically.

Edited by gavinksong
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