Art Recipe 2: Detective postscript

Yesterday I posted the Art Recipe 2 but there was a mistake in the recipe I'd posted.  The first take away from this is that I should make actual notes while making the art rather than type it up after an evening of creative experimentation.

Down to the detective work to find out where I'd gone wrong.  I knew I had the gears right because I had a photo of the setup that I'd used.  And I had a photo that I'd take on the first half of the design where it is easier to see the patterns.

Reference photo that I am trying to re-create

I was pretty sure that it wasn't many steps that I was doing with the hoop and with the gear and while I thought I know which direction I was moving each gear I could no longer be sure.

To begin my investigation I set about testing a few simple combinations.

Each test helped me get a better intuitive feel for how the different combinations of movements changed the design.
Also, I quickly came up with a simple notation for myself at the top of each page after realizing that I was going to loose track of what I was doing almost immediately if I didn't.

After doing a few of these I started to think more critically and try and work out what I could deduce because even if I was only searching in the space of three steps or less that was 36 different combinations to test and that was going to take far too long; especially if I could be clever and make it easier.

My first realization was that I could look at the reference picture to see how many times I had iterated the base 9 pointed design.  This would tell me how many times I was shifting the hoop to the right.  Every two steps of the bottom of the hoop is like translating it one step right in terms of hoop position.

The red marks count the number of repetitions of the base design

Picking one of the fans created by shifting and rotating the design I counted 24 instances of it.  Combined with the fact that I knew that the design had been concluded somewhere in the 20s on the oblong this helped me conclude that it must have been made with 2 step increments on one side of the hoop.  That nailed down the parameter as 2(right).

Next I looked at the designs that I'd made and the one I was trying to copy and saw that on the same fan of petals that I counted when I stepped the gear to the right it did not make that counter clockwise progressing swoop.  From there I determined that I would explore the left stepping solution space.  Instead of trying the few combinations at hand now I was able to make one more deduction.

If I step the hoop two steps right and the gear 1 steps left it is very  similar (but not identical) to translating the hoops top and bottom both 1 right.  The movements almost cancel each other out as I'd seen in hoop 2 right and gear 1 left.  I'd also tried hoop 1 right and gear 1 left and I figured that would make something similar to hoop 2 right and gear 2 left and that wasn't the shape I was looking for either so I opted to try hoop 2 right gear 3 left which worked out.

Other things to note take away from this is that the translation of the center of each 9 pointed design is shifted to the right with each iteration but because of the 3 left steps of the gear the pattern itself rotates to the left making the characteristic fan shape that was counted.