### reading off external angles from binary decomposition images

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• 3f
• Posts: 1455

#### reading off external angles from binary decomposition images

« on: May 11, 2019, 10:27:43 PM »
To find the external angles of an island:

1. draw a line along the binary decomposition from a tip of a filament until you reach the middle of a cell
2. draw similar lines from the neighbouring filaments one level deeper in the period doubling rings
3. fill in the numbers on the paths between them
4. these two bitstrings are possibly-rotated copies of the periodic external angles, next is to find the rotation offset
5. if the two bitstrings are opposite, the island is on the real axis and the angles must be between the Feigenbaum/Morse angle ".0 1 10 1001 10010110 ..." and its complement - rotate so that this is satisfied
6. otherwise, the last digit must differ, and the first digit must be the same - rotate so that this is satisfied
7. it remains to be proven that this always works
8. its tedious once the period gets above 20, because the cells in the period doubling rings typically get small in the pinch points

animation showing the process:

examples also attached

• 3f
• Posts: 1455

#### Re: reading off external angles from binary decomposition images

« Reply #1 on: May 11, 2019, 10:28:19 PM »
another example

• Fractal Furball
• Posts: 288

#### Re: reading off external angles from binary decomposition images

« Reply #2 on: May 12, 2019, 06:00:51 PM »

• 3f
• Posts: 1455

#### Re: reading off external angles from binary decomposition images

« Reply #3 on: May 12, 2019, 06:13:04 PM »
sure, both use binary decomposition, but those articles start with angles and find the location that matches, while mine starts with an unknown location and finds the external angles from the image.

• Fractal Furball
• Posts: 288

#### Re: reading off external angles from binary decomposition images

« Reply #4 on: May 12, 2019, 06:18:19 PM »
maybe you can do something like exponential map along ray to have all binary numbers on one image ?

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