Iterated Offset

[claude]

rendered with kalles fraktaler 2:

http://mathr.co.uk/mandelbrot/2014-11-10_iterated_offset.ogv (44MB, 2m00s, no sound)

Not too deep or special, colours could do with more work, but the process behind it is perhaps interesting:

The first minibrot island passed along the way has period 4 and external angles .(0111) .(1000) (my current explorer project gui can trace the external ray from a given point, to find the corresponding angle)

The island at the center of the embedded julia set that appears has period 107 and external angles .(01111000011110000111100001111000011110000111100001111000011110000111100001111000011110000111100001110111100) .(01111000011110000111100001111000011110000111100001111000011110000111100001111000011110000111100001111000011)

The island offset one step to the side in the embedded julia set has period 111, and the external angles are formed by concatenating the angles of the previous two: 111(lo,hi) = 107(lo,hi) + 4(hi,lo)

Repeating the zoom in and offset step gives a period 218 island, with angles 218(lo,hi) = 111(lo,hi) + 107(hi,lo)

This looked like a pattern (append the angles of the previous two islands, with the lower one swapped), so I wrote a bit of Haskell code to automate it:

Code:

rays
  = ("0111", "1000")
  : ("01111000011110000111100001111000011110000111100001111000011110000111100001111000011110000111100001110111100", "01111000011110000111100001111000011110000111100001111000011110000111100001111000011110000111100001111000011")
  : zipWith append (tail rays) rays
  where
    append (alo, ahi) (blo, bhi) = (alo ++ bhi, ahi ++ blo)

main = putStrLn . (".("++) . (++")") . fst . (!! 11) $ rays

the 11th island in the list has period 9743.  I stuck its external angles into some other code of mine that traces the external ray and finds the coordinates of the island, along with a size estimate (around 1e-50).  I copy/pasted the results into a freshly saved default .kfr and rendered the zoom out sequence.

had some problems with the keyframe movie maker (corrupt AVIs), but i managed to use it to save the 170 keyframes to a 6-second uncompressed avi, converted to ppm files with avconv and used my own zoom video assembler program.