Message Boards Message Boards

Curlicue Fractals

Posted 9 years ago

The CurlicueFractal can be simplified with AnglePath.

Graphics[Line[AnglePath[N[ (7 Sqrt[7] Khinchin Pi E EulerGamma) Range[-20000, 20000]]]]]

curlicue fractal

Who else can find some nice ones?

POSTED BY: Ed Pegg
11 Replies

Here is a case which finds after a rather long path into repetition

Graphics[Line[AnglePath[Partition[N[Flatten[
      SubstitutionSystem[Rule[#[[1]], {#[[2]]}] & /@ Partition[First[RealDigits[N[E, 130]]], 2], 
       First [RealDigits[N[1/EulerGamma, 150]]], 146]]], 2]]]]

enter image description here

Want to look into the evil eye? Try

Graphics[Line[AnglePath[Partition[N[Flatten[
      SubstitutionSystem[Rule[#[[1]], {#[[2]]}] & /@ Partition[First[RealDigits[N[\[Pi], 30]]], 2], 
       First [RealDigits[N[1/E, 10]]], 206]]], 2]]]]

or

Graphics[Line[AnglePath[Partition[N[Flatten[
      SubstitutionSystem[Rule[#[[1]], {#[[2]]}] & /@ Partition[First[RealDigits[N[\[Pi], 30]]], 2], 
       First [RealDigits[N[\[Pi]/E, 10]]], 144]]], 2]]]]
POSTED BY: Udo Krause

Take some negative numbers too

Graphics[Line[AnglePath[N[Flatten[SubstitutionSystem[
      Rule[#[[1]], {RandomChoice[{-1, 1}] #[[2]]}] & /@ 
       Partition[First[RealDigits[N[E, 30]]], 2], 
      First[RealDigits[N[E \[Pi], 133]]], 31]]]]]]

enter image description here

and one symmetrical (despite the pesudo-randomness in the rule, it's only active during rule creation)

Graphics[Line[
  AnglePath[
   N[Flatten[
     SubstitutionSystem[
      Rule[#[[1]], {RandomChoice[{-1, 1}] #[[2]]}] & /@ 
       Partition[First[RealDigits[N[Pi, 30]]], 2], {7, 3, 2, 8}, 
      304]]]]]]

giving

enter image description here

replace $\pi$ again with $e$

Graphics[Line[AnglePath[N[Flatten[SubstitutionSystem[
          Rule[#[[1]], {RandomChoice[{-1, 1}] #[[2]]}] & /@ 
           Partition[First[RealDigits[N[E, 30]]], 2], {7, 3, 2, 8}, 304]]]]]]

and hit the return key a few times to see

enter image description here

POSTED BY: Udo Krause
POSTED BY: Todd Rowland

Just one more thing (.... I'm sorry about this ....)

Graphics[Line[AnglePath[N[Flatten[SubstitutionSystem[
      Rule[#[[1]], {#[[2]]}] & /@ Partition[First[RealDigits[N[Pi, 30]]], 2], {7, 3, 2, 8}, 304]]]]]]

enter image description here

after seven steps it gets into repetition.

POSTED BY: Udo Krause
POSTED BY: Udo Krause
POSTED BY: Marco Thiel
POSTED BY: Marco Thiel

This is so cool

POSTED BY: Eduardo Serna

I'm game. I'd add a bit of Opacity to see structure in concentrated places. Some of these could probably make nice tattoos ;-)

Multicolumn[
Graphics[{Opacity[.5], Line[AnglePath[N[# Range[-20000, 20000]]]]}, 
    ImageSize -> 500] & /@ {51 E, 350, 30 E, 37 E, 90 E, 406 E}, 2]  

enter image description here

POSTED BY: Vitaliy Kaurov

Following cool ideas from Marco's answer I am adding Collatz to my gallery. The longest progression for any initial starting number less than 10 billion it is 9,780,657,631, with 1132 steps. I plot its neighbor 9,780,657,630 too and for 1200 to see the cycle.

data0 = NestList[If[EvenQ[#], #/2, 3 # + 1] &, 9780657630, 1200];
data1 = NestList[If[EvenQ[#], #/2, 3 # + 1] &, 9780657631, 1200];
Graphics[{
  Red, BSplineCurve[Reverse /@ AnglePath[Log@N@data0]],
  Blue, BSplineCurve[Reverse /@ AnglePath[Log@N@data1]]}, 
 ImageSize -> 900, Background -> GrayLevel[.1]]

enter image description here

Do not dwell too long on Collatz though... "Mathematics may not be ready for such problems." ~ Paul Erd?s about the Collatz conjecture

enter image description here

POSTED BY: Vitaliy Kaurov
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract