2-fold Embedded Julia Set  

Robert P. Munafo, 2008 Feb 17.



An embedded Julia set with 2-fold (or "order 2") rotational symmetry.

All of the following embedded Julia sets are 2-fold:

-1.770 666 447 810 813 192 662 +0.016 335 275 874 991 552 073i  2.8e-18
-1.770 666 447 810 813 192 662 +0.016 335 275 874 991 552 073i @ 2.8e-18
This one is a sparse Cantor dust.

-1.254 558 754 876 718 04 + 0.381 722 466 793 449 24 i   5.0e-15
-1.254 558 754 876 718 04 + 0.381 722 466 793 449 24 i @ 5.0e-15
A somehat less sparse Cantor dust.

-0.7454291059302 +0.1130087933933 i  1.8e-10
-0.7454291059302 +0.1130087933933 i @ 1.8e-10
_ -m\v@L@@*^Fmm-_:@@F^7@*"Fam@@bc_ aea- .a@rv.@@@(@/F\emmd^**Y@@]@(be%F\d@F@bm _d@C[ /@@bm@@@@@@@b((F"*` '^`'md$@@@c _dT,.@@@@[ ^YCb@@@@@@@@@@CF^ '^@C@ ^-".d@@@@[ '*b@@@@@@@C@"" :ammme :@@ ._, ._dCF@@@[ '^"***~^ _:dF^-m/` a@[ .d@@b_ad"$L@^@$[ ._, _ . ._.@@C, ._@@ '@@@@@@"\C%`*@mc e/@bmv@mdbmmme_m_.e__@@@F*m mem@CC ^*@@@^*@/~m@b@[ mm@Yb@**"^^^^*Y@bY@**_""a*]L^*%"@@__ . -d@@` e@@%`' ..__, .$@@ C,@F@b$@@Tdb~m@@@@mbme,, _c Y@"^ ]@`-* .,'%%a@/@@@@@@@@@@F"_@*'^._/Yb ./[ /\_, 'Y@memm@@;*-@@@@@@@@@@@@^^mm, "@Ld" _:%%; d@@F ^^"Y@YT)(^@@b$Y@7YL"`T@Fc_,_ ,._.@Y@([ @%F*\,@mcm:__ /@@mmme"[-*:@~"d@@d@@7@m@m@L@@dC@@[ $@c_@F/%@b@@@@, _@@d"*" *md@@@/^`" `^'^^^"^*^^"` ` @\@*@^d@@Y@@@@` _d@" ._ ^T@CF^` b@d@$"` "***` :@" :dT$ a@@F` ,.__e-m:,a,_ @@@"___ _]@@` '*\mm@* ._/dY$@@@@@@(@\m @F aF(F __a$FCF, ,/m@YF@Y@@@@@@@@@$ae ^ Yb_,_,,'@@d@@b@C . .(@b*@@@@@@@@@@@\FT@ -$@Y*@*b(7*@@[r@.mme. mmm@v@F@d@@@@@@@(^T@@@b '^^'`'^^` ^`^`^^^^'^ '^^^^^^^^^^^^^^^'^^^` -0.7454291059302 + 0.1130087933933 i @ 1.8e-10, Nmax: 20000
Rollo Silver's "Concha Quad"

-1.747 310 71 +0.000 000 00i  0.000 078 13
-1.747 310 71 +0.000 000 00i @ 0.000 078 13
'r, ', . '^ .- _mc` '-,, ..,-.ed_. ] . -. .` ._-' ^~.,'.b$F(7**\_ ' //dm*vLa,` ..-' _ .,/-/" :^ '""" c '") "F^%@`_-'` ^_ed"* ^:d, [ .L,,-$a,_. ^---._@e)a--'"'^--_.mm/-"/-'""L.;'*[ _.. -d@L,` ,. '\"` 'T^d,^^^` ----"da,a)d- ' .__\,aYFL`_ --~**m/F- --- --Fdc@~L, '^'' ` __'_., `' ' ____a__________________dd7$d_______________________, ' -""""- ' ._'-/a,c. .'_ ..ca_a _- .__,v)@*%7@' T :L@T;*\c_-- _^b$/ '"" ._],_ '`` .*@c- ___._@e_e_. .__]Cb[__, , _'/%",^^---. ^^ "@L~' ^:d/^`` '^^" ''`:d)c e/F`. -^\")*,-'T _, [ __, -[ _L"d, ..-':$_/m,'"Tb ` /CC_"\_mF- ^-\, ._,~ ."/"\d@*(` ] ''"^*^\' ` '~_- ,/` _/ `: " ` '\, '-., '^` ' ' ` ^` -1.747 310 71 +0.000 000 00i @ 0.000 078 13, Nmax: 20000
One of the largest embedded Julia sets, found in R2F(1/2B1)S.C(0)

-1.7499254344 +0.0000000003i  2.975e-7
-1.7499254344 +0.0000000003i @ 2.975e-7
Even though ones like this appear "connected", they are also Cantor dusts.

-0.744 539 860 355 905  +0.121 723 773 894 425 i  7.5e-12
-0.744 539 860 355 905 +0.121 723 773 894 425 i @ 7.5e-12

-1.74876455 +0.00000001i  +0.00002441
-1.74876455 +0.00000001i @ +0.00002441

Compare the above to the pictures in 4-fold embedded Julia set.

The following image is actually a 2-fold embedded Julia set tuned by another 2-fold embedded Julia set, the latter of which is a very sparse Cantor dust. I usually call it a paramecium rather than an embedded Julia set:

-1.7487590765 +0.0000047386i  +0.0000007155
-1.7487590765 +0.0000047386i @ +0.0000007155
A paramecium

It is important to distinguish "2-fold embedded Julia sets" from second-order embedded Julia sets. See that entry and embedded Julia set for clarification.

See also N-fold embedded Julia set and bifurcation.




From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.     Mu-ency index
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