Distortion  

Robert P. Munafo, 2010 Oct 20.



Some features of the Mandelbrot set are often said to be "distorted" when compared to other features1.

One common example are islands in cusps. In these illustrations, the first island is the slightly asymmetrical R2F(1/3B1)S. The second is R2F(1/15B1)S, located fairly close to the cusp R2.C(0):

-0.15842 +1.03335 i  0.03048
-0.15842 +1.03335 i @ 0.03048

Figure 1. R2F(3)S . Coordinates: -0.15842 +1.03335 i @ 0.03048
(-, ./, , . /^ `YL "__-` `r ._ \_cr'\ ..,._7 _( -*; /c)@ -* ' '\v. \,_ "v,a/` "__._--% '` ._), _ _a[` ,-d` . ., .a^ ` ' '` "\(L_/' -, ^c^ , :-- ^c-' .,_:'$@m.'.'/`,:dc- .a[- ^\" ., -r`. -``"''"~Y@Cdb_L(@m:_-`,.\ 7(,_- \ d\$m/ , ., :..d)$@^^*^^ ^"@m\\.La\@$[a,dmb@@b);``"- ^`\ ,. `,"'@* '@_)**^^`'^"*Y` :@@--: . , ,_---)'-e___._^dd@L, -"` ^($L--..,__/ , ._-' '` ." - `~^-^@be____ "@aLm%'^ `:(--^` ' :/-~`--"^%F :_@Y;- ^~^ "-c,e@` 'Y)-_- ,, '-\.____ -dT@, .@*% ^~/` -"` /^'^-Y@__ .r""~ '~ ,'`'"(@c, :[^"c__ -*-mv.%m*"-. __@`_ ''e_,_er . a,:@b,__/)e_.._-__m_@~, `"F, _ __/ ,. ._-________/)b@mY@F'/"C*bed('C' '^ ^[,m_, ,_-r(^' -\a"-_____7c'`][ '`'*"\7`.` ' \/(*"" ^-- ]`^-\_'*`` ' :"` '~ _/^- '- .*. '` :\_ :; v~ " d$`, ' '` -".,- ^%^ :," ^~- -""v- ~~, ^$^ ^@. . .:r-'_ .:'^'-,d ' '"-- ' ^"*
0.2925294 +0.0149698 i  0.0006675
0.2925294 +0.0149698 i @ 0.0006675

Figure 1. R2F(15)S . Coordinates: 0.2925294 + 0.0149698 i @ 0.0006675
d^'Y/^e@Fa@~*@mYr]@)"dY^@)v` _^^^ .vmmc.mF@bC.:bbF :7@*"`-"@@@F^ `^m, .%)@^$d@m'` 'Y@dmem[ ___ .Cb)_, :@*^^ ^`""_c__/^@^@%c ^ Te, /*"m(be, ___,__,_ 'Y., .m@@C@CTF*` d(L\e,._.,___L_a$@@@['^^Y@(d*( _._,___ /$*YeF '^m_, ____)_$$@@'^^Y*(FY$d@-""- dC^"^@ TvY@@@@b-e_.___L.@@^\_.__"T@b"^\YF@b$)^"___r@*F` *L ._FmcdF@_dF^ *Y"b*C("d"^"^*$)(@@mY*/$$F"F:m\@"@ :___@@L@@e@___$F'$F)d@_/*"^^ bd@C,CF_d\, ..%Yd@F@d@@@.mmF@@*@@@_m"F"a@m@@@$mm@Cdrd@@F$^F _F^ ^@*%FF-'*YbCY@- '*@/*"^` ""*YCaY@@*^^ ^"Y*"Y@@"@F^(d[____e @L_m,.@m$@,_ :m(@@e__ ~` '^*d` -a@YY%F^'Y(.d**"` ^^^^ @@@Fmd@b-^`@mT/*` '- .@@drY)-~****^ 'F^@;/dCmc-@@@^ ~_____.m@Y@@L],._^c '@F^ '*%@abF* ~@)"@Y@F:*`d@[^@*; '~-_a~"$@@e_c 'dbd*v@Y` 'C$m@@ :@@)@b_d%C@b- *@L*-"F**-\mY-F"\-_, T%@e-*@m^)YYmm.,___ ./m@F"L(` 'TY@e___@@a,_ $)` '@7*"@*Y%bTe_ _ -"@@/"-FLre-_a/m$b(*F"~@m/ , '\c .'- .amd)-(*@bm,_eem@_,:@YYb^^%^ '`*Y@@FC@r~'^^ ._/ed@bm_ _/@F*@m@mb,]m)Y$@@@/($@@@@@@dL/$ -c ."*m"\@YCC\m\_/^*YC~^^^` :c_d_//C`Ta)*-*`_bY"-e,_:m" m_.db(@^^Yd@FYbe(@%mem@c,._]@b@'7*'^'"Y;, T*@-a@"]*^ @@rd@bm@@ ._@rb@d@7Y/%)"""` ""d@` .ammmFvC@@mC* m**d@@`"**''*""*****'^ ^"- ./@d^%@@**"` d__@@m@F .,_ dFLmm/^YY^F :-_ @%^$@*" `'b 'L/LFb-@*^._, .a@L **" '^-- "*" '""**~`"-*"*-"*""-

As the R2.C(0) article illustrates, the cardioid is symmetrical, yet the north and south radicals of this island are of quite different sizes and the .1/2 child mu-atom is clearly off-center.


footnotes

1 : (In fact, provided that the software is accurate, no Mandelbrot set feature is truly distorted — they are all exctly the shape that they "should" be).




From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.     Mu-ency index


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