Paramecia, Mu-Ency at MROB

Paramecia

Robert P. Munafo, 2008 Jan 31.

My colloquial name for the lumpy objects found inside embedded Julia sets, particularly those found near the C(0) cusp of the influencing island mu-molecule.

We will start with this embedded Julia set, which is influenced by R2F(1/2B1)S:

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

~.. -, -( , .- '"., T ] [ . ./ __*- `., \ e,c____.. [ c_'_-.' .ac` `-_ -_)$""^/""*Ye:\_)/(@"$d\e,~ ..-' .,'-.@m- T` -@L[]-Ymbc `r-~"Y)_.-'` `-.dF"@--._", '' ] '^' [ '"FL%e, . `--.,_,'.FF`, _%Y)\. ,'_ .__Ym,`'*'$m,^ ^'_@;"Y"^^ ^`-.d@$)^'' ' '-vm'-FL_,---'' '"(b\/^ , -^^' ',:$e_ ^^^^:C.[ .ab(,- [ ._a___ T`@^-,- --'*/e'**/F- -) ,dC:d`a(/`L '`'^""F"' .__[,, -^Y(**`^ -.--:----:-------------,----]T@@---.----------.--c-:-m--,.,- _ .a,,-.. -'"F" ,e:a__-_ -__L\F^L%Ye. .L -aF%$^")mr- ____vm"/ '@@d"- [ -**-"; T:b`--- ._$F/L `' _`., :"~YF- _"bca)_, ._.-@$(-__ .."d-(@"---.,_ _,,--^^^mC[d^`'^-:e-^^` ~^^ '^^-am~ __`v@% '^ ` _."@L.m,_'*^` . ] _. %`'./F""` ` ` '^^"/m T, adme'-dmd- aL_dT$``-_, _,^ .Ymv.^-\_^%)[]'YmL,v`:r/F"/ '~_, _-~ /`""FF-"""^ -c `/^^^^', ' '`\/, __-` . ] '[ ' '\ `-_, .'" a` % '` ^-. Figure 1: -1.74876455 +0.00000001i @ +0.00002441, Nmax: 1000

The structure is almost exactly symmetrical on both the vertical and horizontal axes. Looking at one quadrant:

-1.74876109 +0.00000471i 0.00001010
-1.74876109 +0.00000471i @ 0.00001010

_e,a~ . v- ' : ] , / .^ . '^Fmm_.~ ' . ..\_rva/a-re__L .` , . . :_-^ '^--""/e-` r -,.)@F*Ta ^`^`/`'d\:-__L_~, .-` ../'^` .,.. ~FL-, [ '.dF"-^^`^`--F` '"^~Fm'~*YL,~ a_-' -\d*$(c]$( .; .$d"`:ee, ^_, ,:. ,.]m/"*b/_,- ._-'^ -/$_^^,/F- ] -"Y\/*":@^` 'db*"^'` `'^"/@/_, ._-^ .-'"*/]^^` . ^~*/a_)( ` -[ _.__'(-- __-' " ] ^^ .` ' '` \YdCaF^.,-^ T` '[ '''.-'`._L_- _, $` C .`_ . a,-^ .F%^"^rm,'. _._- / ^.d7(-__-` -'"@F` ~""Y/__--^ B , ' _ , __-^^(C^@d- ' ._.TbL. - '\.e_mv\, -___,-^^ .`^^-, , ._-"~-mT[--^ ^-/.,_ :m(*d_/\@`-----^'" /`.__ ._~` a_,(F-- .. `_F_'"*d$F- '^.$@C._.'*""____-.---^'' '^Fba"r**`-, ^"F~ .mre\ -` ' ' ` ' D ]c '^`d@_^ ]- ^..,,--rY,^^^` -- -_ . -d^,/ad@`- -- .'v.,'_ a v :d\)--_.___,_ _, '- ^^`^-a/T$7@$e/_, __,-ad`- ' ^^^^^ _- '"@"v'..a_^b\- :'^cm/" `` '^**^ .,-^^@~( '"CLd*- .,m-_$`'-.,_ __ ' \^@/-__' ._.-/.m/"`` ^`^ A :; ]` ~^*C(@./%/Ld*~` '--_ . ] . ` L^`/^'\' '-- '-v:)r__-(c--` ' ` ' . ^dY".Ld,"Fb` . , , _ --^^^^_Y^]@@["FL^^^^-^^^`^*^^'[^^^^^-^-^`^-^^^--^^-^-^^`^"^^^'^^'`^-^` ''^^ ' ^^` Figure 2. -1.74876109 +0.00000471i @ +0.00001010, Nmax: 1000

One can easily discern three types of objects in the embedded Julia set:

- The "shell", made of self-similar pieces that resemble the letter C with spiral curls at each end,

- The filaments, which are all very long and straight,

- The "lumps" that exist at every place where two filaments cross.

These "lumps" are the 'nucleus' and the paramecia. They all have a structure relatively similar to the entire Embedded Julia Set, including the same three components.

-1.748764596 +0.000000002i +0.000001450
-1.748764596 +0.000000002i @ +0.000001450

'\ , ' . '` , [ _ /` _, . , '\.e/__:.:,: . :- \.e,[__c. . . ., ._ '`-,_ ^..d/-(aC^\:""^F"e-. -c .,b_/"7"*`)^$cLv\___~ _.-'` '^-._ -_r*\['^:'_."`^'^^'Yd- : -b@"^^``'", `/^`]/*m.- ../^- '-_,Y@`;, ^.**^ -)vb,@~ .- ~F'm/d, ^"*,^ .'"@,..-^ .,.a\-/"Y" . '`*mbm"' ], `"bdm*/` c 'F"(-',,., '^-_-d("@@c`-, _." ' ' '` : ''-- ` '" . _-':FC"$mc_-^ --:@7( ` vmCdc- '; -:)Cm` ''`.)Cbe- ---.d*\-a)____,-'Y"-_ -- ._-"-^-.____(_-("__--- _\m*-.7^ `\_ . '` ._/~ "" -*m/_ . ,"C\m- ^~, ,'c__' __.`_ ..'^ -m/$" ^'^^\(^a /_d,- `.,:drF(bv^(mm;_,' -.___ ]e^)^^`^` ---Ym\/*b~@-- _ -me(FF.~.__,~ ]T)am , --@~d\_/mF--- '''"/--YT' '^$dF._:a@$bbdCb,_-\@C^` `FF--\"``` _a________________________@m_a@d@*"`'@@@__m@__________________________ ..,.e.. -(F^_'\$*@bmm@\F\,^Y- ,,a,. __'_aY^^L7Y-. -^^Ydac-^""d@")`-.cmF^^- '-F(d^"Fe_._, .$/_` "@L*%' _\-@%m(._.,T,$F'/ ^$"a@* '_\$- ^^^^v@-_, ` . _/' ^'^*^^'*""^` `\, ' ._-@c^^^^ --(b.^`. ..*` ' ', '",, . ,'^,d"- ____-^C;-mb--.,,.:,_,'` ]` '`v,.e,_.,--Y)-\%^-.___ ` ./(L,~ :d$b), -- _dd%b; `./~), _/Td) ame .-'""; ., "- . '""`-, _me.eb*^, '` ."YY"*,,\, : -./_e., -` .,a_L,- / ._L/~-F"-- ^` _,-\d"Y^ :_, '-@^m^@_ : _@^a/@`` -._ ""~b-^._ .,-` %Fb-\_-^Y*-\ '"^,$, :- .\\^"` ../-"F^^e/-@"` '\, _,-` `:^*ebam(m .*c^vcabF- :` --"dm_-%-*,-m/de//"\, "-, -,` /~ "''-'""((Y"Y*(`' .- `'"^""^)"FY`\` ~ ^. '~~ .` ' : ' ., ` [ - '\ ` ' '` ' Figure 3. The nucleus (a 4-fold embedded Julia Set). This is feature 'A' from figure 2. View: -1.748764596 +0.000000002i @ +0.000001450, Nmax: 2000

Notice that the nucleus has 4-fold rotational symmetry. This is an example of bifurcation.

-1.7487590765 +0.0000047386i +0.0000007155
-1.7487590765 +0.0000047386i @ +0.0000007155

-*, [ ,/` ` '-_ . '[ .~` `e -_\a_\_,/ :/` , ..-a@/`%%'*)c, /- --,_ -LYF*"~\-` :mrF`-_,m,.- ._ '`^-$c\c--*^, ^^/\)%^()b/_.--' __%]Le',_, '\a_' ^^ .e/bC- ' -Yd,]Fm`..-^^~__ ,:-'`-/,^^ . ' -""^"\-^` L@[._*""___...__-:c----^^ '^---mc__a,________a___--^^.a--m,- , .@~ee, - '` ` _d.@`T@(*- .'_,__($-__, ___-@F\ ' .."*;._\@" _-C^*-., . ./^ :b7^.,.______ .____,_$Ym- -_m(,-_,__:~@" `^ ' ,vr-^`%:@m*C@` :\)^^ --d@\c--"@b*C)b_^T$@------ , _ _ , '.C:, ' / '^'-_.,Yc`_ , ^'^^^^^^^^^:(Y^ ._` '^FF^` ,__.__a__________ab -vbC"` \dr, deae'd) ..,--'^^ ' '" __-'Tc\-^^^- .*(%(^.-^.,_ ,^C;ea \ ' .~:m@*mC_-- _''%*`--'@@.___.,-` '"^ /Y@L____, ___--'^ -/F@` ./^ "FF-__..-"^T, ._.--^` ''*L--a_ .e_,. . '"*md-* ^ . __--'^` -''^"F)_L.%Y/-cm$F __-^@c`-.,_ _--" .~ ""7*(^^] ("m_Lm**`- '`r-._, a- .` ' ] ' ] '-, ''` -` ( ~- '^ ' Figure 4. A paramecium, feature 'D' from figure 2. View: -1.7487590765 +0.0000047386i @ +0.0000007155

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index

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