Reverse Bifurcation
Robert P. Munafo, 2002 Apr 18.
Normally, bifurcation in rotational symmetry around islands occurs in a pattern that doubles as you get in closer to the island. However, with special effort, one can also find features in which nested embedded Julia sets appear to bifurcate in the reverse order — with doubling that occurs as you go out. To the best of my knowledge, reverse bifurcation in embedded Julia sets was discovered by Jonathan Leavitt.
A fine example can be seen here:
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"., ., \ , [ . .[ : [ , .` . ..-
^-,. ~, -.c/-.e:(mLe_(e-, [ ,,__dvL/v:e.-. . ./` ._,-
-^"., '-:rb*bd@m ~a/""""\b,[:b***--ma^-d@dFY)(.~, . a-^`
'`-_ '-.b"*\/ `^~._/` -:F@(c[dmCb" ~,a.,^ :_*"\L-' .,-^
-. ^\F^"^``^'^*@` -^*v" ['*""` -*@`^'``^^`FL- .-~
`-,(@amb_ ', -[ ./ .a)m/@-.-`
._, `-aL_med/7"*Y- -:- 'F -:- -T(F";daeea;.- _.-
'`-.\"Y/^/dbTe`., 'c , -[ , c .-`d\rc'"$*e_--^
-_.d*(_ '^7"' '-, -_.,`.:,.`. -[ , \ -a/_a_, _/` ''"^~ :_*YL.,
_,_ -@(_ ^_ ^.m"F*)"Y*)v,. [ .:a-"C^*/"FYL-` .c`.%Yb.___
^\d@^-ade `:dF^edbc-'^^`^b, [/aF%^^^-vdde/^Fe%` \me,"TC_^`
'-eF,^ _^"^^^-----__dbc"YF":, .dT@]F ['@:@Te :'Yb*vm@__,.---^`^*"L ^^(L--
---'b/Y-` .-@`_~ ..'-,'*/^` [ '*r"`,-, '\,TF. ^("aF---
__$L/c _e, .,-"@%,:mdvc "-_ . [, ..-`dd(/c ]]F`-_ :,e, r_*L_
.'@L]C%@(` -')L_$YLF- '.mYC\m' 'b$"La)"-- .m(%).$F. '
_____`"~"-"L_._______,___`'^^^'____"Ymbd*^____'^^^L`___,_____,___^~*-"``____
,a/v-ae ' ` . ,.___. ad*FYm_ ..___e,_ ` ` ,_/m-a,,
__'.@F'C%@(- , -.)F^$d^b` .'*dC/*., :F$r^")e-` . '*(%)'$b'_.
dF~` `` '`^e@% '*Y"`_"^ [ '-,"Y(\` ]]bc^^ '`` ."^m-
^^^.F~d-,, ,'-@,^\. ./-`.m\_e [..m"e,`-, ,:~`/b' , __(r"L^^^
--"L`_ La__----^^^^^F`$dba: 'Y/@]b [:@'@/" ::dFm%*@^^`'^---,_ma^ __(F--
.___/^@_^"Y" ,'YL_"YF`-.__,_F` ['"Lm___-^YY"\_b"-, ~*"`T/C^_,_,
^` -@(^__^ _-'*eLm)edm)"`' [ ':"-vC_m\ebdF-, '`,'%dF ^
^^'Ym(^, /_)e. ..-` - '`,'' ', '[ / -'',' ` ~,, ..eL/ '^mdF'`
_a,-'/rd\_/YF/",'` :` '[ ` .'-,Y/"`.a$m" --__
' ,-'^^*"^~)vmd- ''` 'L ''; -/(bv;^"""";'- '`
_,-.(@"*F^ ,:` ,. '[ . , \,, '")*\@-'-,
_-' -~bLe_,,_.-m@c -_m"e [.mve, dm@c-.,,__cLF- '-\,
.,-^ .-'Fem/T e_/' ~, ^'b@(`[Y*CF" _/`:'__,:^me/F-. '`-_
_a' .-:"FmFY@*_/"\Tvre/F ['Fmmm-%*"_-Y@Ybd)('\ '-_c
__-`' ./` - `:-'"'(*F"^(`- .[ `^^Y"F\"'" - ' ~, `-_
"' " ` ` ' ` ' ` ` ' ' '-
-1.74928893611435556407228
+0.00000000000000000000000 i
@ +0.00000000000000000006600, Nmax: 20000
mini-polaftis (2-4-8 reverse bifurcation) |
History
In April 2002, in the course of locating this and other similar images I discovered how useful the exponential map coordinate transform can be for locating the coordinates of a mystery image.
revisions: 20020418 oldest on record; 20101205 Add link to exponential map
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2012. Mu-ency index
Robert Munafo's home pages on HostMDS © 1996-2012 Robert P. Munafo. about contact
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