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Reverse Bifurcation

Robert P. Munafo, 2023 Jul 1.

"Reverse bifurcation" is a property of certain high-order multi-level embedded Julia sets, involving tuning by sparse Fatou dusts alternating with successively longer zoomes in through the nested paramecia until getting close to the central island, then repeating.

This exploration method, a type of Leavitt navigation, is commonly called Julia morphing (or "building a tree" by Claude Heiland-Allen).

Bifurcation vs. Reverse Bifurcation

Normally, bifurcation in the rotational symmetry around islands occurs with a pattern that doubles its degree of symmetry as you zoom in closer to the island. For example, look at the shepherd crook article and note the last two pictures, showing two-way rotational symmetry and four-way rotational symmetry (with the 4-way "inside" the 2-way).

Reverse bifurcation refers to features in which nested embedded Julia sets appear to bifurcate in the reverse order — with doubling as you zoom out.

The images are located by repeated application of Julia morphing to locate a multi-level embedded Julia paramecium with a shape that includes two squaring preimages of the larger paramecium in which it is embedded; which in turn has two squaring preimages of an earlier and even larger paramecium.

To the best of my knowledge, reverse bifurcation in embedded Julia sets was discovered by Jonathan Leavitt.

A fine example can be seen here:

 -1.74928893611435556407228 + 0 i @ 6.6e-20 "., ., \ , [ . .[ : [ , .` . ..- ^-,. ~, -.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)

This image is based on an image that Leavitt called "Polaftis".

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.

It is an esoteric learned skill, which I call Leavitt navigation.

revisions: 20020418 oldest on record; 20101205 Add link to exponential map; 20230620 "Leavitt navigation"; 20230701 complete rewrite of the text above the image

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jul 01. s.27