Fractals/Iterations in the complex plane/island t

How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accesible from main cardioid ( M0) by a finite number of boundary crossing ?

How to describe island ? edit

Criteria for classifications ( measures):

Usually more then one measure can be used:

Islands by period edit

const roots = [
   [0, 0],
   [-1.98542425305421, 0,'Needle Far Left'],
   [-1.86078252220485, 0,'Needle Not So Far Left'],
   [-1.6254137251233, 0,'Needle Near'],
   [-1.25636793006818, -0.380320963472722, "Biggest Minibrot Lower Left"],
   [-1.25636793006818 , 0.380320963472722, "Biggest Minibrot Upper Left"],
   [-0.504340175446244 ,-0.562765761452982, "Bulb MainLeftLower"],
   [-0.504340175446244 ,0.562765761452982, "Bulb MainLeftUpper"],
   [-0.0442123577040706 ,-0.986580976280893,"Minibrot Lower Right"],
   [-0.0442123577040706 , 0.986580976280893,"Minibrot Upper Right"],
   [-0.198042099364254 ,-1.1002695372927,'#Deeper Minibrot Lower Left'],
   [-0.198042099364254 , 1.1002695372927,'#Deeper Minibrot Upper Left'],
   [0.379513588015924 ,-0.334932305597498,"Bulb MainRightLower"],
   [0.379513588015924 ,+ 0.334932305597498,"Bulb MainRightUpper"],
   [0.359259224758007 ,-0.642513737138542,"Minibrot MainRightLower Back"],
   [0.359259224758007 , 0.642513737138542,"Minibrot MainRightUpper Back"]
 ]

period 3 island edit

 
Wakes near the period 3 island in the Mandelbrot set. Boundary of the Mandelbrot set rendered with distance estimation (exterior and interior). Labelled with periods (blue), internal angles and rays (green) and external angles and rays (red).

find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7)

Plane description :[1]

-1.76733 +0.00002 i @ 0.05

One can check it using program Mandel by Wolf Jung :

The angle  3/7  or  p011 has  preperiod = 0  and  period = 3.
The conjugate angle is  4/7  or  p100 .
The kneading sequence is  AB*  and the internal address is  1-2-3 .
The corresponding parameter rays are landing at the root of a primitive component of period 3.


The largest mini on the antenna has:

  • internal adress 1 1/2 2 1/2 3[2]
  • external angles (3/7,4/7) which in binary is (.(011),.(100))

Period 5 islands edit

2/5 edit

  • The 2/5-wake of the main cardioid is bounded by the parameter rays with the angles 9/31 or p01001 and 10/31 or p01010 .

the center of the satellite component c = -0.504340175446244 +0.562765761452982 i period = 5


in the 1/3-sublimb of the period-2 component edit

the primitive component of period 5 in the 1/3-sublimb of the period-2 component

  • center c = -1.256367930068181 +0.380320963472722 i period = 5
  • The angle 11/31 or p01011 has preperiod = 0 and period = 5. The conjugate angle is 12/31 or p01100 .
  • The kneading sequence is ABAB* and the internal address is 1-2-5 .
  • internal address
 


on the main antenna edit

 
Wakes along the main antenna in the Mandelbrot set. Boundary of the Mandelbrot set rendered with distance estimation (exterior and interior). Labelled with periods (blue), internal addresses (green) and external angles and rays (red).

There are 3 period 5 componenets on the main antenna ( checked with program Mandel by Wolf Jung ) :

  • The angle 13/31 or p01101 has preperiod = 0 and period = 5. The conjugate angle is 18/31 or p10010 . The kneading sequence is ABAA* and the internal address is 1-2-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5.
  • The angle 14/31 or p01110 has preperiod = 0 and period = 5. The conjugate angle is 17/31 or p10001 . The kneading sequence is ABBA* and the internal address is 1-2-3-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5.
  • The angle 15/31 or p01111 has preperiod = 0 and period = 5. The conjugate angle is 16/31 or p10000 . The kneading sequence is ABBB* and the internal address is 1-2-3-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5. On the next root land 2 rays 16/33 i 17/33



1-3-4-5 edit

Angled internal address in the form used by Claude Heiland-Allen:[3]

1 1/2 2 1/2 3 1/2 4 1/2 5

or in the other form :

 

Where

  •   denotes Sharkovsky ordering which describes what is going on between period 1 and 3 on the real axis. Its first part is period doubling scenario from period 1 :   denotes  
  •   denotes period p component and infinite number of boundary crossing along 1/2 internal rays, for example   denotes  

So going from period 1 to period 5 on the main antenna means infinite number of boundary crossing ! It is to much so one has to start from main component of period 5 island.

External angles of this componnet can be computed by other algorithms.[4]

period 7 edit

period 9 island edit

 
Part of parameter plane with Minimandelbrot sets for periods 1, 3, 9, 27, 81, 243. Also external rays are seen.
  • the period 9 island in the antenna of the period 3 island

Check with Mandel:

The angle  228/511  or  p011100100 has  preperiod = 0  and  period = 9.
The conjugate angle is  283/511  or  p100011011 .
The kneading sequence is  ABBABAAB*  and the internal address is  1-2-3-6-9 .
The corresponding parameter rays are landing at the root of a primitive component of period 9.

period 18 edit

Period 18 island with angled internal address


  


whose:

  • upper external angle is .(010101010101100101) [5]
  • center ( nucleus) c = -0.814158841137593 +0.189802029306573 i

Info from progrm Mandel :

The angle  87397/262143  or  p010101010101100101 has  preperiod = 0  and  period = 18.
The conjugate angle is  87386/262143  or  p010101010101011010 .
The kneading sequence is  ABABABABABABABAAA*  and the internal address is  1-2-16-18 .
The corresponding parameter rays land at the root of a primitive component of period 18.

period 16 edit

  • +0.2925755 -0.0149977i @ +0.0005 [6]
  • c = 0.292503753234193 -0.014925068998344 i period = 16 (precise value of the period 16 center computed with Mandel by Wolf Jung)

period 25 edit

Choose

  

First compute external angles for r/s wake :

 
 

and root of the island ( using program Mandel ) :

The angle  13/31  or  p01101
has  preperiod = 0  and  period = 5.
The conjugate angle is  18/31  or  p10010 .
The kneading sequence is  ABAA*  and
the internal address is  1-2-4-5 .
The corresponding parameter rays are landing
at the root of a primitive component of period 5.

 
 

then in   replace :

  • digit 0 by block of length q from  
  • digit 1 by block of length q from  

Result is :

 
 
theta_minus = 0.(0110101101011010110110010)
theta_plus  = 0.(0110101101011011001001101)

One can check it using program Mandel by Wolf Jung :

The angle  14071218/33554431  or  p0110101101011010110110010
has  preperiod = 0  and  period = 25.
The conjugate angle is  14071373/33554431  or  p0110101101011011001001101 .
The kneading sequence is  ABAABABAABABAABABAABABAA*  and
the internal address is  1-2-4-5-25 .
The corresponding parameter rays are landing
at the root of a satellite component of period 25.
It is bifurcating from period 5.
Do you want to draw the rays and to shift c
to the corresponding center?

33 edit

  • c = 0.181502832839439 -0.582826014844503 i period = 33 center
 
Julia set for z*z+ 0.181502832839439 -0.582826014844503I


36 edit

  • period 36 island with center c = -0.763926983955582 +0.092287538419582 i located in the wake 1/34 of period 2 component
 
period 36 island with center c = -0.763926983955582 +0.092287538419582 i

period 44 edit

Plane parameters :[7]

-0.63413421522307309166332840960 + 0.68661141963581069380394003021 i @ 3.35e-24

and external rays :

.(01001111100100100100011101010110011001100011)
.(01001111100100100100011101010110011001100100)

One can check it with program Mandel by Wolf Jung :

The angle  5468105041507/17592186044415  or  p01001111100100100100011101010110011001100011
has  preperiod = 0  and  period = 44.
The conjugate angle is  5468105041508/17592186044415  or  p01001111100100100100011101010110011001100100 .
The kneading sequence is  AAAABBBBABAABAABAABAABBBABABABAAABAAABABAAB*  and
the internal address is  1-5-6-7-8-10-13-16-19-22-23-24-26-28-30-34-38-40-43-44 .
The corresponding parameter rays are landing
at the root of a primitive component of period 44.

period 49 edit

  • center c = -0.748427377115632 +0.067417674789180 i period = 49
  • distorted
  • in the wake of c = -0.747115035379558 +0.066741875885198 i period = 47

period 52 edit

Plane parameters :[8]

  -0.22817920780250860271129306628202459167994 +   1.11515676722969926888221122588497247465766 i @ 2.22e-41

and external rays :

.(0011111111101010101010101011111111101010101010101011)
.(0011111111101010101010101011111111101010101010101100)

One can check it with program Mandel by Wolf Jung :

The angle  1124433913621163/4503599627370495  or  p0011111111101010101010101011111111101010101010101011
has  preperiod = 0  and  period = 52.
The conjugate angle is  1124433913621164/4503599627370495  or  p0011111111101010101010101011111111101010101010101100 .
The kneading sequence is  AABBBBBBBBBABABABABABABABABBBBBBBBBABABABABABABABAB*  and
the internal address is  1-3-4-5-6-7-8-9-10-11-13-15-17-19-21-23-25-27-28-29-30-31-32-33-34-35-37-39-41-43-45-47-49-51-52 .
The corresponding parameter rays are landing
at the root of a primitive component of period 52.

render using MPFR ( double precision is not enough)

period 61 edit

  • center c = -0.749007413067268 +0.053603465229520 i period = 61
  • distorted
The 29/59-wake of the main cardioid is bounded by the parameter rays with the angles
192153584101141161/576460752303423487  or  p01010101010101010101010101010101010101010101010101010101001  and
192153584101141162/576460752303423487  or  p01010101010101010101010101010101010101010101010101010101010 .
Do you want to draw the rays and to shift c to the center of the satellite component?
c = -0.748168212862783  +0.053193574107985 i    period = 59

period 116 edit

 
Island with 116 period

It is inside 5/11 wake

size 1000 1000
view 53 -7.2398344555005190e-01 2.8671972540880530e-01 8.0481388661397700e-07
text 53 -7.2398348100841969e-01 2.8671974646855508e-01 116
ray_in 2000 .(01010101001101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001010101010)
ray_in 3000 .(01010101001101001010101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001)

Angled internal address :

 

Above approach above address seems true but not practical.

Visual analysis gives full path inside Mandelbrot set ( more precisely inside main cardioid and 5/11-limb) :

  • start with center of period 1 ( c=0)
  • go along interna ray 5/11 to root ( bond)
  • go to the period 11 center
  • go along escape route 1/2 (thru period doubling cascade , Myrberg-Feigenbaum point and chotic part ) to principal Misiurewicz point of 5/11 wake: M_{11,1} = c = -0.724112682973574 +0.286456567676711 i [/li]
  • turn into 3 branch
  • go "straight" along the branch until center of period 116


 


There are ininite number of hyperbolic componnets inside branch, chaotic part and period doubling cascade so ther is no need to list them.

period 134 edit

  • a period 134 island, which like the above example is within an embedded Julia set near R2F(1/2B1)S.
 
Period 134 island
size 2000 1000
view 54 -1.74920463345912691e+00 -2.8684660237361114e-04 2.158333333333333e-12
ray_in 2000 .(10010010010010010010010010010010010010010010001101101101101101101101101101101101101101101101101101101101101101101101101101101101101101)
text 63 -1.7492046334590113301e+00 -2.8684660234660531403e-04 134
text 62 -1.7492046334594190961e+00 -2.8684660260955536656e-04 268
ray_in 2000 .(10010010010010010010010010010010010010010001110010010010010010010010010010010010010010010010010010010010010010010010010010010010010010)

period 275 edit

  • center of main pseudocardioid c = -1.985467748182376 +0.000003464322064 i period = 275
  • distorted
  • in the wake of c = -1.985424253054205 +0.000000000000000 i period = 5

period 3104 edit

Description:[9]

  • Real number position: 0,25000102515011806826817597033225524583655
  • Imaginary number position: 0,0000000016387052819136931666219461
  • Zoom: 6,871947673*(10^10)
  • bits = 38 , use mpfr type
  • wake 1/3103


period 418864 edit

  • location
    • x = -1.7697970032213981159127251304389983279942336949906874604031232136913947627989973432768538410642493843143927357668033073370

49665460755808389013248912202462392189032875057823197659362732380873696894875347373595161248407157606303961329755736109322011630746 286872455033371782761711152485963814840985495119858112247809563217001440012335481392958891277404641915770292234769570579423526083615 869119473397655144269230554048451408287129839729482745812536821304009849356175786421926754317166054095017677737478909629824101459411 484678651540446085496579356154087444768864107144068903495747107840142587494964830790373105466387017637804940200093226948331098336564 024101191304782846009251093956024054859850114380942506295799272703040122491695848188554900910110348500660088142142935996917999415780 4134090723185056583183709863897144993893599460179220543896055493072398638187712235171179588280308584482354373699407785045486558094140 86286410278094103602829312453365743012069479897322687170061953674357190866700112517607208995688167519085493168568587128984804788006359 59347100781293499250828473881321840106718612921692041981341359850708691437845116651465935653020129685931665064112991181637664436069589 91219786468762583523133485646097250073032150797026331458996316635041742470636626183572017944917556643345811610632517182664699299968048 382369034487284966906681433196008740895151252917642683455349811749762919778556988057469252293997296152251096052453458307226555176061477 44507997235610446150765888279849316729036292301646101698262415387848655551453813389172582295590171380746790465457505657035692901532708877 91912366870238890702486377674493961627842425415072641536223340784982438486048756109238181153075391103742999718461989487988255182749425809 658290851105686957800331487046619356847741786931568734133797812990312933679468689355633257241932332586807751783991361005487951858068862626 8278755133144450865524035720261352693414152655639148956133170959450801291112496173999474719515703746019410263225865758896785340001484117155 48247602090863178460885536238487047026969052782268862081294620522011538188275677933094565746782844741895263212598E+00

  • y = 4.503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919752 32128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529943 75685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998508 35523341852444210373993799979274072458522457971439601401283190488219977380751679864657632594486990141780409069050808535 33679083210095437351400022620788443700681865056074859184889623921225508741770547501475133877301147491846294015630493195 94413147950329230917914373568299313895801070552430312839787385413077643433921434686758800882730741386718858427487804801 73527152642383437688144097648231731279522222357988455250353865370120443546331395472996006556618614941953429666058354649 10451202485512530423175907298924572677884684325102852936015719933302605823099958630951988450410491580664701963842251461 35190645341340161891884063141465638742680614101092435645795624718302058131414609501281021540435472453888874524109018147 02121578711328524425442226752168664749086242203613749999027884515745350840633982861734634138141253642303937961493945458 38176191438823739844915158113285022936463789829746280707055929391192625872076997627990447836359937976951672647199177818 72517689037585583899463944250055017306480718807197254236743510423432718914191161718864625412816080818679138546319519759 89748541205329675986737013154577653006827691952880225127567357459621316524513472420563020300861878311519895655738526548 29737784116356975937395880502857287215780402078167418834602295096014173047038182390355477059048628119343002217338189674 84428900612407421285966391654470156922336601567981570299684787648714514350236588685564191491795576963451396365624203611 89623693814216660262167258794137460777065623334381376669587093792227710384619914833779522355034279775231366236846879929 65077410226071259699613708732240144706025226046260403350230398624904436384826525344982234790191805054228954439652523570

302168603714064304338213704267058855190821114715226120756650420403035424042014899248446596447E-03

  • Zoom depth = 2.0E-2105[10]

Old Wood Dish edit

The atom periods of the center of Old Wood Dish are:[11]

1, 2, 34, 70, 142, 286, 574, 862, 1438, 2878, 5758


The angled internal address of Old Wood Dish starts:

 


and the pattern can be extended indefinitely by

 


milionaire edit

Mandelbrot set deep zooms with central period just over a million by Claude Heiland-Allen[12]

Re: -1.941564847210618381782745533146630687852577330811479185328717110626315465313888984406570091271861776378826092790143826203994152325590923147877133022224438450505595392332442169268786604880239682848013406897983579432062702292199644932564206420775763033730026410960393034024379448558313295127784426381592278080925192198166506414945985414913745366605657655610477078243223433128650561902149109766955341541448889252090006440504495875324697439205551007663522598546938799920069758806395662880415099380114727803945598174113344976815709788824810872243858870025811047073266393172169520770249454031205263249410283959479169565468406337528155043698920579273678870784676542455819793013621475835287373620100519033551698084870044144096525907756214603649878765768441725598786715107648812695912688272348358202539017931213566557756771117546689787437119363273090858225103068635520748447418748363430805526175228812153552404870337873296242637654897774106552491179507233830264867055720154027738114532472834129907542036414627198070205428671288600626717940810743065719692081657257083298414914079629307719877169697203460540630000679002070296933515367765096894637520233387261677527116574909499106836689943282145414983901962836972429294354792030773990246030933771660915959463839410311609001092258001208772024174367234004812961533343197123692106177497640839672883719696626288402363726247440183295029163880397919214040826126900073973638637566578208702814548391703168474392383593212772787731464838088077224699638406743366046222299920539039887163949934166963836811009496709145476491269052150307331295997696598642224921758611196703647774310100824454754453378692238473281876068395860361747421509077890568367923248938440919450666764746563667104471327430234809386514744994479578918258139825168762910680781831023955275492781814592422214938019500942282403152718152583429320091988757597326162896044423940280436579379250758238150181677659582319810124929896915790686630777656868121991116553823614967588334809071895616642606935606074858069732264297184172026997781642831813555710815432177033080251973441185057582367440091110843860622138414561875643370900646057697961216473136674094515585359492045093031169458552950861210067868990069649613018250078461502572888267902093886429413235954091998533512387698508224032840007461108905888878936712481329924920758423963101423671524810383755418536530931147569126085854905997546750284390836145218644767026524860219389612672917184135093515351440137017875343267106105093234535923345453588257553484550210988965614138849027483452997361327494579395325872160214974105239331592524594369150188797359380002999428260744648273685801485062771062283341261252665204132897101670705129030433332038916860978021784527372660253641186001797960631162225700735543442809212529421959800859631684925688086957903450031876903213378895363984669656174378394848828662491287275427562355094329881843892371905891363815917013435735261628338481776645199230206051992093463701679670012828167982422504379893524492879191497084894922576575660921357705998236585956378644035589226542323286665436731208546815423007982122742733894034678760552296794329535587849467738317885329863463242761164667692358223018142882123247539116527159532753920850365440722610461795764889919310185260171054544985137369235154554304940059632171722414684286138383501773415057939488043213172235441555106648355044355391233758480556732598344113015309927936212966784974691525847156004581824315522927394900768552759061458508079647172411453020446899544906575567336056418575049261413172131851152494947470184383652557573651992514409529812895846029166547850176356488097864860289314225499677990887360532931687650438535765399000166232522594559391892684925740039704185368242880536277639758993317174680558415652951942514607673069357919857260486354
Im: 2.348911956401652748611382363072520535146733491918842206389055226478822558334356028474458306453568269131543696797365302213154106976514279082244760267169482925324526783567612979671556935057632055950984996909780142673870494806718441563468971222881465156907737846885411815804623686136775248121351602452938196791632141551203544924477065181043689768585002934501366247348894440025575034790977798556673982209118819387316634056673728437905475480824207093789985152660660796470895526541440245169605192293780704054201356420547490025338952432606049964709328857846861417513600552731799643681595245395686988951646887256885954913669780792964184025852007185490455600079530313065015412120431544281411000883436175700100755643502134003127400266634841554627987192002123927402658620084127543742083778598017547508760673625017745837047226871893523527022399890081945911197605364730161342705278848485124574682491279788530067609533079049478398986047847983972001764819156565755354326002905542507480820059290426742712804028817087523369562937215212612904336088048132302802862775437161150812264724605689069081436863515240452173801300714588231927754167001145055783695030502517679091867645972152131281950436820800642430650719709799248997373662802383522383728708100167105045934741758120563240619508429409263325664232101394865918891717788286392682273910844038755619719694482789478765835921982258456504697071599084602547626988072659073902294817850999295146301151819189581096894966914306782148725101047973857971183966368556392489984001268762215576350231765055323286514244799060484573201272893610318786886204290069662111659708122739712189774743739800965469849720836828331398655933538341163498137309170029696829049759241035466935137380840598501596696433658234571517949705876880775966141832184491036484520614953456138004895628751874368118806676048052933590152010351393305876747506539949321504627614276077826614282883826502801546997144217149427591454981918422414700754950892289586365073462657884225261119072856209897972217681362290126057381673109065004859492884983392588329325943196101413715919534526662966518996029715522705281433766162245585921066836784151039282692733266581776821803392615463278063762569154398096421583781961425272467224400238786777787057691570645817689820989087807507202607204424924302873613904218059784818247676395238645472434259554102514159552240730691322340413842241268213834149709528706514553724664567801903402240125384283406177463810865382078416066041162205457202040097571654039068900436565607579688861751386273437147633837175759423123782121059992340843638976542491619616721240707699182762901902457881956252753755542525046656795781387399414211410058657062996651489499230059912393101353702379101252993636688212173092017002441988691600905387288953613012271760014041471507305899461467237026040155865673294436686288489064573310042362571214740808656550235500893082338024464752705630639598923599812755067568406644418410908656796336604324227361637640201957166044187263630622072489236137199511921096807717330636805940632361331614384427249172810219683522407075518367730649165243792872682230339286009707120948066977912801945301971489666691152738504000234264492829861082007755878186353657391575969037890793507812419299941828403592000286654117164037545076892235511998963155488596897447316433466284271702397024720026567764282966538060228938320243333174656442494289469553689264266824656421054000462281567678086050788756644012025683226050038399480067529754327429973435343599699135593109676352382173193844221842544770748622726551353163716194488271418173093929303360824571352118582549565180371595453272352097817496528144543662534792783260636316303412950338514389922496456633319004613765453940022755497086172440543892771156330673123788549821094421341129110235193999814306839803150313852972941e-4
Zoom: 4e2804
Iterations: 10100100
Period: 1137764


shallow zoom edit

shallow locations with high iteration counts by Claude Heiland-Allen[13]

  • Re: 3.56992006738525396399695724115347205e-01
  • Im: 6.91411005282446050826514373514151521e-02
  • Zoom: 1e19 so it needs arbitrary precision numbers ( like MPFR) with 72 binary bits when using simple method ( without perturbation)
  • Period: 1000000
  • Iterations: 1100100100
  • it is a dense set
  • The angled internal address of the mini in the middle is (morally, if not factually) something like 1 1/9 10 2/9 100 4/9 1000 5/9 10000 7/9 100000 8/9 1000000, so the external angles could be computed relatively easily I suppose.

See also edit

References edit

  1. R2F(1/2B1)S by Robert P. Munafo, 2008 Feb 28.
  2. fractalforums.org : help with ideas for fractal art
  3. Patterns of periods in the Mandelbrot set by Claude Heiland-Allen
  4. Parameter rays of root points of period p components
  5. atom domains and newton basins by Claude Heiland-Allen
  6. R2.C(0) by Robert P. Munafo, 2012 Apr 16.
  7. Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen
  8. Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen
  9. fractalforums.com: how-distorted-can-a-minibrot-be ?
  10. fractalforums.com: *continued*-superfractalthing-arbitrary-precision-mandelbrot-set-rendering-in-java
  11. old wood dish by Claude Heiland-Allen
  12. Millionaires by Claude Heiland-Allen
  13. fractalforums.org : challenging-locations-for-current-acceleration-methods