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 accessible from main cardioid ( M0) by a finite number of boundary crossing ?

How to describe island ?

Criteria for classifications ( measures):

Usually more then one measure can be used:

Islands by period

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

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

2/5

• 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

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 .
${\displaystyle 1\quad \xrightarrow {1/2} \ 2\quad \xrightarrow {1/3} \ 6\quad ...\ 5}$

on the main antenna

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

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 :

${\displaystyle 1{\xrightarrow {Sharkovsky}}3{\xrightarrow {1/2}}\cdots {\xrightarrow {}}4{\xrightarrow {1/2}}\cdots {\xrightarrow {}}5}$


Where

• ${\displaystyle 1{\xrightarrow {Sharkovsky}}3}$ 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 : ${\displaystyle 1{\xrightarrow {1/2}}\cdots }$ denotes ${\displaystyle 1{\xrightarrow {1/2}}2{\xrightarrow {1/2}}4{\xrightarrow {1/2}}8{\xrightarrow {1/2}}16{\xrightarrow {1/2}}32{\xrightarrow {1/2}}1*2^{n}....}$
• ${\displaystyle p{\xrightarrow {1/2}}\cdots }$ denotes period p component and infinite number of boundary crossing along 1/2 internal rays, for example ${\displaystyle 3{\xrightarrow {1/2}}\cdots }$ denotes ${\displaystyle 3{\xrightarrow {1/2}}6{\xrightarrow {1/2}}12{\xrightarrow {1/2}}24{\xrightarrow {1/2}}48{\xrightarrow {1/2}}3*2^{n}....}$

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 9 island

• 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

Period 18 island with angled internal address

 ${\displaystyle 1\xrightarrow {1/2} 2\xrightarrow {1/8} 16\xrightarrow {1/2} ...\xrightarrow {} 18}$



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

• +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

Choose

 ${\displaystyle 1{\xrightarrow {Sharkovsky}}3{\xrightarrow {1/2}}\cdots {\xrightarrow {}}4{\xrightarrow {1/2}}\cdots {\xrightarrow {}}5{\xrightarrow {1/5}}25}$


First compute external angles for r/s wake :

${\displaystyle \theta _{-}(r/s)=\theta _{-}(1/5)=0.(00001)}$
${\displaystyle \theta _{+}(r/s)=\theta _{+}(1/5)=0.(00010)}$


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.


${\displaystyle \theta _{-}(island)=0.({\color {Blue}01101})}$
${\displaystyle \theta _{+}(island)=0.({\color {Red}10010})}$


then in ${\displaystyle \theta (r/s)}$ replace :

• digit 0 by block of length q from ${\displaystyle \theta _{-}(island)}$
• digit 1 by block of length q from ${\displaystyle \theta _{+}(island)}$

Result is :

${\displaystyle \theta _{-}(island,r/s)=\theta _{-}(5,1/5)=0.({\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Red}10010})}$
${\displaystyle \theta _{+}(island,r/s)=\theta _{+}(5,1/5)=0.({\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Red}10010}\ {\color {Blue}01101})}$

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

• c = 0.181502832839439 -0.582826014844503 i period = 33 center

36

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

period 44

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

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

period 52

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

• 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

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)


${\displaystyle 1\xrightarrow {5/11} 11\xrightarrow {1/2} 22\xrightarrow {1/2} 33\xrightarrow {1/2} 44\xrightarrow {1/2} 55\xrightarrow {1/2} 66\xrightarrow {1/2} 77\xrightarrow {1/2} 88\xrightarrow {1/2} 99\xrightarrow {1/2} 110\xrightarrow {1/2} 116}$

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) :

• 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

${\displaystyle 1\xrightarrow {5/11} 11\xrightarrow {1/2} M_{11,1}\to ThirdBranch\to 116}$

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

• a period 134 island, which like the above example is within an embedded Julia set near R2F(1/2B1)S.
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

• 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

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

• 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

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:

${\displaystyle 11/2216/17331/2341/3691/2701/31411/21421/32851/2286...}$

and the pattern can be extended indefinitely by

${\displaystyle ...1/3(p-1)1/2p1/3(2p+1)1/2(2p+2)...}$


milionaire

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

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.