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):
 * period of main pseudocardioid
 * localization
 * internal adress
 * wake
 * center of main pseudocardioid ( nucleus)
 * viewport: rectangle part (window) of parameter plane used to show the island. It is usually descibed by radius and center
 * angles of external rays that land on cusp of the pseudocardiod
 * orientation
 * size
 * size of main pseudocardioid
 * shape
 * distortion

Usually more then one measure can be used:
 * biggest island of the wake
 * biggest island with period

=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 : -1.76733 +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
 * external angles (3/7,4/7) which in binary is (.(011),.(100))

2/5
the center of the satellite component c = -0.504340175446244 +0.562765761452982 i    period = 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.

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.
 * internal address


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

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:

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

or in the other form : $$1 \xrightarrow{Sharkovsky} 3 \xrightarrow{1/2} \cdots \xrightarrow{} 4 \xrightarrow{1/2}  \cdots \xrightarrow{} 5$$

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

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.

period 7

 * 2013-09-30 islands in hairs by Claude

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 $$1 \xrightarrow{1/2} 2 \xrightarrow{1/8} 16 \xrightarrow{1/2}  ... \xrightarrow{} 18$$ whose:
 * upper external angle is .(010101010101100101)
 * 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
 * c = 0.292503753234193 -0.014925068998344 i    period = 16 (precise value of the period 16 center computed with Mandel by Wolf Jung)

period 25
Choose

$$1 \xrightarrow{Sharkovsky} 3 \xrightarrow{1/2} \cdots \xrightarrow{} 4 \xrightarrow{1/2}  \cdots \xrightarrow{} 5 \xrightarrow{1/5} 25$$ First compute external angles for r/s wake :

$$\theta_-(r/s) =\theta_-(1/5) = 0.(00001) $$ $$\theta_+(r/s) =\theta_+(1/5) = 0.(00010) $$

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.

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

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


 * digit 0 by block of length q from $$\theta_-(island) $$
 * digit 1 by block of length q from $$\theta_+(island)$$

Result is :

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

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 :

-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 :

-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) Angled internal address :

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

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) :
 * 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

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

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:
 * 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
49665460755808389013248912202462392189032875057823197659362732380873696894875347373595161248407157606303961329755736109322011630746 286872455033371782761711152485963814840985495119858112247809563217001440012335481392958891277404641915770292234769570579423526083615 869119473397655144269230554048451408287129839729482745812536821304009849356175786421926754317166054095017677737478909629824101459411 484678651540446085496579356154087444768864107144068903495747107840142587494964830790373105466387017637804940200093226948331098336564 024101191304782846009251093956024054859850114380942506295799272703040122491695848188554900910110348500660088142142935996917999415780 4134090723185056583183709863897144993893599460179220543896055493072398638187712235171179588280308584482354373699407785045486558094140 86286410278094103602829312453365743012069479897322687170061953674357190866700112517607208995688167519085493168568587128984804788006359 59347100781293499250828473881321840106718612921692041981341359850708691437845116651465935653020129685931665064112991181637664436069589 91219786468762583523133485646097250073032150797026331458996316635041742470636626183572017944917556643345811610632517182664699299968048 382369034487284966906681433196008740895151252917642683455349811749762919778556988057469252293997296152251096052453458307226555176061477 44507997235610446150765888279849316729036292301646101698262415387848655551453813389172582295590171380746790465457505657035692901532708877 91912366870238890702486377674493961627842425415072641536223340784982438486048756109238181153075391103742999718461989487988255182749425809 658290851105686957800331487046619356847741786931568734133797812990312933679468689355633257241932332586807751783991361005487951858068862626 8278755133144450865524035720261352693414152655639148956133170959450801291112496173999474719515703746019410263225865758896785340001484117155 48247602090863178460885536238487047026969052782268862081294620522011538188275677933094565746782844741895263212598E+00 302168603714064304338213704267058855190821114715226120756650420403035424042014899248446596447E-03
 * location
 * x = -1.7697970032213981159127251304389983279942336949906874604031232136913947627989973432768538410642493843143927357668033073370
 * y = 4.503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919752 32128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529943 75685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998508 35523341852444210373993799979274072458522457971439601401283190488219977380751679864657632594486990141780409069050808535 33679083210095437351400022620788443700681865056074859184889623921225508741770547501475133877301147491846294015630493195 94413147950329230917914373568299313895801070552430312839787385413077643433921434686758800882730741386718858427487804801 73527152642383437688144097648231731279522222357988455250353865370120443546331395472996006556618614941953429666058354649 10451202485512530423175907298924572677884684325102852936015719933302605823099958630951988450410491580664701963842251461 35190645341340161891884063141465638742680614101092435645795624718302058131414609501281021540435472453888874524109018147 02121578711328524425442226752168664749086242203613749999027884515745350840633982861734634138141253642303937961493945458 38176191438823739844915158113285022936463789829746280707055929391192625872076997627990447836359937976951672647199177818 72517689037585583899463944250055017306480718807197254236743510423432718914191161718864625412816080818679138546319519759 89748541205329675986737013154577653006827691952880225127567357459621316524513472420563020300861878311519895655738526548 29737784116356975937395880502857287215780402078167418834602295096014173047038182390355477059048628119343002217338189674 84428900612407421285966391654470156922336601567981570299684787648714514350236588685564191491795576963451396365624203611 89623693814216660262167258794137460777065623334381376669587093792227710384619914833779522355034279775231366236846879929 65077410226071259699613708732240144706025226046260403350230398624904436384826525344982234790191805054228954439652523570
 * Zoom depth = 2.0E-2105

Old Wood Dish
The atom periods of the center of Old Wood Dish are:

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

The angled internal address of Old Wood Dish starts:

$$ 1 1/2 2 16/17 33 1/2 34 1/3 69 1/2 70 1/3 141 1/2 142 1/3 285 1/2 286 ...$$

and the pattern can be extended indefinitely by

$$... 1/3 (p-1) 1/2 p 1/3 (2p+1) 1/2 (2p+2) ...$$

milionaire
Mandelbrot set deep zooms with central period just over a million by Claude Heiland-Allen 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
 * 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=
 * How to compute external angles of principal Misiurewicz poin of wake p/q ?
 * fractalforums.com : the-mandelbrot-polynomial-roots-challenge
 * fractalforums.org: the-mandelbrot-set-root-challenge-10-years-later
 * distortion of the islands ( mini Mandelbrot sets )

=References=