# 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 described 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 than one measure can be used:
* biggest island of the wake
* biggest island with period

# Islands by period
## 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 :
One can check it using program Mandel by Wolf Jung :
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))

## 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 .
* 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:
or in the other form :
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:

## period 18

Period 18 island with angled internal address
whose:
* upper external angle is .(010101010101100101)
* center ( nucleus) c = -0.814158841137593 +0.189802029306573 i
Info from progrm Mandel :

## 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
First compute external angles for r/s wake :
and root of the island ( using program Mandel ) :
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 :
One can check it using program Mandel by Wolf Jung :

## 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 :
and external rays :
One can check it with program Mandel by Wolf Jung :

## 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 :
and external rays :
One can check it with program Mandel by Wolf Jung :
render using MPFR ( double precision is not enough)

## period 61

* center c = -0.749007413067268 +0.053603465229520 i period = 61
* distorted

## period 116

It is inside 5/11 wake
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.

## 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

* 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

## Old Wood Dish

The atom periods of the center of Old Wood Dish are:
The angled internal address of Old Wood Dish starts:
$11/2216/17331/2341/3691/2701/31411/21421/32851/2286...$
and the pattern can be extended indefinitely by

## milionaire

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

### 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

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