# Fractals/Iterations in the complex plane/p misiurewicz

How to compute external angles of principal Misiurewicz point[1] of wake p/q using Devaney's algorithm ?

# names

• a principal Misiurewicz points[2] of the wake ( or the limb or the shrub )
• the main node of the shrub [3]
• the hub = center part of shrub ( Pastor notation), the point where spokes join
• a junction point of q spokes which is attached directly to the p/q bulb ( Devaney notation )[4]
• "the first dominating α-Misiurewicz point in M p/q , i.e., the one of lowest pre-period" [5]
• Eye of elephant resting on internal angle 1/4 of main cardioid ( Curtis McMullen)

# notes

Principal misiurewicz point of p/q-wake is ${\displaystyle c=M_{q,1}}$

• it has q arms ( spokes, branches) numbered from 0 to q-1 in a clockwise direction
• it is a landing point for q external angles
• critical point has preperiod q and period p = 1 under complex quadratic map for ${\displaystyle c=M_{q,1}}$

External angles of q rays landing on ${\displaystyle c=M_{q,p}}$

• in the binary expansion length of preperiodic and periodic part is q
• period and preperiod of angle under doubling map is q

Important differences:

• Romero-Pastor notation uses q/p not p/q
• Preperiod: the usual convention is to use the preperiod of the critical value ${\displaystyle z_{cv}=c}$, not preperiod of critical point ${\displaystyle z_{cr}=0}$. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane

# introduction

How to work with the shift map ?

If length of string s is q then

  ${\displaystyle \sigma ^{q}s=\sigma ^{0}s=s}$


## shifting q digits in blocks of b digits

Note that

  ${\displaystyle b


### q=5 b=1

 ${\displaystyle \sigma ^{0}(s)=\sigma ^{0}(00001)={\color {Red}00001}}$
${\displaystyle \sigma ^{1}(s)=\sigma ^{1}(00001)=00010}$
${\displaystyle \sigma ^{2}(s)=\sigma ^{2}(00001)=00100}$
${\displaystyle \sigma ^{3}(s)=\sigma ^{3}(00001)=01000}$
${\displaystyle \sigma ^{4}(s)=\sigma ^{4}(00001)=10000}$
${\displaystyle \sigma ^{5}(s)=\sigma ^{5}(00001)={\color {Red}00001}}$


### q=5 b=2

 ${\displaystyle \sigma ^{0}(s)=\sigma ^{0}(00001)={\color {Red}00001}}$
${\displaystyle \sigma ^{2}(s)=\sigma ^{2}(00001)=00100}$
${\displaystyle \sigma ^{4}(s)=\sigma ^{4}(00001)=10000}$


### q=5 b=3

 ${\displaystyle \sigma ^{0}(s)=\sigma ^{0}(00001)={\color {Red}00001}}$
${\displaystyle \sigma ^{3}(s)=\sigma ^{3}(00001)=01000}$


### q=5 b=4

 ${\displaystyle \sigma ^{0}(s)=\sigma ^{0}(00001)={\color {Red}00001}}$
${\displaystyle \sigma ^{4}(s)=\sigma ^{4}(00001)=10000}$


# Algorithm

Algorithm is based on the Theorem 5.3 in: Geometry of the Antennas in the Mandelbrot Set by R L Devaney and M Moreno-Rocha, April 11, 2000[6]

External Angles of Hub ( see section 3.9 of the Book by Claude) or spoke [7]

The ${\displaystyle p/q}$ bulb ( = hyperbolic component) has 2 external angles landing on it's root point (bond) :

${\displaystyle \theta _{\color {blue}-}(p/q)=0.({\color {blue}s_{-}})}$
${\displaystyle \theta _{\color {red}+}(p/q)=0.({\color {red}s_{+}})}$


such that :

 ${\displaystyle \theta _{\color {blue}-}<\theta _{\color {red}+}}$


These angles have :

• repeating binary expansion denoted by round brackets or overline
• length of repeating ( periodic ) part is ${\displaystyle q}$

Other names of these angles are angles of the wake.

The junction point of its hub ( principal Misiurewicz point) ${\displaystyle M}$ has external angles in increasing order

 ${\displaystyle 0.s_{-}(s_{+})}$
${\displaystyle 0.s_{-}(\sigma ^{b}s_{+})}$
${\displaystyle \vdots }$
${\displaystyle .s_{-}(\sigma ^{(q-p-1)b}s_{+})}$
${\displaystyle .s_{+}(\sigma ^{(q-p)b}s_{+})}$
${\displaystyle \vdots }$
${\displaystyle 0.s_{+}(s_{-})}$


where

• s is a finite string of q binary digits = s consist of q binary digits = length(s)= q
• ${\displaystyle \sigma }$ is the shift map
• ${\displaystyle {\frac {p}{q}}}$ fraction has Farey parents a/b and r/s
• b is a denominator of lower Farey parent

${\displaystyle {\begin{cases}{\frac {a}{b}}<{\frac {p}{q}}<{\frac {r}{s}}\\{\frac {a}{b}}\oplus {\frac {r}{s}}={\frac {p}{q}}\\\end{cases}}}$

Implementation:

// https://gitlab.com/adammajewski/wake_gmp
printf("p/q = %d/%d\tb=%d\n\n", p, q, b); // input

printf("(s-)\n"); // first wake ray
printf("s-(s+)\n"); // first Misiurewicz ray

for (j = 1; j< q-1; j++){ // there are q rays ( from 0 to q-1) but only (q-2) has to be computed

n = (j*b) % q;

if (j< q-p)
{printf("s-(d^%d(s+))\n", n);}
else printf("s+(d^%d(s+))\n", n);

}

printf("s+(s-)\n"); // last Misiurewicz ray
printf("(s+)\n"); // last wake ray


## input and output

• input : 2 external angles of the wake ${\displaystyle p/q}$
• output : ${\displaystyle q}$ external angles of principal Misiurewicz point ( hub)

## steps

• input = ${\displaystyle p/q}$
• check input
• both p and q are:
• integers
• > 0
• proper fraction : p < q
• irreducible fraction = in lowest terms ( An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 A fraction is in lowest terms when the greatest common factor (GCF) of the numerator and denominator is 1 )
• if input is good then there are ${\displaystyle q}$ angles to compute
• compute 2 angles of the wake : ${\displaystyle s_{-}(p/q)}$ and ${\displaystyle s_{+}(p/q)}$
• compute first 2 of q angles : ${\displaystyle 0.s_{-}(s_{+})}$ and ${\displaystyle 0.s_{+}(s_{-})}$
• compute last ${\displaystyle q-2}$ angles
• compute Farey parents of ${\displaystyle p/q}$
• compute ${\displaystyle q-p}$
• ( to do )
-- Haskell code by Claude Heiland-Allen
-- http://mathr.co.uk/blog/
-- http://math.bu.edu/people/bob/papers/monica.pdf
-- Geometry of the Antennas in the Mandelbrot Set
-- by R L Devaney and M Moreno-Rocha, April 11, 2000
-- computa a list of external angles from internal angle
hub :: InternalAngle -> [ExternalAngle]
hub pq =
-- List comprehension
[ (sm, shift k sp) | k <- [0, b .. (q - p - 1) * b] ] ++
[ (sp, shift k sp) | k <- [(q - p) * b, (q - p + 1) * b .. (q - 1) * b] ]
where
p = numerator pq
q = denominator pq
-- compute tuple of wake angles = bulb,
-- sm=s- < sp=s+
(([], sm), ([], sp)) = bulb pq -- preperiod is 0 so empty list :  pre = []
(ab, cd) = parents pq -- Farey parents
b = denominator ab
shift k = genericTake q . genericDrop k . cycle  -- shift map


# Examples by wake

wake            angles of the wake      angle of principal Misiurewicz point    angles that land on z=0 on the dynamical plane                  period(c)               c
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k/r = 1/2	wake 1 ; 2/3		Mis 5/12				zcr 5 ; 17/24	                     				period_landing = 1	c -0.2281554936539618 ; 1.115142508039937
k/r = 1/3	wake 1 ; 2/7		Mis 9/56				zcr 9 ; 65/112  						period_landing = 1	c -0.1010963638456222 ; 0.9562865108091415
k/r = 1/4	wake 1 ; 2/15		Mis 17/240				zcr 17 ; 257/480						period_landing = 1	c -0.01718797733835019 ; 1.037652343793215
k/r = 1/5	wake 1 ; 2/31		Mis 33/992				zcr 33 ; 1025/1984						period_landing = 1	c -0.01660571692147523 ; 1.006001828834065
k/r = 1/6	wake 1 ; 2/63		Mis 65/4032				zcr 65 ; 4097/8064						period_landing = 1	c 0.002241106093233115 ; 1.006987004324957
k/r = 1/7	wake 1 ; 2/127		Mis 129/16256				zcr 129 ; 16385/32512						period_landing = 1	c -0.001369133815686842 ; 1.002755660363466
k/r = 1/8	wake 1 ; 2/255		Mis 257/65280				zcr 257 ; 65537/130560						period_landing = 1	c 0.001159450074256577 ; 1.000609019839529
k/r = 1/9	wake 1 ; 2/511		Mis 513/261632				zcr 513 ; 262145/523264 					period_landing = 1	c 0.0001701882004481036 ; 1.000517331884371
k/r = 1/10	wake 1 ; 2/1023		Mis 1025/1047552			zcr 1025 ; 1048577/2095104       				period_landing = 1	c 0.0002217350415235168 ; 0.9999309294242422
k/r = 1/11	wake 1 ; 2/2047		Mis 2049/4192256			zcr 2049 ; 4194305/8384512       				period_landing = 1	c 8.600871635354104e-05 ; 1.000043520609493
k/r = 1/12	wake 1 ; 2/4095		Mis 4097/16773120			zcr 4097 ; 16777217	/33546240				period_landing = 1	c 1.907198794976112e-05 ; 0.9999636227152136
k/r = 1/13	wake 1 ; 2/8191		Mis 8193/67100672			zcr 8193 ; 67108865	/134201344				period_landing = 1	c 1.619607246569189e-05 ; 0.9999946863543573
k/r = 1/14	wake 1 ; 2/16383	Mis 16385/268419072			zcr 16385 ; 268435457	/536838144				period_landing = 1	c -2.164159763572468e-06 ; 0.9999930692712914
k/r = 1/15	wake 1 ; 2/32767	Mis 32769/1073709056			zcr 32769 ; 1073741825	/2147418112				period_landing = 1	c 1.36020585022823e-06 ; 0.9999973111035358
k/r = 1/16	wake 1 ; 2/65535	Mis 65537/4294901760			zcr 65537 ; 4294967297	/8589803520				period_landing = 1	c -1.136844998313359e-06 ; 0.9999994042152635
k/r = 1/17	wake 1 ; 2/131071	Mis 131073/17179738112			zcr 131073 ; 17179869185	/34359476224			period_landing = 1	c -1.660928890362016e-07 ; 0.9999994938657326
k/r = 1/18	wake 1 ; 2/262143	Mis 262145/68719214592			zcr 262145 ; 68719476737	/137438429184			period_landing = 1	c -2.165774171377629e-07 ; 1.000000067631949
k/r = 1/19	wake 1 ; 2/524287	Mis 524289/274877382656			zcr 524289 ; 274877906945	/549754765312			period_landing = 1	c -8.402826966472988e-08 ; 0.9999999574950604
k/r = 1/20	wake 1 ; 2/1048575	Mis 1048577/1099510579200		zcr 1048577 ; 1099511627777	/2199021158400			period_landing = 1	c -1.861820421561348e-08 ; 1.000000035526125
k/r = 1/21	wake 1 ; 2/2097151	Mis 2097153/4398044413952		zcr 2097153 ; 4398046511105	/8796088827904			period_landing = 1	c -1.581664298449309e-08 ; 1.000000005190412
k/r = 1/22	wake 1 ; 2/4194303	Mis 4194305/17592181850112		zcr 4194305 ; 17592186044417	/35184363700224			period_landing = 1	c 2.11348855536603e-09 ; 1.000000006768042
k/r = 1/23	wake 1 ; 2/8388607	Mis 8388609/70368735789056		zcr 8388609 ; 70368744177665	/140737471578112		period_landing = 1	c -1.32827905765734e-09 ; 1.000000002625882
k/r = 1/24	wake 1 ; 2/16777215	Mis 16777217/281474959933440		zcr 16777217 ; 281474976710657	/562949919866880		period_landing = 1	c 1.110191297822782e-09 ; 1.000000000581819
k/r = 1/25	wake 1 ; 2/33554431	Mis 33554433/1125899873288192		zcr 33554433 ; 1125899906842625	/2251799746576384		period_landing = 1	c 1.62200284270896e-10 ; 1.00000000049427
k/r = 1/26	wake 1 ; 2/67108863	Mis 67108865/4503599560261632		zcr 67108865 ; 4503599627370497	/9007199120523264		period_landing = 1	c 2.115013311798569e-10 ; 0.9999999999339535
k/r = 1/27	wake 1 ; 2/134217727	Mis 134217729/18014398375264256		zcr 134217729 ; 18014398509481985	/36028796750528512	period_landing = 1	c 8.205882795347896e-11 ; 1.000000000041509
k/r = 1/28	wake 1 ; 2/268435455	Mis 268435457/72057593769492480		zcr 268435457 ; 72057594037927937	/144115187538984960	period_landing = 1	c 1.818186256603596e-11 ; 0.9999999999653065
k/r = 1/29	wake 1 ; 2/536870911	Mis 536870913/288230375614840832	zcr 536870913 ; 288230376151711745	/576460751229681664	period_landing = 1	c 1.544590637441404e-11 ; 0.9999999999949313
k/r = 1/30	wake 1 ; 2/1073741823	Mis 1073741825/1152921503533105152	zcr 1073741825 ; 1152921504606846977	/2305843007066210304	period_landing = 1	c -2.063955458366402e-12 ; 0.9999999999933906
k/r = 1/31	wake 1 ; 2/2147483647	Mis 2147483649/4611686016279904256	zcr 2147483649 ; 4611686018427387905	/9223372032559808512	period_landing = 1	c 1.29718610843552e-12 ; 0.9999999999974356
k/r = 1/32	wake 1 ; 2/4294967295	Mis 4294967297/18446744069414584320	zcr 4294967297 ; 1	/18446744065119617024			period_landing = 1	c -1.084197223871117e-12 ; 0.9999999999994318
k/r = 1/33	wake 1 ; 2/8589934591	pow  error



## 1/2

So here are 4 angles (q+2) in increasing order :

• 2 rays landing on the root point ( s+ and s- )
• q=2 rays landing on the Misiurewicz point

Farey parents of 1/2 are 0/1 and 1/1

  0/1 < 1/2 < 1/1 	 0.0000000000000000 < 0.5000000000000000 < 1.0000000000000000


The denominator of smaller parent :

 ${\displaystyle b=1}$


  {\displaystyle {\begin{aligned}0.(s_{-})=0.(01)={\frac {1}{3}}=0.33333333333333333333\\0.s_{-}(s_{+})=01(10)={\frac {5}{12}}=0.41666666666666666666\\0.s_{+}(s_{-})=10(01)={\frac {7}{12}}=0.5833333333333333333\\0.(s_{+})=0.(10)={\frac {2}{3}}=0.66666666666666666666\\\end{aligned}}}


The angle  5/12  or  01p10 has  preperiod = 2  and  period = 2. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 2 and period dividing 2.


${\displaystyle M_{2,1}=c=-1.543689012692076+0.000000000000000i}$

Compare with

• ${\displaystyle \mathbf {MF} _{1/2}}$ is the Myrberg-Feigenbaum point c = −1.401155 with external angles = (0.412454... , 0,58755...)

${\displaystyle {\begin{cases}0.(s_{-})=0.(01)={\frac {1}{3}}={\frac {4}{12}}=0.(3)=wake\\0.s_{-}(s_{+})=0.01(10)={\frac {5}{12}}=0.41(6)=PrincipalMis=M_{2,2}\\0.0(1)={\frac {1}{2}}={\frac {6}{12}}=0.5=tip=M_{1,1}=c=-2\\0.s_{+}(s_{-})=0.10(01)={\frac {7}{12}}=0.58(3)=PrincipalMis=M_{2,2}\\0.(s_{+})=0.(10)={\frac {2}{3}}={\frac {8}{12}}=0.(6)=wake\\\end{cases}}}$

## 1/3

The ${\displaystyle {\frac {p}{q}}={\frac {1}{3}}}$ bulb ( = period 3 hyperbolic component) has 2 external angles landing on it's root point (bond) :

 ${\displaystyle \theta _{-}(1/3)=0.({\color {Blue}s_{-}})=0.({\color {Blue}001})}$
${\displaystyle \theta _{+}(1/3)=0.({\color {Red}s_{+}})=0.({\color {Red}010})}$


such that :

 ${\displaystyle \theta _{-}<\theta _{+}}$


Principal Misiurewicz point ${\displaystyle M_{3,1}}$ of ${\displaystyle {\frac {1}{3}}}$ wake is a landing point for ${\displaystyle q=3}$ external angles. It is denoted by

 ${\displaystyle c=M_{3,1}=-0.101096363845622+0.956286510809142i}$


where :

• first number denotes preperiod
• second number denotes period

Two of them one can easly compute from angles the wake :

 ${\displaystyle \theta _{-}(M)=0.{\color {Blue}s_{-}}({\color {Red}s_{+}})=0.{\color {Blue}001}({\color {Red}010})}$
${\displaystyle \theta _{+}(M)=0.{\color {Red}s_{+}}({\color {Blue}s_{-}})=0.{\color {Red}010}({\color {Blue}001})}$


such that :

 ${\displaystyle {\color {Blue}s_{-}}<{\color {Blue}s_{-}}({\color {Red}s_{+}})<{\color {Red}s_{+}}({\color {Blue}s_{-}})<{\color {Red}s_{+}}}$


So the problem is to compute only 1 ray.

First find Farey parents[8] of ${\displaystyle {\frac {1}{3}}}$

 ${\displaystyle {\frac {0}{1}}\oplus {\frac {1}{2}}={\frac {0+1}{1+2}}={\frac {1}{3}}}$


such that :

  ${\displaystyle {\frac {0}{1}}<{\frac {1}{3}}<{\frac {1}{2}}}$


Take denominator of smaller parent :

 ${\displaystyle b=1}$


and compute last fraction.

First find periodic part :

• remember that shift map works on the infinite sequence
• take only first q digits from result of shift map
 ${\displaystyle \sigma ^{b}({\color {Red}s_{+}})=\sigma (\color {Red}010\ 010\ 010\color {Black}...)=\color {Green}100}$


then last angle is :

  ${\displaystyle 0.{\color {Blue}s_{-}}({\color {Green}\sigma ^{b}s_{+}})=0.{\color {Blue}001}({\color {Green}100})}$


So here are 5 angles (q+2) in increasing order :

   ${\displaystyle {\begin{cases}&\theta _{-}(1/3)&=&\ 0.({\color {Blue}s_{-}})&=&0.({\color {Blue}001})&&{\text{ lower angle of the wake}}\\&\theta _{-}(M)&=&\ 0.\ {\color {Blue}s_{-}}({\color {Red}s_{+}})&=&0.\ {\color {Blue}001}\ ({\color {Red}010})={\frac {9}{56}}&&=M_{3,1}\\&\theta _{m}(M)&=&\ 0.\ {\color {Blue}s_{-}}({\color {Green}\sigma s_{+}})&=&0.\ {\color {Blue}001}\ ({\color {Green}100})={\frac {11}{56}}&&=M_{3,1}\\&\theta _{+}(M)&=&\ 0.\ {\color {Red}s_{+}}({\color {Blue}s_{-}})&=&0.\ {\color {Red}010}\ ({\color {Blue}001})={\frac {15}{56}}&&=M_{3,1}\\&\theta _{+}(1/3)&=&\ 0.({\color {Red}s_{+}})&=&0.({\color {Red}010})&&{\text{upper angle of the wake}}\\\end{cases}}}$


One can check it with Mandel:

The angle  9/56  or  001p010
has  preperiod = 3  and  period = 3.
The corresponding parameter ray is landing
at a Misiurewicz point of preperiod 3 and
period dividing 3.
Do you want to draw the ray and to shift c
to the landing point?


## 1/4

The ${\displaystyle {\frac {p}{q}}={\frac {1}{4}}}$ bulb ( = period 4 hyperbolic component) has 2 external angles landing on it's root point (bond) :

 ${\displaystyle \theta _{-}(1/4)=0.(s_{-})=0.({\color {Blue}0001})}$
${\displaystyle \theta _{+}(1/4)=0.(s_{+})=0.({\color {Red}0010})}$


Principal Misiurewicz point ${\displaystyle M}$ of ${\displaystyle {\frac {1}{4}}}$ wake is a landing point for ${\displaystyle q=4}$ external angles.

 ${\displaystyle M_{4,1}=c=0.366362983422764+0.591533773261445i}$


Two of them one can easly compute from angles the wake :

 ${\displaystyle \theta _{-}(M)=0.s_{-}(s_{+})=0.{\color {Blue}0001}({\color {Red}0010})}$
${\displaystyle \theta _{+}(M)=0.s_{+}(s_{-})=0.{\color {Red}0010}({\color {Blue}0001})}$


So the problem is to compute only ${\displaystyle q-2=2}$ rays.

First find Farey parents of ${\displaystyle {\frac {1}{4}}}$

 ${\displaystyle {\frac {0}{1}}\oplus {\frac {1}{3}}={\frac {0+1}{1+3}}={\frac {1}{4}}}$


Take denominator of lower parent :

 ${\displaystyle b=1}$


and compute last fractions.

First find periodic parts for n :

${\displaystyle \sigma ^{1}(s_{+})=\sigma ^{1}({\color {Red}0010\ 0010\ 0010}...)=\color {Green}0100}$
${\displaystyle \sigma ^{2}(s_{+})=\sigma ^{2}({\color {Red}0010\ 0010\ 0010}...)=\color {Magenta}1000}$


then 2 last angles are :

  ${\displaystyle 0.s_{-}(\sigma ^{1}s_{+})=0.{\color {Blue}0001}({\color {Green}0100})}$
${\displaystyle 0.s_{-}(\sigma ^{2}s_{+})=0.{\color {Red}0010}({\color {Magenta}1000})}$


So here are ${\displaystyle q+2=6}$ angles in increasing order :

  {\displaystyle {\begin{aligned}&\theta _{-}(1/4)&=&0.(s_{-})&=&0.({\color {Blue}0001})&&{\text{ lower angle of the wake}}\\&\theta _{-}(M)&=&0.s_{-}\ (\quad s_{+})&=&0.\ {\color {Blue}0001}\ ({\color {Red}0010})&&{\text{ lower angle of M}}\\&\theta _{m-}(M)&=&\ 0.s_{-}(\sigma ^{1}s_{+})&=&0.\ {\color {Blue}0001}\ ({\color {Green}0100})&&{\text{ middle angle of M}}\\&\theta _{m+}(M)&=&\ 0.s_{-}(\sigma ^{2}s_{+})&=&0.\ {\color {Blue}0001}\ ({\color {Magenta}1000})&&{\text{ middle angle of M}}\\&\theta _{+}(M)&=&0.s_{+}\ (\quad s_{-})&=&0.\ {\color {Red}0010}\ ({\color {Blue}0001})&&{\text{ upper angle of M}}\\&\theta _{+}(1/4)&=&0.(s_{+})&=&0.({\color {Red}0010})&&{\text{upper angle of the wake}}\\\end{aligned}}}


## 2/5

The ${\displaystyle {\frac {p}{q}}={\frac {2}{5}}}$ bulb ( = period 5 hyperbolic component) has 2 external angles landing on it's root point (bond) :

 ${\displaystyle \theta _{-}(2/5)=0.(s_{-})=0.({\color {Blue}01001})={\frac {9}{31}}=0.(290322580645161)}$
${\displaystyle \theta _{+}(2/5)=0.(s_{+})=0.({\color {Red}01010})={\frac {10}{31}}=0.(322580645161290)}$


Farey parents of 2/5 are 1/3 and 1/2

  1/3 < 2/5 < 1/2
0.333333 < 0.400000 < 0.500000


${\displaystyle {\frac {1}{3}}\oplus {\frac {1}{2}}={\frac {1+1}{2+3}}={\frac {2}{5}}}$

so denominator of smaller parent is b = 3.

Angles in the symbolic form

(s-)
s-(s+)
s-(d^3(s+))
s-(d^1(s+))
s+(d^4(s+))
s+(s-)
(s+)


*Main> :main 2 5
bulb:
p01001 = 9 % 31
p01010 = 10 % 31

hub:
01001p01010 = 289 % 992
01001p10010 = 297 % 992
01001p10100 = 299 % 992
01010p00101 = 315 % 992
01010p01001 = 319 % 992


The angle  289/992  or  01001p01010 has  preperiod = 5  and  period = 5.
The corresponding parameter ray is landing at a Misiurewicz point of preperiod 5 and period dividing 5.


## 1/7

The wake 1/7 of main cardioid

${\displaystyle M_{7,1}=c=0.397391822296541+0.133511204871878i}$ = principal Misiurewicz
c = 0.367375134418445  +0.147183763188559 i = root of the wake 1/7
c = 0.376008681846768  +0.144749371321633 i = period 7 center


External rays:

• 1/127 = 0.(0000001) = 0.0078740157480315 = wake
• 129 /16256 = 0.0000001(0000010) = 0.00793553149606299 = pM_{7,1}
• 131 /16256 = 0.0000001(0000100) = 0.00805856299212598 = pM
• 135 /16256 = 0.0000001(0001000 = 0.00830462598425197 = pM
• 143 /16256 = 0.0000001(0010000) = 0.00879675196850394 = pM
• 159 /16256 = 0.0000001(0100000) = 0.00978100393700787 = pM
• 191 /16256 = 0.0000001(1000000) = 0.01174950787401575 = pM
• 255 /16256 = 0.0000010(0000001) = 0.0156865157480315 = pM
• 1/64 = 0.000000(1) = 0.015625 = M_{6,1} = longest tip
• 2/127 = 0.(0000010) = 0.01574803149606299 = wake

## 3/7

Wake 3/7 and its principal Misiurewicz point (hub)

${\displaystyle M_{7,1}=c=-0.670209187903254+0.458060975296946i}$


*Main> :main 3 7
bulb:
p0101001 = 41 % 127
p0101010 = 42 % 127

hub:
0101001p0101010 = 5249 % 16256
0101001p1001010 = 5281 % 16256
0101001p1010010 = 5289 % 16256
0101001p1010100 = 5291 % 16256
0101010p0010101 = 5355 % 16256
0101010p0100101 = 5371 % 16256
0101010p0101001 = 5375 % 16256


Check with Mandel

The angle  5249/16256  or  0101001p0101010
has  preperiod = 7  and  period = 7.
The corresponding parameter ray is landing
at a Misiurewicz point of preperiod 7 and
period dividing 7.


## 5/11

ghci
GHCi, version 8.10.7: https://www.haskell.org/ghc/  :? for help
Prelude> :l bh.hs
[1 of 1] Compiling Main             ( bh.hs, interpreted )
*Main> :main 5 11
internal angle p/q = 5 / 11
internal angle in lowest terms =
5 % 11
rays of the bulb:
(01010101001) = 681 % 2047
(01010101010) = 682 % 2047

rays of the principle hub:
01010101001(01010101010) = 1394689 % 4192256
01010101001(10010101010) = 1395201 % 4192256
01010101001(10100101010) = 1395329 % 4192256
01010101001(10101001010) = 1395361 % 4192256
01010101001(10101010010) = 1395369 % 4192256
01010101001(10101010100) = 1395371 % 4192256
01010101010(00101010101) = 1396395 % 4192256
01010101010(01001010101) = 1396651 % 4192256
01010101010(01010010101) = 1396715 % 4192256
01010101010(01010100101) = 1396731 % 4192256
01010101010(01010101001) = 1396735 % 4192256


Check with Mandel:

The 5/11-wake of the main cardioid is
bounded by the parameter rays with the angles
681/2047  or  p01010101001  and
682/2047  or  p01010101010 .
Do you want to draw the rays and to shift c to the center of the satellite component?


The result is a center of period 11 satelite component c = -0.697838195122425 +0.279304134101366 i period = 11

The angle  1394689/4192256  or  01010101001p01010101010 has  preperiod = 11  and  period = 11.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 11 and period dividing 11. Do you want to draw the ray and to shift c to the landing point?


The result is a principal Misiurewicz point of wake 5/11 M_{11,1} = c = -0.724112682973574 +0.286456567676711 i

## 12/25

${\displaystyle {\begin{cases}0.(0101010101010101010101001)_{2}={\frac {11184809}{33554431}}=0.33333329359690229883498844012583613770711832365..._{10}=wake\\0101010101010101010101001(0101010101010101010101010)=375299913023489\%1125899873288192=principalMis\\0101010101010101010101001(1001010101010101010101010)=375299921412097\%1125899873288192=principalMis\\0101010101010101010101001(1010010101010101010101010)=375299923509249\%1125899873288192=principalMis\\0101010101010101010101001(1010100101010101010101010)=375299924033537\%1125899873288192=principalMis\\0101010101010101010101001(1010101001010101010101010)=375299924164609\%1125899873288192=principalMis\\0101010101010101010101001(1010101010010101010101010)=375299924197377\%1125899873288192=principalMis\\0101010101010101010101001(1010101010100101010101010)=375299924205569\%1125899873288192=principalMis\\0101010101010101010101001(1010101010101001010101010)=375299924207617\%1125899873288192=principalMis\\0101010101010101010101001(1010101010101010010101010)=375299924208129\%1125899873288192=principalMis\\0101010101010101010101001(1010101010101010100101010)=375299924208257\%1125899873288192=principalMis\\0101010101010101010101001(1010101010101010101001010)=375299924208289\%1125899873288192=principalMis\\0101010101010101010101001(1010101010101010101010010)=375299924208297\%1125899873288192=principalMis\\0101010101010101010101001(1010101010101010101010100)=375299924208299\%1125899873288192=principalMis\\0101010101010101010101010(0010101010101010101010101)=375299940985515\%1125899873288192=principalMis\\0101010101010101010101010(0100101010101010101010101)=375299945179819\%1125899873288192=principalMis\\0101010101010101010101010(0101001010101010101010101)=375299946228395\%1125899873288192=principalMis\\0101010101010101010101010(0101010010101010101010101)=375299946490539\%1125899873288192=principalMis\\0101010101010101010101010(0101010100101010101010101)=375299946556075\%1125899873288192=principalMis\\0101010101010101010101010(0101010101001010101010101)=375299946572459\%1125899873288192=principalMis\\0101010101010101010101010(0101010101010010101010101)=375299946576555\%1125899873288192=principalMis\\0101010101010101010101010(0101010101010100101010101)=375299946577579\%1125899873288192=principalMis\\0101010101010101010101010(0101010101010101001010101)=375299946577835\%1125899873288192=principalMis\\0101010101010101010101010(0101010101010101010010101)=375299946577899\%1125899873288192=principalMis\\0101010101010101010101010(0101010101010101010100101)=375299946577915\%1125899873288192=principalMis\\0101010101010101010101010(0101010101010101010101001)=375299946577919\%1125899873288192=principalMis\\0.(0101010101010101010101010)_{2}={\frac {11184810}{33554431}}=0.3333333233992255747087471100314590344267795809=wake\\\end{cases}}}$


*Main> :main 12 25
internal angle p/q = 12 / 25
internal angle in lowest terms =
12 % 25
rays of the bulb:
(0101010101010101010101001) = 11184809 % 33554431
(0101010101010101010101010) = 11184810 % 33554431

rays of the principle hub:
0101010101010101010101001(0101010101010101010101010) = 375299913023489 % 1125899873288192
0101010101010101010101001(1001010101010101010101010) = 375299921412097 % 1125899873288192
0101010101010101010101001(1010010101010101010101010) = 375299923509249 % 1125899873288192
0101010101010101010101001(1010100101010101010101010) = 375299924033537 % 1125899873288192
0101010101010101010101001(1010101001010101010101010) = 375299924164609 % 1125899873288192
0101010101010101010101001(1010101010010101010101010) = 375299924197377 % 1125899873288192
0101010101010101010101001(1010101010100101010101010) = 375299924205569 % 1125899873288192
0101010101010101010101001(1010101010101001010101010) = 375299924207617 % 1125899873288192
0101010101010101010101001(1010101010101010010101010) = 375299924208129 % 1125899873288192
0101010101010101010101001(1010101010101010100101010) = 375299924208257 % 1125899873288192
0101010101010101010101001(1010101010101010101001010) = 375299924208289 % 1125899873288192
0101010101010101010101001(1010101010101010101010010) = 375299924208297 % 1125899873288192
0101010101010101010101001(1010101010101010101010100) = 375299924208299 % 1125899873288192
0101010101010101010101010(0010101010101010101010101) = 375299940985515 % 1125899873288192
0101010101010101010101010(0100101010101010101010101) = 375299945179819 % 1125899873288192
0101010101010101010101010(0101001010101010101010101) = 375299946228395 % 1125899873288192
0101010101010101010101010(0101010010101010101010101) = 375299946490539 % 1125899873288192
0101010101010101010101010(0101010100101010101010101) = 375299946556075 % 1125899873288192
0101010101010101010101010(0101010101001010101010101) = 375299946572459 % 1125899873288192
0101010101010101010101010(0101010101010010101010101) = 375299946576555 % 1125899873288192
0101010101010101010101010(0101010101010100101010101) = 375299946577579 % 1125899873288192
0101010101010101010101010(0101010101010101001010101) = 375299946577835 % 1125899873288192
0101010101010101010101010(0101010101010101010010101) = 375299946577899 % 1125899873288192
0101010101010101010101010(0101010101010101010100101) = 375299946577915 % 1125899873288192
0101010101010101010101010(0101010101010101010101001) = 375299946577919 % 1125899873288192


Landing point = principal Misiurewicz point

The angle  375299913023489/1125899873288192  or  0101010101010101010101001p0101010101010101010101010 has  preperiod = 25  and  period = 25.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 25 and period dividing 25.
Do you want to draw the ray and to shift c to the landing point?
c = -0.745846774741742  +0.124374904775875 i

m-describe 112 100 10000 -0.745846774741742  +0.124374904775875 4
the input point was -7.4584677474174200000000000000000001e-01 + 1.2437490477587499999999999999999999e-01 i
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at 0.00000e+00 + 0.00000e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are -nan to the east
the atom coordinates of the input point are -0.74585 + 0.12437 i
the atom coordinates in polar form are 0.75615 to the west
the nucleus is 7.56146e-01 to the east of the input point

- a period 2 circle
with nucleus at -1.00000e+00 + 0.00000e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are 0.25415 + 0.12437 i
the atom domain coordinates in polar form are 0.28295 to the east-north-east
the atom coordinates of the input point are 0.50831 + 0.24875 i
the atom coordinates in polar form are 0.56591 to the east-north-east
the nucleus is 2.82954e-01 to the west-south-west of the input point
external angles of this component are:
.(01)
.(10)
the point escaped with dwell 4217.96435

nearby Misiurewicz points to the input point:

- 26p4
with center at -7.45846774741742277327028259457753e-01 + 1.24374904775875452739596099543026e-01 i
the Misiurewicz domain has size 7.57002e-04
the Misiurewicz domain coordinate radius is 7.0135e-13
the center is 5.30927e-16 to the north-north-west of the input point
the multiplier has radius 1.030029879100029796e+00 and angle -0.078808321127835692 (in turns)


## 1/31

*Main> :main 1 31
internal angle p/q = 1 / 31
internal angle in lowest terms =
1 % 31
rays of the bulb:
(0000000000000000000000000000001) = 1 % 2147483647
(0000000000000000000000000000010) = 2 % 2147483647

rays of the principle hub:
0000000000000000000000000000001(0000000000000000000000000000010) = 2147483649 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000000000100) = 2147483651 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000000001000) = 2147483655 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000000010000) = 2147483663 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000000100000) = 2147483679 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000001000000) = 2147483711 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000010000000) = 2147483775 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000000100000000) = 2147483903 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000001000000000) = 2147484159 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000010000000000) = 2147484671 % 4611686016279904256
0000000000000000000000000000001(0000000000000000000100000000000) = 2147485695 % 4611686016279904256
0000000000000000000000000000001(0000000000000000001000000000000) = 2147487743 % 4611686016279904256
0000000000000000000000000000001(0000000000000000010000000000000) = 2147491839 % 4611686016279904256
0000000000000000000000000000001(0000000000000000100000000000000) = 2147500031 % 4611686016279904256
0000000000000000000000000000001(0000000000000001000000000000000) = 2147516415 % 4611686016279904256
0000000000000000000000000000001(0000000000000010000000000000000) = 2147549183 % 4611686016279904256
0000000000000000000000000000001(0000000000000100000000000000000) = 2147614719 % 4611686016279904256
0000000000000000000000000000001(0000000000001000000000000000000) = 2147745791 % 4611686016279904256
0000000000000000000000000000001(0000000000010000000000000000000) = 2148007935 % 4611686016279904256
0000000000000000000000000000001(0000000000100000000000000000000) = 2148532223 % 4611686016279904256
0000000000000000000000000000001(0000000001000000000000000000000) = 2149580799 % 4611686016279904256
0000000000000000000000000000001(0000000010000000000000000000000) = 2151677951 % 4611686016279904256
0000000000000000000000000000001(0000000100000000000000000000000) = 2155872255 % 4611686016279904256
0000000000000000000000000000001(0000001000000000000000000000000) = 2164260863 % 4611686016279904256
0000000000000000000000000000001(0000010000000000000000000000000) = 2181038079 % 4611686016279904256
0000000000000000000000000000001(0000100000000000000000000000000) = 2214592511 % 4611686016279904256
0000000000000000000000000000001(0001000000000000000000000000000) = 2281701375 % 4611686016279904256
0000000000000000000000000000001(0010000000000000000000000000000) = 2415919103 % 4611686016279904256
0000000000000000000000000000001(0100000000000000000000000000000) = 2684354559 % 4611686016279904256
0000000000000000000000000000001(1000000000000000000000000000000) = 3221225471 % 4611686016279904256
0000000000000000000000000000010(0000000000000000000000000000001) = 4294967295 % 4611686016279904256


## 8/47 = 16/94

*Main> :main 16 94
internal angle p/q = 16 / 94
internal angle in lowest terms =
8 % 47
rays of the bulb:
(00001000001000001000001000001000001000001000001) = 4467856773185 % 140737488355327
(00001000001000001000001000001000001000001000010) = 4467856773186 % 140737488355327

rays of the hub:
00001000001000001000001000001000001000001000001(00001000001000001000001000001000001000001000010)
00001000001000001000001000001000001000001000001(00001000001000001000001000001000001000010000010)
00001000001000001000001000001000001000001000001(00001000001000001000001000001000010000010000010)
00001000001000001000001000001000001000001000001(00001000001000001000001000010000010000010000010)
00001000001000001000001000001000001000001000001(00001000001000001000010000010000010000010000010)
00001000001000001000001000001000001000001000001(00001000001000010000010000010000010000010000010)
00001000001000001000001000001000001000001000001(00001000010000010000010000010000010000010000010)
00001000001000001000001000001000001000001000001(00010000010000010000010000010000010000010000010)
00001000001000001000001000001000001000001000001(00010000010000010000010000010000010000010000100)
00001000001000001000001000001000001000001000001(00010000010000010000010000010000010000100000100)
00001000001000001000001000001000001000001000001(00010000010000010000010000010000100000100000100)
00001000001000001000001000001000001000001000001(00010000010000010000010000100000100000100000100)
00001000001000001000001000001000001000001000001(00010000010000010000100000100000100000100000100)
00001000001000001000001000001000001000001000001(00010000010000100000100000100000100000100000100)
00001000001000001000001000001000001000001000001(00010000100000100000100000100000100000100000100)
00001000001000001000001000001000001000001000001(00100000100000100000100000100000100000100000100)
00001000001000001000001000001000001000001000001(00100000100000100000100000100000100000100001000)
00001000001000001000001000001000001000001000001(00100000100000100000100000100000100001000001000)
00001000001000001000001000001000001000001000001(00100000100000100000100000100001000001000001000)
00001000001000001000001000001000001000001000001(00100000100000100000100001000001000001000001000)
00001000001000001000001000001000001000001000001(00100000100000100001000001000001000001000001000)
00001000001000001000001000001000001000001000001(00100000100001000001000001000001000001000001000)
00001000001000001000001000001000001000001000001(00100001000001000001000001000001000001000001000)
00001000001000001000001000001000001000001000001(01000001000001000001000001000001000001000001000)
00001000001000001000001000001000001000001000001(01000001000001000001000001000001000001000010000)
00001000001000001000001000001000001000001000001(01000001000001000001000001000001000010000010000)
00001000001000001000001000001000001000001000001(01000001000001000001000001000010000010000010000)
00001000001000001000001000001000001000001000001(01000001000001000001000010000010000010000010000)
00001000001000001000001000001000001000001000001(01000001000001000010000010000010000010000010000)
00001000001000001000001000001000001000001000001(01000001000010000010000010000010000010000010000)
00001000001000001000001000001000001000001000001(01000010000010000010000010000010000010000010000)
00001000001000001000001000001000001000001000001(10000010000010000010000010000010000010000010000)
00001000001000001000001000001000001000001000001(10000010000010000010000010000010000010000100000)
00001000001000001000001000001000001000001000001(10000010000010000010000010000010000100000100000)
00001000001000001000001000001000001000001000001(10000010000010000010000010000100000100000100000)
00001000001000001000001000001000001000001000001(10000010000010000010000100000100000100000100000)
00001000001000001000001000001000001000001000001(10000010000010000100000100000100000100000100000)
00001000001000001000001000001000001000001000001(10000010000100000100000100000100000100000100000)
00001000001000001000001000001000001000001000001(10000100000100000100000100000100000100000100000)
00001000001000001000001000001000001000001000010(00000100000100000100000100000100000100000100001)
00001000001000001000001000001000001000001000010(00000100000100000100000100000100000100001000001)
00001000001000001000001000001000001000001000010(00000100000100000100000100000100001000001000001)
00001000001000001000001000001000001000001000010(00000100000100000100000100001000001000001000001)
00001000001000001000001000001000001000001000010(00000100000100000100001000001000001000001000001)
00001000001000001000001000001000001000001000010(00000100000100001000001000001000001000001000001)
00001000001000001000001000001000001000001000010(00000100001000001000001000001000001000001000001)
00001000001000001000001000001000001000001000010(00001000001000001000001000001000001000001000001)


c output

~/book/code/bin$./mandelbrot_describe_external_angle '.00001000001000001000001000001000001000001000001(00001000001000001000001000001000001000001000010)' binary: .00001000001000001000001000001000001000001000001(00001000001000001000001000001000001000001000010) decimal: 628794940589397270782279681/19807040628565943660897632256 preperiod: 47 period: 47  ## 34/89 a@zelman:~/haskell/hub$ ghci
GHCi, version 8.0.2: http://www.haskell.org/ghc/  :? for help
Prelude> :l bh.hs
[1 of 1] Compiling Main             ( bh.hs, interpreted )
*Main> :main 34 89
internal angle p/q = 34 / 89
internal angle in lowest terms =
34 % 89
rays of the bulb:
(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001) = 179622968672387565806504265 % 618970019642690137449562111
(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010) = 179622968672387565806504266 % 618970019642690137449562111

rays of the principle hub:
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010) = 111181232447426046807770849175978166730445345710407681 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010010100101001001010010100100101001001010010100100101001001010010100100101001010) = 111181232447426046807770849175978166766474142729371649 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010010100101001001010010100100101001001010010100100101001010010010100100101001010) = 111181232447426046807770849175978166766474142731468801 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010100100101001001010010100100101001001010010100100101001010010010100100101001010) = 111181232447426046807770849176053724630200057054887937 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010100100101001001010010100100101001010010010100100101001010010010100100101001010) = 111181232447426046807770849176053724630204455101399041 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010100100101001001010010100100101001010010010100100101001010010010100101001001010) = 111181232447426046807770849176053724630204455101399297 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010100100101001010010010100100101001010010010100100101001010010010100101001001010) = 111181232447426046807770849176053733853576491956175105 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01001010010100100101001010010010100100101001010010010100101001001010010010100101001001010) = 111181232447426046807770849176053733853576492493046017 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100100101001010010010100100101001010010010100101001001010010010100101001001010) = 111181232447426046807770849195396546967410559288344833 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100100101001010010010100101001001010010010100101001001010010010100101001001010) = 111181232447426046807770849195396546968536459195187457 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100100101001010010010100101001001010010010100101001001010010100100101001001010) = 111181232447426046807770849195396546968536459195252993 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100101001001010010010100101001001010010010100101001001010010100100101001001010) = 111181232447426046807770849195398908151777894017859841 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100101001001010010010100101001001010010100100101001001010010100100101001001010) = 111181232447426046807770849195398908151778031456813313 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100101001001010010010100101001001010010100100101001001010010100100101001010010) = 111181232447426046807770849195398908151778031456813321 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100101001001010010100100101001001010010100100101001001010010100100101001010010) = 111181232447426046807770849195398908440008407608525065 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010010100101001001010010100100101001001010010100100101001010010010100100101001010010) = 111181232447426046807770849195398908440008407625302281 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010100100101001001010010100100101001001010010100100101001010010010100100101001010010) = 111181232447426046807770849196003371349815722212655369 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010100100101001001010010100100101001010010010100100101001010010010100100101001010010) = 111181232447426046807770849196003371349850906584744201 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010100100101001001010010100100101001010010010100100101001010010010100101001001010010) = 111181232447426046807770849196003371349850906584746249 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010100100101001010010010100100101001010010010100100101001010010010100101001001010010) = 111181232447426046807770849196003445136827201422952713 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(01010010100100101001010010010100100101001010010010100101001001010010010100101001001010010) = 111181232447426046807770849196003445136827205717920009 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100100101001010010010100100101001010010010100101001001010010010100101001001010010) = 111181232447426046807770849350745950047499740080310537 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100100101001010010010100101001001010010010100101001001010010010100101001001010010) = 111181232447426046807770849350745950056506939335051529 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100100101001010010010100101001001010010010100101001001010010100100101001001010010) = 111181232447426046807770849350745950056506939335575817 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100101001001010010010100101001001010010010100101001001010010100100101001001010010) = 111181232447426046807770849350764839522438417916430601 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100101001001010010010100101001001010010100100101001001010010100100101001001010010) = 111181232447426046807770849350764839522439517428058377 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100101001001010010010100101001001010010100100101001001010010100100101001010010010) = 111181232447426046807770849350764839522439517428058441 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100101001001010010100100101001001010010100100101001001010010100100101001010010010) = 111181232447426046807770849350764841828282526641752393 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010010100101001001010010100100101001001010010100100101001010010010100100101001010010010) = 111181232447426046807770849350764841828282526775970121 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001001010010100100101001001010010100100101001010010010100100101001010010010) = 111181232447426046807770849355600545106741043474794825 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001001010010100100101001010010010100100101001010010010100100101001010010010) = 111181232447426046807770849355600545107022518451505481 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001001010010100100101001010010010100100101001010010010100101001001010010010) = 111181232447426046807770849355600545107022518451521865 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001010010010100100101001010010010100100101001010010010100101001001010010010) = 111181232447426046807770849355601135402832877157173577 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001010010010100100101001010010010100101001001010010010100101001001010010010) = 111181232447426046807770849355601135402832911516911945 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001010010010100100101001010010010100101001001010010010100101001001010010100) = 111181232447426046807770849355601135402832911516911947 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001010010010100101001001010010010100101001001010010010100101001001010010100) = 111181232447426046807770849355601135474890505554839883 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100100101001010010010100101001001010010010100101001001010010100100101001001010010100) = 111181232447426046807770849355601135474890505559034187 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100101001001010010010100101001001010010010100101001001010010100100101001001010010100) = 111181232447426046807770849355752251202342334205872459 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100101001001010010010100101001001010010100100101001001010010100100101001001010010100) = 111181232447426046807770849355752251202351130298894667 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100101001001010010010100101001001010010100100101001001010010100100101001010010010100) = 111181232447426046807770849355752251202351130298895179 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100101001001010010100100101001001010010100100101001001010010100100101001010010010100) = 111181232447426046807770849355752269649095204008446795 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10010100101001001010010100100101001001010010100100101001010010010100100101001010010010100) = 111181232447426046807770849355752269649095205082188619 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001001010010100100101001001010010100100101001010010010100100101001010010010100) = 111181232447426046807770849394437895876763338672786251 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001001010010100100101001010010010100100101001010010010100100101001010010010100) = 111181232447426046807770849394437895879015138486471499 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001001010010100100101001010010010100100101001010010010100101001001010010010100) = 111181232447426046807770849394437895879015138486602571 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001010010010100100101001010010010100100101001010010010100101001001010010010100) = 111181232447426046807770849394442618245498008131816267 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001010010010100100101001010010010100101001001010010010100101001001010010010100) = 111181232447426046807770849394442618245498283009723211 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001010010010100100101001010010010100101001001010010010100101001001010010100100) = 111181232447426046807770849394442618245498283009723227 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001010010010100101001001010010010100101001001010010010100101001001010010100100) = 111181232447426046807770849394442618821959035313146715 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100100101001010010010100101001001010010010100101001001010010100100101001001010010100100) = 111181232447426046807770849394442618821959035346701147 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100101001001010010010100101001001010010010100101001001010010100100101001001010010100100) = 111181232447426046807770849395651544641573664521407323 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100101001001010010010100101001001010010100100101001001010010100100101001001010010100100) = 111181232447426046807770849395651544641644033265584987 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100101001001010010010100101001001010010100100101001001010010100100101001010010010100100) = 111181232447426046807770849395651544641644033265589083 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100101001001010010100100101001001010010100100101001001010010100100101001010010010100100) = 111181232447426046807770849395651692215596622942002011 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001(10100101001001010010100100101001001010010100100101001010010010100100101001010010010100100) = 111181232447426046807770849395651692215596631531936603 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001001010010100100101001001010010100100101001010010010100100101001010010010100101) = 111181232447426046807770849705136702036941700256717659 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001001010010100100101001010010010100100101001010010010100100101001010010010100101) = 111181232447426046807770849705136702054956098766199643 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001001010010100100101001010010010100100101001010010010100101001001010010010100101) = 111181232447426046807770849705136702054956098767248219 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001010010010100100101001010010010100100101001010010010100101001001010010010100101) = 111181232447426046807770849705174480986819055928957787 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001010010010100100101001010010010100101001001010010010100101001001010010010100101) = 111181232447426046807770849705174480986821254952213339 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001010010010100100101001010010010100101001001010010010100101001001010010100100101) = 111181232447426046807770849705174480986821254952213467 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001010010010100101001001010010010100101001001010010010100101001001010010100100101) = 111181232447426046807770849705174485598507273379601371 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00100101001010010010100101001001010010010100101001001010010100100101001001010010100100101) = 111181232447426046807770849705174485598507273648036827 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010010100101001001010010010100101001001010010100100101001001010010100100101) = 111181232447426046807770849714845892155424307045686235 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010010100101001001010010100100101001001010010100100101001001010010100100101) = 111181232447426046807770849714845892155987256999107547 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010010100101001001010010100100101001001010010100100101001010010010100100101) = 111181232447426046807770849714845892155987256999140315 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010100100101001001010010100100101001001010010100100101001010010010100100101) = 111181232447426046807770849714847072747607974410443739 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010100100101001001010010100100101001010010010100100101001010010010100100101) = 111181232447426046807770849714847072747608043129920475 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010100100101001001010010100100101001010010010100100101001010010010100101001) = 111181232447426046807770849714847072747608043129920479 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010100100101001010010010100100101001010010010100100101001010010010100101001) = 111181232447426046807770849714847072891723231205776351 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001001010010100100101001010010010100100101001010010010100101001001010010010100101001) = 111181232447426046807770849714847072891723231214164959 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001010010010100100101001010010010100100101001010010010100101001001010010010100101001) = 111181232447426046807770849715149304346626888507841503 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001010010010100100101001010010010100101001001010010010100101001001010010010100101001) = 111181232447426046807770849715149304346644480693885919 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001010010010100100101001010010010100101001001010010010100101001001010010100100101001) = 111181232447426046807770849715149304346644480693886943 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001010010010100101001001010010010100101001001010010010100101001001010010100100101001) = 111181232447426046807770849715149341240132628112990175 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(00101001010010010100101001001010010010100101001001010010100100101001001010010100100101001) = 111181232447426046807770849715149341240132630260473823 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010010100101001001010010010100101001001010010100100101001001010010100100101001) = 111181232447426046807770849792520593695468897441669087 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010010100101001001010010100100101001001010010100100101001001010010100100101001) = 111181232447426046807770849792520593699972497069039583 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010010100101001001010010100100101001001010010100100101001010010010100100101001) = 111181232447426046807770849792520593699972497069301727 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010100100101001001010010100100101001001010010100100101001010010010100100101001) = 111181232447426046807770849792530038432938236359729119 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010100100101001001010010100100101001010010010100100101001010010010100100101001) = 111181232447426046807770849792530038432938786115543007 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010100100101001001010010100100101001010010010100100101001010010010100101001001) = 111181232447426046807770849792530038432938786115543039 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010100100101001010010010100100101001010010010100100101001010010010100101001001) = 111181232447426046807770849792530039585860290722390015 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001001010010100100101001010010010100100101001010010010100101001001010010010100101001001) = 111181232447426046807770849792530039585860290789498879 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001010010010100100101001010010010100100101001010010010100101001001010010010100101001001) = 111181232447426046807770849794947891225089549138911231 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001010010010100100101001010010010100101001001010010010100101001001010010010100101001001) = 111181232447426046807770849794947891225230286627266559 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001010010010100100101001010010010100101001001010010010100101001001010010100100101001001) = 111181232447426046807770849794947891225230286627274751 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001010010010100101001001010010010100101001001010010010100101001001010010100100101001001) = 111181232447426046807770849794948186373135465980100607 % 383123885216472214589586756168607276261994643096338432
01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001) = 111181232447426046807770849794948186373135483159969791 % 383123885216472214589586756168607276261994643096338432
*Main>


## 15/94

*Main> :main 15 94
internal angle p/q = 15 / 94
internal angle in lowest terms =
15 % 94
rays of the bulb:
(0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001) = 314396870629096754623553665 % 19807040628566084398385987583
(0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010) = 314396870629096754623553666 % 19807040628566084398385987583

rays of the hub:
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000010000010000010000010000001000001000001000000100000100000100000100000010000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000010000010000010000010000001000001000001000001000000100000100000100000010000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000010000010000010000010000001000001000001000001000000100000100000100000100000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000010000010000010000001000001000001000001000000100000100000100000100000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000010000010000010000010000001000001000001000000100000100000100000100000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000010000010000010000010000001000001000001000001000000100000100000100000010000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000010000010000010000010000001000001000001000001000000100000100000100000100000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000010000010000010000001000001000001000001000000100000100000100000100000010000010)
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0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000010000001000001000001000000100000100000100000100000010000010000010000010000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000010000001000001000001000001000000100000100000100000010000010000010000010000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001)


First angle of the hub is:

  6227271590044554501136183694529415329491604978647695361% 392318858461667547739736838930672110377831130880616169472


# Code

-- Haskell code by Claude Heiland-Allen
-- http://mathr.co.uk/blog/
-- http://math.bu.edu/people/bob/papers/monica.pdf
-- Geometry of the Antennas in the Mandelbrot Set
-- by R L Devaney and M Moreno-Rocha, April 11, 2000

import Data.List (genericTake, genericDrop, intercalate)
import Data.Fixed (mod')
import Data.Ratio ((%), numerator, denominator)
import System.Environment (getArgs)

type InternalAngle = Rational -- let pq = p % q

type ExternalAngle = ([Bool], [Bool]) -- tuple of lists ([preperiodic], [periodic])

-- convert bool value to char 0 or 1
bit :: Bool -> Char
bit False = '0'
bit True = '1'

-- convert list of bool values to string of bits
bits :: [Bool] -> String
bits = map bit

-- converts tuple of the lists (pre, per) to the string
pretty :: ExternalAngle -> String
pretty (pre, per) =  bits pre ++ "(" ++ bits per ++")"

-- converts a list of bits to an Integer, by parsing a String.
binary :: [Bool] -> Integer
binary [] = 0
binary s = case readInt 2 (elem"01") (\c -> case c of '0' -> 0 ; '1' -> 1) (bits s) of
[(b, "")] -> b

-- external angle from tuple form to rational number
rational :: ExternalAngle -> Rational
rational (pre, per) = (binary pre % 2^p) + (binary per % (2^p * (2^q - 1)))
where
p = length pre
q = length per

-- compute a tuple of external angles from internal angle
-- rays for such angles land on the root of pq wake =  wake angles = bulb
bulb :: InternalAngle -> (ExternalAngle, ExternalAngle)
bulb pq = (([], bs ++ [False, True]), ([], bs ++ [True, False]))
where
q = denominator pq
bs
= genericTake (q - 2)
. map (\x -> 1 - pq < x && x < 1)
. iterate (\x -> (x + pq) mod' 1)
$pq -- parents in the Farey tree -- http://mathr.co.uk/blog/2016-10-31_finding_parents_in_the_farey_tree.html parents :: InternalAngle -> (InternalAngle, InternalAngle) parents pq = go q 1 0 p 0 1 where p = numerator pq q = denominator pq go r1 s1 t1 r0 s0 t0 | r0 == 0 = let ab = - s1 % t1 a = numerator ab b = denominator ab c = p - a d = q - b cd = c % d in (min ab cd, max ab cd) | otherwise = let (o, r) = divMod r1 r0 s = s1 - o * s0 t = t1 - o * t0 in go r0 s0 t0 r s t -- computa a list of external angles from internal angle hub :: InternalAngle -> [ExternalAngle] hub pq = -- List comprehension [ (sm, shift k sp) | k <- [0, b .. (q - p - 1) * b] ] ++ [ (sp, shift k sp) | k <- [(q - p) * b, (q - p + 1) * b .. (q - 1) * b] ] where p = numerator pq q = denominator pq -- compute tuple of wake angles = bulb, -- sm=s- < sp=s+ (([], sm), ([], sp)) = bulb pq -- preperiod is 0 so empty list : pre = [] (ab, cd) = parents pq -- Farey parents b = denominator ab shift k = genericTake q . genericDrop k . cycle -- shift map main :: IO () main = do -- read the input [sp, sq] <- getArgs p <- readIO sp q <- readIO sq -- compute let pq = p % q (lo, hi) = bulb pq hs = hub pq -- output the results putStrLn$ "internal angle p/q = " ++ sp ++ " / " ++ sq
putStrLn $"internal angle in lowest terms = " print pq putStrLn$ "rays of the bulb:"
putStrLn $pretty lo ++ " = " ++ show (rational lo) putStrLn$ pretty hi ++ " = " ++ show (rational hi)
putStrLn $"" putStrLn$ "rays of the principle hub:"
forM_ hs $\h -> putStrLn$ pretty h  ++ " = " ++ show (rational h)


Save it as a bh.hs and use it from console in an interactive way :

ghci
GHCi, version 7.10.3: http://www.haskell.org/ghc/  :? for help
Prelude> :l bh.hs
[1 of 1] Compiling Main             ( bh.hs, interpreted )

*Main> :main 1 2
bulb:
p01 = 1 % 3
p10 = 2 % 3

hub:
01p10 = 5 % 12
10p01 = 7 % 12

*Main> :main 1 3
bulb:
p001 = 1 % 7
p010 = 2 % 7

hub:
001p010 = 9 % 56
001p100 = 11 % 56
010p001 = 15 % 56

*Main> :main 1 4
bulb:
p0001 = 1 % 15
p0010 = 2 % 15

hub:
0001p0010 = 17 % 240
0001p0100 = 19 % 240
0001p1000 = 23 % 240
0010p0001 = 31 % 240

:main 1 5
bulb:
p00001 = 1 % 31
p00010 = 2 % 31

hub:
00001p00010 = 33 % 992
00001p00100 = 35 % 992
00001p01000 = 39 % 992
00001p10000 = 47 % 992
00010p00001 = 63 % 992

*Main> :main 1 6
bulb:
p000001 = 1 % 63
p000010 = 2 % 63

hub:
000001p000010 = 65 % 4032
000001p000100 = 67 % 4032
000001p001000 = 71 % 4032
000001p010000 = 79 % 4032
000001p100000 = 95 % 4032
000010p000001 = 127 % 4032

*Main> :main 1 7
bulb:
p0000001 = 1 % 127
p0000010 = 2 % 127

hub:
0000001p0000010 = 129 % 16256
0000001p0000100 = 131 % 16256
0000001p0001000 = 135 % 16256
0000001p0010000 = 143 % 16256
0000001p0100000 = 159 % 16256
0000001p1000000 = 191 % 16256
0000010p0000001 = 255 % 16256

*Main> :main 1 8
bulb:
p00000001 = 1 % 255
p00000010 = 2 % 255

hub:
00000001p00000010 = 257 % 65280
00000001p00000100 = 259 % 65280
00000001p00001000 = 263 % 65280
00000001p00010000 = 271 % 65280
00000001p00100000 = 287 % 65280
00000001p01000000 = 319 % 65280
00000001p10000000 = 383 % 65280
00000010p00000001 = 511 % 65280

*Main> :main 1 9
bulb:
p000000001 = 1 % 511
p000000010 = 2 % 511

hub:
000000001p000000010 = 513 % 261632
000000001p000000100 = 515 % 261632
000000001p000001000 = 519 % 261632
000000001p000010000 = 527 % 261632
000000001p000100000 = 543 % 261632
000000001p001000000 = 575 % 261632
000000001p010000000 = 639 % 261632
000000001p100000000 = 767 % 261632
000000010p000000001 = 1023 % 261632

*Main> :main 1 10
bulb:
p0000000001 = 1 % 1023
p0000000010 = 2 % 1023

hub:
0000000001p0000000010 = 1025 % 1047552
0000000001p0000000100 = 1027 % 1047552
0000000001p0000001000 = 1031 % 1047552
0000000001p0000010000 = 1039 % 1047552
0000000001p0000100000 = 1055 % 1047552
0000000001p0001000000 = 1087 % 1047552
0000000001p0010000000 = 1151 % 1047552
0000000001p0100000000 = 1279 % 1047552
0000000001p1000000000 = 1535 % 1047552
0000000010p0000000001 = 2047 % 1047552

*Main> :main 1 5
bulb:
p00001 = 1 % 31
p00010 = 2 % 31

hub:
00001p00010 = 33 % 992
00001p00100 = 35 % 992
00001p01000 = 39 % 992
00001p10000 = 47 % 992
00010p00001 = 63 % 992

*Main> :main 3 5
bulb:
p10101 = 21 % 31
p10110 = 22 % 31

hub:
10101p10110 = 673 % 992
10101p11010 = 677 % 992
10110p01011 = 693 % 992
10110p01101 = 695 % 992
10110p10101 = 703 % 992
*Main>  :main 4 5
bulb:
p11101 = 29 % 31
p11110 = 30 % 31

hub:
11101p11110 = 929 % 992
11110p01111 = 945 % 992
11110p10111 = 953 % 992
11110p11011 = 957 % 992
11110p11101 = 959 % 992
*Main>

*Main> :main 1  7
bulb:
p0000001 = 1 % 127
p0000010 = 2 % 127

hub:
0000001p0000010 = 129 % 16256
0000001p0000100 = 131 % 16256
0000001p0001000 = 135 % 16256
0000001p0010000 = 143 % 16256
0000001p0100000 = 159 % 16256
0000001p1000000 = 191 % 16256
0000010p0000001 = 255 % 16256

*Main> :main 2  7
bulb:
p0010001 = 17 % 127
p0010010 = 18 % 127

hub:
0010001p0010010 = 2177 % 16256
0010001p0100010 = 2193 % 16256
0010001p0100100 = 2195 % 16256
0010001p1000100 = 2227 % 16256
0010001p1001000 = 2231 % 16256
0010010p0001001 = 2295 % 16256
0010010p0010001 = 2303 % 16256

*Main> :main 4  7
bulb:
p1010101 = 85 % 127
p1010110 = 86 % 127

hub:
1010101p1010110 = 10881 % 16256
1010101p1011010 = 10885 % 16256
1010101p1101010 = 10901 % 16256
1010110p0101011 = 10965 % 16256
1010110p0101101 = 10967 % 16256
1010110p0110101 = 10975 % 16256
1010110p1010101 = 11007 % 16256

*Main> :main 5  7
bulb:
p1101101 = 109 % 127
p1101110 = 110 % 127

hub:
1101101p1101110 = 13953 % 16256
1101101p1110110 = 13961 % 16256
1101110p0110111 = 14025 % 16256
1101110p0111011 = 14029 % 16256
1101110p1011011 = 14061 % 16256
1101110p1011101 = 14063 % 16256
1101110p1101101 = 14079 % 16256

*Main> :main 6  7
bulb:
p1111101 = 125 % 127
p1111110 = 126 % 127

hub:
1111101p1111110 = 16001 % 16256
1111110p0111111 = 16065 % 16256
1111110p1011111 = 16097 % 16256
1111110p1101111 = 16113 % 16256
1111110p1110111 = 16121 % 16256
1111110p1111011 = 16125 % 16256
1111110p1111101 = 16127 % 16256
*Main>

:main 1 65
bulb:
p00000000000000000000000000000000000000000000000000000000000000001 = 1 % 36893488147419103231
p00000000000000000000000000000000000000000000000000000000000000010 = 2 % 36893488147419103231

hub:
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000000000010 = 36893488147419103233 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000000000100 = 36893488147419103235 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000000001000 = 36893488147419103239 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000000010000 = 36893488147419103247 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000000100000 = 36893488147419103263 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000001000000 = 36893488147419103295 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000010000000 = 36893488147419103359 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000000100000000 = 36893488147419103487 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000001000000000 = 36893488147419103743 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000010000000000 = 36893488147419104255 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000000100000000000 = 36893488147419105279 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000001000000000000 = 36893488147419107327 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000010000000000000 = 36893488147419111423 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000000100000000000000 = 36893488147419119615 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000001000000000000000 = 36893488147419135999 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000010000000000000000 = 36893488147419168767 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000000100000000000000000 = 36893488147419234303 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000001000000000000000000 = 36893488147419365375 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000010000000000000000000 = 36893488147419627519 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000000100000000000000000000 = 36893488147420151807 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000001000000000000000000000 = 36893488147421200383 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000010000000000000000000000 = 36893488147423297535 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000000100000000000000000000000 = 36893488147427491839 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000001000000000000000000000000 = 36893488147435880447 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000010000000000000000000000000 = 36893488147452657663 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000000100000000000000000000000000 = 36893488147486212095 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000001000000000000000000000000000 = 36893488147553320959 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000010000000000000000000000000000 = 36893488147687538687 % 1361129467683753853816604941579653742592
00000000000000000000000000000000000000000000000000000000000000001p00000000000000000000000000000000000100000000000000000000000000000 = 36893488147955974143 % 1361129467683753853816604941579653742592
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