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
[edit | edit source]- 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
[edit | edit source]Principal misiurewicz point of p/q-wake is
- 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
External angles of q rays landing on
- 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 , not preperiod of critical point . 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
[edit | edit source]How to work with the shift map ?
If length of string s is q then
shifting q digits in blocks of b digits
[edit | edit source]Note that
q=5 b=1
[edit | edit source]
q=5 b=2
[edit | edit source]
q=5 b=3
[edit | edit source]
q=5 b=4
[edit | edit source]Algorithm
[edit | edit source]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 bulb ( = hyperbolic component) has 2 external angles landing on it's root point (bond) :
such that :
These angles have :
- repeating binary expansion denoted by round brackets or overline
- length of repeating ( periodic ) part is
Other names of these angles are angles of the wake.
The junction point of its hub ( principal Misiurewicz point) has external angles in increasing order
where
- s is a finite string of q binary digits = s consist of q binary digits = length(s)= q
- is the shift map
- fraction has Farey parents a/b and r/s
- b is a denominator of lower Farey parent
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
[edit | edit source]- input : 2 external angles of the wake
- output : external angles of principal Misiurewicz point ( hub)
steps
[edit | edit source]- input =
- 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 )
- both p and q are:
- if input is good then there are angles to compute
- check input
- compute 2 angles of the wake : and
- compute first 2 of q angles : and
- compute last angles
- compute Farey parents of
- compute
- ( 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
[edit | edit source]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
[edit | edit source]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 :
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.
Compare with
- is the Myrberg-Feigenbaum point c = −1.401155 with external angles = (0.412454... , 0,58755...)
1/3
[edit | edit source]The bulb ( = period 3 hyperbolic component) has 2 external angles landing on it's root point (bond) :
such that :
Principal Misiurewicz point of wake is a landing point for external angles. It is denoted by
where :
- first number denotes preperiod
- second number denotes period
Two of them one can easly compute from angles the wake :
such that :
So the problem is to compute only 1 ray.
First find Farey parents[8] of
such that :
Take denominator of smaller parent :
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
then last angle is :
So here are 5 angles (q+2) in increasing order :
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
[edit | edit source]The bulb ( = period 4 hyperbolic component) has 2 external angles landing on it's root point (bond) :
Principal Misiurewicz point of wake is a landing point for external angles.
Two of them one can easly compute from angles the wake :
So the problem is to compute only rays.
First find Farey parents of
Take denominator of lower parent :
and compute last fractions.
First find periodic parts for n :
then 2 last angles are :
So here are angles in increasing order :
2/5
[edit | edit source]The bulb ( = period 5 hyperbolic component) has 2 external angles landing on it's root point (bond) :
Farey parents of 2/5 are 1/3 and 1/2
1/3 < 2/5 < 1/2 0.333333 < 0.400000 < 0.500000
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
[edit | edit source]The wake 1/7 of main cardioid
= 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
[edit | edit source]Wake 3/7 and its principal Misiurewicz point (hub)
*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
[edit | edit source]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 )
Ok, one module loaded.
*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
[edit | edit source]
*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
[edit | edit source]*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
[edit | edit source]Haskell output
*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
[edit | edit source]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 )
Ok, modules loaded: Main.
*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
[edit | edit source]*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)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000010000010000010000010000001000001000001000000100000100000100000100000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000010000010000010000010000001000001000001000001000000100000100000100000010000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000010000010000010000010000001000001000001000001000000100000100000100000100000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000010000010000010000001000001000001000001000000100000100000100000100000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000010000010000010000010000001000001000001000000100000100000100000100000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000010000010000010000010000001000001000001000001000000100000100000100000010)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000010000010000010000010000001000001000001000001000000100000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000100000010000010000010000001000001000001000001000000100000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000100000010000010000010000010000001000001000001000000100000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0000100000100000100000100000010000010000010000010000001000001000001000001000000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000000100000100000100000010000010000010000010000001000001000001000001000000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000000100000100000100000100000010000010000010000001000001000001000001000000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000000100000100000100000100000010000010000010000010000001000001000001000000100000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000000100000100000100000100000010000010000010000010000001000001000001000001000000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000000100000100000100000010000010000010000010000001000001000001000001000000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000000100000100000100000100000010000010000010000001000001000001000001000000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000000100000100000100000100000010000010000010000010000001000001000001000000100000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000000100000100000100000100000010000010000010000010000001000001000001000001000000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000000100000100000100000010000010000010000010000001000001000001000001000000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000000100000100000100000100000010000010000010000001000001000001000001000000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000000100000100000100000100000010000010000010000010000001000001000001000000100)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000000100000100000100000100000010000010000010000010000001000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000001000000100000100000100000010000010000010000010000001000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000001000000100000100000100000100000010000010000010000001000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0001000001000001000001000000100000100000100000100000010000010000010000010000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000001000001000001000000100000100000100000100000010000010000010000010000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000001000001000001000001000000100000100000100000010000010000010000010000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000001000001000001000001000000100000100000100000100000010000010000010000001000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000001000001000001000001000000100000100000100000100000010000010000010000010000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000001000001000001000000100000100000100000100000010000010000010000010000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000001000001000001000001000000100000100000100000010000010000010000010000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000001000001000001000001000000100000100000100000100000010000010000010000001000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000001000001000001000001000000100000100000100000100000010000010000010000010000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000001000001000001000000100000100000100000100000010000010000010000010000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000001000001000001000001000000100000100000100000010000010000010000010000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000010000001000001000001000000100000100000100000100000010000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000010000001000001000001000001000000100000100000100000010000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0010000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000010000010000010000010000001000001000001000000100000100000100000100000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000010000010000010000010000001000001000001000001000000100000100000100000010000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000010000010000010000010000001000001000001000001000000100000100000100000100000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000010000010000010000001000001000001000001000000100000100000100000100000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000010000010000010000010000001000001000001000000100000100000100000100000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000010000010000010000010000001000001000001000001000000100000100000100000010000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000010000010000010000010000001000001000001000001000000100000100000100000100000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000010000010000010000001000001000001000001000000100000100000100000100000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000010000010000010000010000001000001000001000000100000100000100000100000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000010000010000010000010000001000001000001000001000000100000100000100000010000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000010000010000010000010000001000001000001000001000000100000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000100000010000010000010000001000001000001000001000000100000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000100000010000010000010000010000001000001000001000000100000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(0100000100000100000100000010000010000010000010000001000001000001000001000000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000000100000100000100000010000010000010000010000001000001000001000001000000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000000100000100000100000100000010000010000010000001000001000001000001000000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000000100000100000100000100000010000010000010000010000001000001000001000000100000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000000100000100000100000100000010000010000010000010000001000001000001000001000000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000000100000100000100000010000010000010000010000001000001000001000001000000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000000100000100000100000100000010000010000010000001000001000001000001000000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000000100000100000100000100000010000010000010000010000001000001000001000000100000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000000100000100000100000100000010000010000010000010000001000001000001000001000000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000000100000100000100000010000010000010000010000001000001000001000001000000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000000100000100000100000100000010000010000010000001000001000001000001000000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000000100000100000100000100000010000010000010000010000001000001000001000000100000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000000100000100000100000100000010000010000010000010000001000001000001000001000000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000001000000100000100000100000010000010000010000010000001000001000001000001000000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000001000000100000100000100000100000010000010000010000001000001000001000001000000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001(1000001000001000001000000100000100000100000100000010000010000010000010000001000001000001000000)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000001000001000001000000100000100000100000100000010000010000010000010000001000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000001000001000001000001000000100000100000100000010000010000010000010000001000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000001000001000001000001000000100000100000100000100000010000010000010000001000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000001000001000001000001000000100000100000100000100000010000010000010000010000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000001000001000001000000100000100000100000100000010000010000010000010000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000001000001000001000001000000100000100000100000010000010000010000010000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000001000001000001000001000000100000100000100000100000010000010000010000001000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000001000001000001000001000000100000100000100000100000010000010000010000010000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000001000001000001000000100000100000100000100000010000010000010000010000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000001000001000001000001000000100000100000100000010000010000010000010000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000001000001000001000001000000100000100000100000100000010000010000010000001000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000001000001000001000001000000100000100000100000100000010000010000010000010000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000010000001000001000001000000100000100000100000100000010000010000010000010000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000010000001000001000001000001000000100000100000100000010000010000010000010000001)
0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000010(0000010000010000010000001000001000001000001000000100000100000100000100000010000010000010000001)
First angle of the hub is:
6227271590044554501136183694529415329491604978647695361% 392318858461667547739736838930672110377831130880616169472
Code
[edit | edit source]Haskell code
[edit | edit source]-- 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 Control.Monad (forM_)
import Data.List (genericTake, genericDrop, intercalate)
import Data.Fixed (mod')
import Data.Ratio ((%), numerator, denominator)
import Numeric (readInt)
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 )
Ok, modules loaded: Main.
*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:
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*Main>
Compare with
[edit | edit source]
References
[edit | edit source]- ↑ wikipedia : Misiurewicz point
- ↑ wikipedia : Misiurewicz point
- ↑ Operating with External Arguments of Douady and Hubbard by G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, and F. Montoya
- ↑ Geometry of the Antennas in the Mandelbrot Set by RL Devaney and M Moreno-Rocha
- ↑ EXTENSIONS OF HOMEOMORPHISMS BETWEEN LIMBS OF THE MANDELBROT SET BODIL BRANNER AND NURIA FAGELLA
- ↑ Geometry of the Antennas in the Mandelbrot Set (2000) by R. L. Devaney , M. Moreno-rocha
- ↑ Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen
- ↑ finding_parents_in_the_farey_tree by Claude Heiland-Allen