Parameter rays of root points
of period p components







Description

on the root points of period p hyperbolic component of Mandelbrot set for fc(z)=z*z +c
land pair of rays for angles which are proper rational rational fraction of form : n / d
    where :
d = ( 2p - 1) is fraction's denominator ( odd number )
     n = integer from 0 to (d-1) is fraction's numerator
angles are measured in turns modulo full turn, it means that for example 0 = 1 mod 1

Algorithm:
; reducible angle is not always angle of lower period 
; for example 9/15 = 3/5 has period 4 
;
; Rules by Lavaures :
; no arc crosses any other arc
; arc connects 2 angles with the same period : first and second
; first angle is the smallest angles not yet connected, and second angle is the next smallest angle not yet connected



See also list of conjugate angles up to period 14


Period = 1
External rays for angles of the form : n / ( 21 - 1) = n / 1
(0/1; 1/1)
landing on the point c= 1/4 which lays on the boundary of main cardioid with internal angle = 0,
which is cusp of main cardioid ( period 1 component)
( There is only 1 component with period 1 )

"The ray that is labeled 0 and 1 has special meaning and is counted twice" ( Dominik Eberlein)

Period = 2
External rays for angles of the form : n / ( 22 - 1) = n / 3
(1/3, 2/3) landing on the point c= - 3/4 which lays on the boundary of main cardioid with internal angle = 1/2,
which is root point of period 2 component
( There is only 1 component with period 2 )

binary expansion :

External angles are periodic under doubling map.
Orbit is {1/3 , 2/3 }

Notice:
divisors(2) = {1 , 2 }
and
0/3 = 0/1
3/3 = 1/1
so rays for these angles land on the root of period 1 component

Period = 3
External rays for angles of the form : n / ( 23 - 1) = n / 7
(1/7,2/7) landing on the root point c=0.64951905283833*%i-0.125 which lays on the boundary of main cardioid with internal angle = 1/3
(3/7,4/7) landing on the point c= -1.75488 = -7/4 , cusp of mini mandelbrot set
(5/7,6/7) landing on the root point c=-0.64951905283833*%i-0.125 which lays on the boundary of main cardioid with internal angle = 2/3
landing on the root points of period 3 components.

binary expansion :

( Number of components with period three = 3 )
External angles are periodic under doubling map.
Number of periodic angles for period = 2 * Number_of_components(period) = 6
Numbers of orbits = Number_of_periodic_angles / period = 6 / 3 = 2
Orbits are :
{1/7 , 2/7 , 4/7 }
{3/7 , 6/7 , 5/7 }

Notice:
divisors(3) = {1 , 3 }
and
0/7 = 0/1
7/7 = 1/1
so rays for these angles land on the root of period 1 component


Period = 4
External rays for angles of form : n / ( 24 - 1) = n / 15
(1/15,2/15) landing on the root point c=0.5*%i+0.25 which lays on the boundary of main cardioid with internal angle = 1/4
(3/15, 4/15) which land on cusp of mini mandelbrot set with center c=-0.15652+ i * 1.03225
(6/15, 9/15) landing on the root point c= -5/4
(7/15, 8/15) which land on cusp of mini mandelbrot set
(11/15,12/15) which land on cusp of mini mandelbrot set
(13/15, 14/15)landing on the root point c=0.25-0.5*%i which lays on the boundary of main cardioid with internal angle = 3/4
landing on the root points of period 4 components.
( Number of components with period four = 6 )
External angles are periodic under doubling map.
Number of periodic angles for period = 2 * Number_of_components(period) = 12
Numbers of orbits = Number_of_periodic_angles / period = 12 / 4 = 3
Orbits are :
{1/15 ,  2/15 ,   4/15 ,   8/15 }
{3/15 ,  6/15 , 12/15 ,   9/15 }
{7/15 , 14/15 , 13/15 , 11/15 }

Notice:
divisors(4) = { 1, 2, 4 }
and 0/15 = 0/1
15/15 = 1/1
so rays for these angles land on the root of period 1 component
5/15 = 1*5/3*5 = 1/3
10/15 = 2*5/3*5 = 2/3
so these are angles of rays landing on root point of period 2 component

Period = 5
External rays for angles of form : n / ( 25 - 1)
(1/31,2/31)which land on the point c=0.32858194507446*%i+0.35676274578121 on boundary of main cardioid with internal angle p/q=1/5

(3/31,4/31); The angle 3/31= p00011 has preperiod = 0 and period = 5. The conjugate angle is 4/31 or p00100 . The kneading sequence is AAAB* and the internal address is 1-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5. It's center is c = 0.359259224758007 +0.642513737138542i and root c = 0.359938765767908 +0.641503731522510i
(5/31, 6/31)
(7/31,8/31)
(9/31,10/31) which land on the point c=0.53165675522002*%i-0.48176274578121 on boundary of main cardioid with internal angle p/q=2/5
(11/31,12/31)
(13/31,18/31)
(14/31,17/31)
(15/31,16/31)
(19/31,20/31)
(21/31,22/31)which land on the point c=-0.53165675522002*%i-0.48176274578121 on boundary of main cardioid with internal angle p/q=3/5
(23/31,24/31)
(25/31,26/31)
(27/31,28/31) ; The angle 28/31 = p11100 has preperiod = 0 and period = 5. The conjugate angle is 27/31 or p11011 . The kneading sequence is AAAB* and the internal address is 1-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5. Root c = 0.359933199059646 -0.641506835663402i ; Center c =0.359259224758007-0.642513737138542i. On the dynamic plane rays land on the point z = 0.360101472087289-0.640020857199482 i
(29/31,30/31) which land on the point on boundary of main cardioid with internal angle p/q=4/5



( Number of components with period five = 15 )
External angles are periodic under doubling map.
Number of periodic angles for period = 2 * Number_of_components(period) = 30
Numbers of orbits = Number_of_periodic_angles / period = 30 / 5 = 6
Orbits are :
{1/31 , 2/31 , 4/31 , 8/31 , 16/31}
{3/31 , 6/31 , 12/31, 24/31 , 17/31}
{5/31 , 10/31 , 20/31 , 9/31 , 18/31}
{7/31 , 14/31 , 28/31 , 25/31 , 19/31}
{11/31 , 22/31 , 13/31 , 26/31 , 21/31}
{15/31 , 30/31 , 29/31 , 27/31 , 23/31}

Notice:
divisors(5) = {1 , 5 }
and
0/31 = 0/1
31/31 = 1/1
so rays for these angles land on the root of period 1 component

Period = 6
External rays for angles of form : n / ( 26 - 1) = n / 63

(  1/63 ,   2/63 )
(  3/63 ,   4/63 )
(  5/63 ,   6/63 )
(  7/63 ,    8/63 )
(10/63 , 17/63 )
(11/63 , 12/63 )
(13/63 , 14/63 )
(15/63 , 16/63 )
(19/63 , 20/63 )

(22/63 , 25/63 )
(23/63 , 24/63 ) inside previous
On main antenna :
* (26/63 , 37/63 )
* (28/63 , 35/63 )
* (29/63 , 34/63 )
* (30/63 , 33/63 )
* (31/63 , 32/63 ) ( !)

(38/63 , 41/63)landing on the point c= 1/4* e(4*Pi*i) / 3 - 1 ( Internal angle = 2/3 of period 2 component)
(39/63 , 40/63 )

(43/63 , 44/63 )
(46/63 , 53/63 )
(47/63 , 48/63 )
(49/63 , 50/63 )
(51/63 , 52/63 )
(55/63 , 56/63 )
(57/63 , 58/63 )
(59/63 , 60/63 )
(61/63 , 62/63 )
Number of components with period six = 27
External angles are periodic under doubling map.
Number of periodic angles for period = 2 * Number_of_components(period) = 54
Numbers of orbits = Number_of_periodic_angles / period = 54 / 6 = 9
Orbits are :
( only numerators )
{ 1, 2, 4, 8,16,32}
{ 3, 6,12,24,48,33}
{ 5,10,20,40,17,34}
{ 7,14,28,56,49,35}
[11,22,44,25,50,37}
[13,26,52,41,19,38}
[15,30,60,57,51,39}
[23,46,29,58,53,43]
[31,62,61,59,55,47]

Notice:
divisors(6) = {1 , 2, 3 , 6 }
and
0/63 = 0/1
63/63 = 1/1
so rays for these angles land on the root of period 1 component
21/63 = 1/3
42/63 = 2/3
so rays for these angles land on the root of period 2 component
9/63 = 1/7
18/63 = 2/7
27/63 = 3/7
36/63 = 4/7
45/63 = 5/7
54/63 = 6/7
so rays for these angles land on the root of period 3 components


Period = 7

External rays for angles of form : n / ( 27 - 1) = n / 127
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32 ,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63, 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95 ,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120, 121,122,123,124,125,126]


Period = 8

External rays for angles of form : n / ( 28 - 1) = n / 255



Period = 9

External rays for angles of form : n / ( 29 - 1) = n / 511
( 74/511 , 81/511) land on c = - 0.3111 + 0.79111*i ( 3 to 9 period )(It is probably slight error . I will check it )



Here are listings from Maxima programs ( *.mac file )

List of Numerators of angles with given period

(%i17) for i:1 thru 10 do disp(GiveEAngleNumerators(i),",");
[0],
[1,2],
[1,2,3,4,5,6],
[1,2,3,4,6,7,8,9,11,12,13,14],
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],
[1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,19,20,22,23,24,25,26,28,29,30,31,32,33,34,35,37,38,39,40,41,43,44,46,47,48,49,50,51,52,53,55,56,57,58,59,60,61,62],
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126],
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254],
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510],
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,331,332,333,334,335,336,337,338,339,340,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,683,684,685,686,687,688,689,690,691,692,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022]


List of Numerators of angles, which period is divisor of given period

(%i20) for i:1 thru 15 do disp(GiveEAngleNumeratorsDiv(i),",");
[],
[0],
[0],
[0,5,10],
[0],
[0,9,18,21,27,36,42,45,54],
[0],
[0,17,34,51,68,85,102,119,136,153,170,187,204,221,238],
[0,73,146,219,292,365,438],
[0,33,66,99,132,165,198,231,264,297,330,341,363,396,429,462,495,528,561,594,627,660,682,693,726,759,792,825,858,891,924,957,990],
[0],
[0,65,130,195,260,273,325,390,455,520,546,585,650,715,780,819,845,910,975,1040,1092,1105,1170,1235,1300,1365,1430,1495,1560,1625,1638,1690,1755,1820,1885,1911,1950,2015,2080,2145,2184,2210,2275,2340,2405,2457,2470,2535,2600,2665,2730,2795,2860,2925,2990,3003,3055,3120,3185,3250,3276,3315,3380,3445,3510,3549,3575,3640,3705,3770,3822,3835,3900,3965,4030],
[0],
[0,129,258,387,516,645,774,903,1032,1161,1290,1419,1548,1677,1806,1935,2064,2193,2322,2451,2580,2709,2838,2967,3096,3225,3354,3483,3612,3741,3870,3999,4128,4257,4386,4515,4644,4773,4902,5031,5160,5289,5418,5461,5547,5676,5805,5934,6063,6192,6321,6450,6579,6708,6837,6966,7095,7224,7353,7482,7611,7740,7869,7998,8127,8256,8385,8514,8643,8772,8901,9030,9159,9288,9417,9546,9675,9804,9933,10062,10191,10320,10449,10578,10707,10836,10922,10965,11094,11223,11352,11481,11610,11739,11868,11997,12126,12255,12384,12513,12642,12771,12900,13029,13158,13287,13416,13545,13674,13803,13932,14061,14190,14319,14448,14577,14706,14835,14964,15093,15222,15351,15480,15609,15738,15867,15996,16125,16254],
[0,1057,2114,3171,4228,4681,5285,6342,7399,8456,9362,9513,10570,11627,12684,13741,14043,14798,15855,16912,17969,18724,19026,20083,21140,22197,23254,23405,24311,25368,26425,27482,28086,28539,29596,30653,31710]

Differences between number of all angles and number of periodic angles for given period p :

L=L2-L1

where:
p is a period under doubling map
L2 = number of all angles ( from 0/d to (d-1)/d ; d/d=1 is not counted because 0/d = d/d mod 1
L1=number of periodic angles
d=(2^p) -1 ; it is a denumerator of periodic angle for given period. It is also equal to number of all angles (L2)

(%i15) for i:1 thru 15 do disp(GiveLengthDifference(i),",");
0,1,1,3,1,9,1,15,7,33,1,75,1,129,37

for p:1 thru 15 do disp("(p=",p,",d=",2^p-1,",L=",GiveLengthDifference(p),"),");
(p=1,d=1,L=0),
(p=2,d=3,L=1),
(p=3,d=7,L=1),
(p=4,d=15,L=3),
(p=5,d=31,L=1),
(p=6,d=63,L=9),
(p=7,d=127,L=1),
(p=8,d=255,L=15),
(p=9,d=511,L=7),
(p=10,d=1023,L=33),
(p=11,d=2047,L=1),
(p=12,d=4095,L=75),
(p=13,d=8191,L=1),
(p=14,d=16383,L=129),
(p=15,d=32767,L=37),




images made with Mandel - program for DOS by Wolf Jung, version 3.5
Probably mathemathically most advanced program for drawing mandel / Fatou / Julia sets

1< enter > ; q = 2 ; e = 0/1 < enter > ; F9 ; < end >
changed to grayscale, resized and converted to jpg using IrfanView


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Autor: Adam Majewski adammaj1-at-o2-dot--pl

Feel free to e-mail me. (:-))
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