Fractals/Mathematics/sequences

Difference between sequences, orders and seriesEdit

types of sequencesEdit

Integer sequencesEdit

Fraction sequencesEdit

Farey sequenceEdit

The Farey sequence of order n is the sequence of completely reduced vulgar fractions between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 (= 0 / 1 ) and 1 (= 1 / 1 ), by taking successive mediants.

Each Farey sequence starts with the value 0, denoted by the fraction 01, and ends with the value 1, denoted by the fraction 11 (although some authors omit these terms).

Farey Addition = the mediant of two fractions :

 

Terms

  • next term = child
  • Previous terms = parents[1]

Farey tree = Farey sequence as a tree

Sorted
 F1 = {0/1,                                                                                                          1/1}
 F2 = {0/1,                                                   1/2,                                                   1/1}
 F3 = {0/1,                               1/3,                1/2,                2/3,                               1/1}
 F4 = {0/1,                     1/4,      1/3,                1/2,                2/3,      3/4,                     1/1}
 F5 = {0/1,                1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5,                1/1}
 F6 = {0/1,           1/6, 1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5, 5/6,           1/1}
 F7 = {0/1,      1/7, 1/6, 1/5, 1/4, 2/7, 1/3,      2/5, 3/7, 1/2, 4/7, 3/5,      2/3, 5/7, 3/4, 4/5, 5/6, 6/7,      1/1}
 F8 = {0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1}

Sequences and orders on the parameter planeEdit

Sequences of Misiurewicz pointsEdit


nEdit

take the Misiurewicz point for Z^n+c and increase n ( propososed by Owen Maresh)

The constants for the quadratic, cubic, and quartic are:

  • (-0.7432918908524301520519705530861564778806 ,0.1312405523087976002753516038253522297699);
  • -0.0649150006787816892861875745218343125883 , 0.76821968591243610206311097043854440463 );
  • (-0.593611822136354943067129147813253628530 ,0.5405019391915187246754930586066158919613 );

Period doubling scenarioEdit

Sharkovsky orderingEdit

It is the infinite sequence of positive integers ( natural numbers). It starts from 3 and ends in 1. It contains infinitely many subsequences.[2]

"The Sharkovski ordering :

  • begins with the odd numbers >= 3,
  • then twice these numbers,
  • then 4 times them,
  • then 8 times them,
  • etc.,
  • ending with the powers of 2 in decreasing order, ending with 2^0 = 1."[3]


 

It is related with structure of the real slice of the Mandelbrot set ( along real exis):

  • chaotic region, which consits of chaotic bands  :
    •  
    •  
    •  
    •  
    •  
  • MF = Myrberg-Feigenbaum point
  • periodic region ( period doubling cascade = 2^n )

sequence of fraction in the elephant valleyEdit

In the elephant valley[4][5] ( from parameter plane ) there is a sequence of componts with period p : from 1/2 to 1/p

Note that :

  • internal ray 0/1 = 1/1
  • internal angle 1/p means that ray goes from period 1 component ( parent period = 1) to period p component ( child period = p)
  • as child period grows computations are harder
  • exponential growth[6] of  . One can easly create a numeric value that is too large to be represented within the available storage space ( integer overflow[7] ). For example   is to big for short ( 16 bit ) and long ( 32 bit) integer.

The upper principal sequence of rotational number around the main cardioid of Mandelbrot set[8]

n rotation number = 1/n parameter c
2 1/2 -0.75
3 1/3 0.64951905283833*i-0.125
4 1/4 0.5*i+0.25
5 1/5 0.32858194507446*i+0.35676274578121
6 1/6 0.21650635094611*i+0.375
7 1/7 0.14718376318856*i+0.36737513441845
8 1/8 0.10355339059327*i+0.35355339059327
9 1/9 0.075191866590218*i+0.33961017714276
10 1/10 0.056128497072448*i+0.32725424859374

See :

sequence of parabolic points on the boundary of main cardioidEdit

 

Here:

  • t = internal angle ( or rotation number) of main cardioid
  • q = number of the critical orbit (star) arms. It means that one have to do q iterations around fixed point to move one point toward fixed point along arm.
  • c is a root point between hyperbolic components of period 1 ( = main cardioid) and period q. This point is at the end ( radius = 1) of internal ray for angle t
k = log10(q)   (double) t  
1 3/10 0.3 +0.047745751406263+0.622474571220695 i
2 33/100 0.33 -0.106920138306109 +0.649235321397436 i
3 333/1000 0.333 -0.123186752260805 +0.649516204880454 i
4 3333/10000 0.3333 -0.124818625550005 +0.649519024348384 i
5 33333/100000 0.33333 -0.124981862061192 +0.649519052553419 i
6 333333/1000000 0.333333 -0.124998186201184 +0.649519052835480 i
7 3333333/10000000 0.3333333 -0.124999818620069 +0.649519052838300 i
8 33333333/100000000 0.33333333 -0.124999981862006 +0.649519052838329 i
9 333333333/1000000000 0.333333333 -0.124999998186201 +0.649519052838329 i
10 3333333333/10000000000 0.3333333333 -0.124999999818620 +0.649519052838329 i

sequence from Siegel disk to Leau-Fatou flowerEdit

  • plain Siegel disk
  • digitated Siegel disk[9]
  • virtual Siegel disk
  • ? Leau-Fatou flower ?

1 over 2Edit

1 over 3Edit

 

n t  
0 0.2763932022500210 +0.1538380639536641 + 0.5745454151066985 i
1 0.3231874668087892 -0.0703924965263780 + 0.6469145331346999 i
2 0.3322326933513446 -0.1190170769366243 + 0.6494880316361160 i
3 0.3332223278292314 -0.1243960357918422 + 0.6495187369145560 i
4 0.3333222232791965 -0.1249395463818515 + 0.6495190496732967 i
5 0.3333322222327929 -0.1249939540657307 + 0.6495190528066729 i
6 0.3333332222223279 -0.1249993954008480 + 0.6495190528380124 i
7 0.3333333222222233 -0.1249999395400276 + 0.6495190528383258 i
8 0.3333333322222222 -0.1249999939540022 + 0.6495190528383290 i
9 0.3333333332222223 -0.1249999993954002 + 0.6495190528383290 i
10 0.3333333333222222 -0.1249999999395400 + 0.6495190528383290 i
11 0.3333333333322222 -0.1249999999939540 + 0.6495190528383290 i

sequence of fractions tending to the golden mean ( Golden Ratio Conjugate )Edit

 
Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers
 
Golden Mean Quadratic Siegel Disc
n =   1 ;  p_n/q_n =  1.0000000000000000000 =                     1 /                    1 
n =   2 ;  p_n/q_n =  0.5000000000000000000 =                     1 /                    2 
n =   3 ;  p_n/q_n =  0.6666666666666666667 =                     2 /                    3 
n =   4 ;  p_n/q_n =  0.6000000000000000000 =                     3 /                    5 
n =   5 ;  p_n/q_n =  0.6250000000000000000 =                     5 /                    8 
n =   6 ;  p_n/q_n =  0.6153846153846153846 =                     8 /                   13 
n =   7 ;  p_n/q_n =  0.6190476190476190476 =                    13 /                   21 
n =   8 ;  p_n/q_n =  0.6176470588235294118 =                    21 /                   34 
n =   9 ;  p_n/q_n =  0.6181818181818181818 =                    34 /                   55 
n =  10 ;  p_n/q_n =  0.6179775280898876404 =                    55 /                   89 
n =  11 ;  p_n/q_n =  0.6180555555555555556 =                    89 /                  144 
n =  12 ;  p_n/q_n =  0.6180257510729613734 =                   144 /                  233 
n =  13 ;  p_n/q_n =  0.6180371352785145888 =                   233 /                  377 
n =  14 ;  p_n/q_n =  0.6180327868852459016 =                   377 /                  610 
n =  15 ;  p_n/q_n =  0.6180344478216818642 =                   610 /                  987 
n =  16 ;  p_n/q_n =  0.6180338134001252348 =                   987 /                 1597 
n =  17 ;  p_n/q_n =  0.6180340557275541796 =                  1597 /                 2584 
n =  18 ;  p_n/q_n =  0.6180339631667065295 =                  2584 /                 4181 
n =  19 ;  p_n/q_n =  0.6180339985218033999 =                  4181 /                 6765 
n =  20 ;  p_n/q_n =  0.6180339850173579390 =                  6765 /                10946 
n =  21 ;  p_n/q_n =  0.6180339901755970865 =                 10946 /                17711 
n =  22 ;  p_n/q_n =  0.6180339882053250515 =                 17711 /                28657 
n =  23 ;  p_n/q_n =  0.6180339889579020014 =                 28657 /                46368 
n =  24 ;  p_n/q_n =  0.6180339886704431856 =                 46368 /                75025 
n =  25 ;  p_n/q_n =  0.6180339887802426829 =                 75025 /               121393 
n =  26 ;  p_n/q_n =  0.6180339887383030068 =                121393 /               196418 
n =  27 ;  p_n/q_n =  0.6180339887543225376 =                196418 /               317811 
n =  28 ;  p_n/q_n =  0.6180339887482036214 =                317811 /               514229 
n =  29 ;  p_n/q_n =  0.6180339887505408394 =                514229 /               832040 
n =  30 ;  p_n/q_n =  0.6180339887496481015 =                832040 /              1346269 
n =  31 ;  p_n/q_n =  0.6180339887499890970 =               1346269 /              2178309 
n =  32 ;  p_n/q_n =  0.6180339887498588484 =               2178309 /              3524578 
n =  33 ;  p_n/q_n =  0.6180339887499085989 =               3524578 /              5702887 
n =  34 ;  p_n/q_n =  0.6180339887498895959 =               5702887 /              9227465 
n =  35 ;  p_n/q_n =  0.6180339887498968544 =               9227465 /             14930352 
n =  36 ;  p_n/q_n =  0.6180339887498940819 =              14930352 /             24157817 
n =  37 ;  p_n/q_n =  0.6180339887498951409 =              24157817 /             39088169 
n =  38 ;  p_n/q_n =  0.6180339887498947364 =              39088169 /             63245986 
n =  39 ;  p_n/q_n =  0.6180339887498948909 =              63245986 /            102334155 
n =  40 ;  p_n/q_n =  0.6180339887498948319 =             102334155 /            165580141 
n =  41 ;  p_n/q_n =  0.6180339887498948544 =             165580141 /            267914296 
n =  42 ;  p_n/q_n =  0.6180339887498948458 =             267914296 /            433494437 
n =  43 ;  p_n/q_n =  0.6180339887498948491 =             433494437 /            701408733 
n =  44 ;  p_n/q_n =  0.6180339887498948479 =             701408733 /           1134903170 
n =  45 ;  p_n/q_n =  0.6180339887498948483 =            1134903170 /           1836311903 

This is a sequence of rational numbers ( Julia sets are parabolic). It's limit is an irrational number ( Julia set has a Siegel disk).

Sequence on the dynamic planeEdit

MoreEdit

ReferencesEdit

  1. Finding parents in the Farey tree by Claude Heiland-Allen
  2. Sharkovskii's theorem in wikipedia
  3. The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1
  4. muency : elephant valley
  5. Visual Guide To Patterns In The Mandelbrot Set by Miqel
  6. integer number in wikipedia
  7. Integer overflow in wikipedia
  8. Mandel Set Combinatorics : Principal Series
  9. scholarpedia : Siegel_disks , Quadratic_Siegel_disks, Digitation