Largest Islands  

Robert P. Munafo, 2010 Oct 21.



The largest islands have many useful properties: because they are large, they are easy to find, easy to compute and are most likely to have been viewed by others. They have associated with them the largest embedded Julia sets.

Because of period scaling, all islands with prime period are part of a primary filament. However, as you go down the list you see that soon the list is dominated by islands with composite periods, most of which are part of a smaller mu-unit and whose period is a product by period scaling. Because of this, the largest island of period 6 (R2.1/2F(1/2B1)S), is larger than the largest of period 5 (R2F(1/4B1)S) and the largest of period 8 (R2.1/2F(1/3B1)S) surpasses the largest of period 7 (R2F(2/5B2)S). The largest of period 15, R2.2/5F(1/2B1)S, comes in at position 20, well before the largest islands of periods 12, 13 and 14.

The largest island mu-molecules in the Mandelbrot Set are listed here, with ties counted as a single entry. For completeness, R2 itself is listed at position 0. The first 28 entries are given, followed by selected higher entries.

Rank R2-Name period Area of Island Coordinates (center @ size)
0 R2 1 1.506591 -0.286768 + 0i @ 3.684481
1 R2F(1/2B1)S 3 5.1023×10-4 -1.759672 + 0i @ 0.067687
2 R2F(1/3B1)S
and R2F(2/3B1)S
4 1.0334×10-4 -0.158428 ± 1.033350i @ 0.030476
3 R2.1/2F(1/2B1)S 6 3.7841×10-5 -1.477333 + 0i @ 0.018443
4 R2F(1/4B1)S
and R2F(3/4B1)S
5 3.4253×10-5 0.358431 ± 0.643507i @ 0.017557
5 R2F(1/2B1)FS[0]S 5 2.4560×10-5 -1.626529 + 0i @ 0.014870
6 R2F(1/3B2)S
and R2F(2/3B2)S
5 1.7627×10-5 -0.043323 ± 0.986304i @ 0.012596
7 R2F(1/5B1)S
and R2F(4/5B1)S
6 1.3700×10-5 0.442990 ± 0.373727i @ 0.011104
8 R2F(1/2(1/3B1)B1)S
and R2F(1/2(2/3B1)B1)S
5 1.2322×10-5 -1.255874 ± 0.380956i @ 0.010542
9 R2.1/2F(1/3B1)S
and R2.1/2F(2/3B1)S
8 1.1143×10-5 -1.186335 ± 0.303122i @ 0.010011
10 R2F(2/5B2)S
and R2F(3/5B2)S
7 9.9227×10-6 -0.530099 ± 0.668181i @ 0.009449
11 R2.1/3F(1/2B1)S
and R2.2/3F(1/2B1)S
9 8.0281×10-6 -0.105379 ± 0.924601i @ 0.008512
12 R2F(2/5B1)S
and R2F(3/5B1)S
6 7.0046×10-6 -0.597425 ± 0.663202i @ 0.007941
13 R2F(2/5B3)S
and R2F(3/5B3)S
8 6.2193×10-6 -0.592352 ± 0.620787i @ 0.007492
14 R2F(1/6B1)S
R2F(5/6B1)S
7 6.1069×10-6 0.432259 ± 0.227315i @ 0.007423
15 R2F(1/3B1)FS[0]S
and R2F(2/3B1)FS[0]S
7 5.6934×10-6 -0.128022 ± 0.987635i @ 0.007156
16 R2F(3/7B2)S
and R2F(4/7B2)S
9 5.5942×10-6 -0.650446 ± 0.478066i @ 0.007095
17 R2.1/2F(1/4B1)S
and R2.1/2F(3/4B1)S
10 4.9632×10-6 -1.008018 ± 0.310908i @ 0.006687
18 R2F(4/9B2)S
and R2F(5/9B2)S
11 3.1238×10-6 -0.694718 ± 0.368459i @ 0.005297
19 R2F(1/7B1)S
and R2F(6/7B1)S
8 2.9532×10-6 0.404879 ± 0.146216i @ 0.005150
20 R2.2/5F(1/2B1)S
and R2.3/5F(1/2B1)S
15 2.5839×10-6 -0.550769 ± 0.626543i @ 0.004824
21 R2F(1/4B2)S
and R2F(3/4B2)S
6 2.5020×10-6 0.396881 ± 0.604168i @ 0.004738
22 R2F(1/3(2/3B1)B1)S
and R2F(2/3(2/3B1)B1)S
7 2.4753×10-6 -0.272135 ± 0.841986i @ 0.004719
23 R2F(1/2(1/3B1)B1)FS[0]S
and R2F(1/2(2/3B1)B1)FS[0]S
7 2.4565×10-6 -1.252823 ± 0.342826i @ 0.004701
24 R2.1/2F(1/5B1)S
and R2.1/2F(4/5B1)S
12 2.4454×10-6 -0.916296 ± 0.277182i @ 0.004695
25 R2F(3/8B3)S
and R2F(5/8B3)S
11 2.4121×10-6 -0.390965 ± 0.646758i @ 0.004664
26 R2.1/4F(1/2B1)S
and R2.3/4F(1/2B1)S
12 2.4111×10-6 0.350915 ± 0.581403i @ 0.004657
27 R2.1/3F(2/3B1)S
and R2.2/3F(2/3B1)S
12 2.4052×10-6 -0.234419 ± 0.826435i @ 0.004654
28 R2F(1/4B3)S
and R2F(3/4B3)S
7 2.2678×10-6 0.386393 ± 0.569007i @ 0.004518
29 R2F(1/4B1)FS[0]S
and R2F(3/4B1)FS[0]S
9 2.0461×10-6 0.360121 ± 0.615092i @ 0.004291
30 R2F(2/7B4)S
and R2F(5/7B4)S
11 1.9481×10-6 0.123677 ± 0.656892i @ 0.004187
31 R2.1/3F(1/3B1)S
and R2.2/3F(1/3B1)S
12 1.9054×10-6 -0.006958 ± 0.806366i @ 0.004141
32 R2F(5/11B2)S
and R2F(6/11B2)S
seahorse valley sequence
13 1.7838×10-6 -0.715175 ± 0.298824i @ 0.004007
33 R2F(1/3B2)FS[0]S
and R2F(2/3B2)FS[0]S
8 1.6047×10-6 -0.074191 ± 0.970449i @ 0.003800
34 R2F(2/7B3)S
and R2F(5/7B3)S
10 1.5937×10-6 0.157283 ± 0.638086i @ 0.003787
35 R2.1/2.1/2F(1/2B1)S 12 1.5448×10-6 -1.417240 + 0i @ 0.003729
36 R2F(1/8B1)S
and R2F(7/8B1)S
9 1.5221×10-6 0.378631 ± 0.098841i @ 0.003704
37 R2F(1/2B1)FS[2]S 4 1.4635×10-6 -1.941076 + 0i @ 0.003627
. . .
50 R2F(3/8B5)S
and R2F(5/8B5)S
13 1.0423×10-6 -0.355706 ± 0.657881i @ 0.003063
57 R2F(1/9B1)S
and R2F(8/9B1)S
10 8.3044×10-7 0.356854 ± 0.069659i @ 0.002734
87 R2F(1/10B1)S
and R2F(9/10B1)S
11 4.7450×10-7 0.339454 ± 0.050823i @ 0.002070
100 R2F(1/2(1/2(1/3B1)B1)B1S
and R2F(1/2(1/2(2/3B1)B1)B1S
7 4.2820×10-7 -1.408358 ± 0.136296i @ 0.001963
122 R2F(1/11B1)S
and R2F(10/11B1)S
12 2.8280×10-7 0.325631 ± 0.038164i @ 0.001596
143 R2.1/5F(2/3B1)S
and R2.4/5F(2/3B1)S
20 2.4298×10-7 0.392053 ± 0.369074i @ 0.001478
190 R2F(1/2B1)SF(1/2B1)S 9 1.6922×10-7 -1.785953 + 0i @ 0.001233
200 R2F(2/11B1)S 12 1.5332×10-7 0.393176 ± 0.274268i @ 0.001175
500 R2F(1/5(1/3B1)B1)FS[0]S 16 4.1960×10-8 0.412200 ± 0.312748i @ 0.000614
546 R2F(1/2B1)SF(1/3B1)S 12 3.6373×10-8 -1.758846 ± 0.019014i @ 0.000572
1000 R2.4/13F(1/3B1)S 52 1.5573×10-8 0.018796 ± 0.640623i @ 0.000375
2000 R2.1/3F(5/14B3)S 51 5.6897×10-9 -0.046212 ± 0.799746i @ 0.000227
2177 R2F(1/2B1)FS[2]FS[2]S 5 5.0830×10-9 -1.985441 + 0i @ 0.000214
3163 R2F(1/2B1)FS[3]S 8 2.9698×10-9 -1.766482 ± 0.041729i @ 0.000163
5000 R2.1/3F(3/11B7)FS[0]S 75 1.5090×10-9 -0.1915867 ± 0.8151886i @ 0.0001165

A listing of the top 707 islands is here: top islands raw data (text file).

See also Enumeration of Features.


Credits

I (Robert Munafo) wrote my own software to locate the largest islands. That software uses pixel counting, and I developed the solutions to the fundamental problems of pixel-counting and statistical rigor during my earlier work on the area. That work on the Mandelbrot set area was inspired by Jay Hill.


revisions: 20080127 oldest on record; 20101021 add link to data text file and R2-names of duplicate (conjugate mirror) islands




From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.     Mu-ency index


Robert Munafo's home pages on HostMDS   © 1996-2017 Robert P. Munafo.
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