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 which therefore are most easily rendered with viewers even when only ordinary machine precision is available..
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 37 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
The original inspiration for conducting a largest-islands survey was a list of large Kuiper belt objects by David Jewitt and a list of trans-Neptunian objects at Harvard-Smithsonian CfA.
I (Robert Munafo) wrote my own software to locate the largest islands, using a design that is thoroughly described in the island area survey article. That software uses pixel counting, using the same solutions (to the fundamental problems of pixel-counting and statistical rigor) that I developed during my earlier work on the Area of the Mandelbrot Set. 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; 20230223 slight clarifications
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jun 26.
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