Largest Islands
Robert P. Munafo, 2008 Jan 27.
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 |
| 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 | 5 | 1.7627×10-5 | -0.043323 + 0.986304i @ 0.012596 |
| 7 | R2F(1/5B1)S | 6 | 1.3700×10-5 | 0.442990 + 0.373727i @ 0.011104 |
| 8 | R2F(1/2(1/3B1)B1)S | 5 | 1.2322×10-5 | -1.255874 + 0.380956i @ 0.010542 |
| 9 | R2.1/2F(1/3B1)S | 8 | 1.1143×10-5 | -1.186335 + 0.303122i @ 0.010011 |
| 10 | R2F(2/5B2)S | 7 | 9.9227×10-6 | -0.530099 + 0.668181i @ 0.009449 |
| 11 | R2.1/3F(1/2B1)S | 9 | 8.0281×10-6 | -0.105379 + 0.924601i @ 0.008512 |
| 12 | R2F(2/5B1)S | 6 | 7.0046×10-6 | -0.597425 + 0.663202i @ 0.007941 |
| 13 | R2F(2/5B3)S | 8 | 6.2193×10-6 | -0.592352 + 0.620787i @ 0.007492 |
| 14 | R2F(1/6B1)S | 7 | 6.1069×10-6 | 0.432259 + 0.227315i @ 0.007423 |
| 15 | R2F(1/3B1)FS[0]S | 7 | 5.6934×10-6 | -0.128022 + 0.987635i @ 0.007156 |
| 16 | R2F(3/7B2)S | 9 | 5.5942×10-6 | -0.650446 + 0.478066i @ 0.007095 |
| 17 | R2.1/2F(1/4B1)S | 10 | 4.9632×10-6 | -1.008018 + 0.310908i @ 0.006687 |
| 18 | R2F(4/9B2)S | 11 | 3.1238×10-6 | -0.694718 + 0.368459i @ 0.005297 |
| 19 | R2F(1/7B1)S | 8 | 2.9532×10-6 | 0.404879 + 0.146216i @ 0.005150 |
| 20 | R2.2/5F(1/2B1)S | 15 | 2.5839×10-6 | -0.550769 + 0.626543i @ 0.004824 |
| 21 | R2F(1/4B2)S | 6 | 2.5020×10-6 | 0.396881 + 0.604168i @ 0.004738 |
| 22 | R2F(1/3(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 | 7 | 2.4565×10-6 | -1.252823 + 0.342826i @ 0.004701 |
| 24 | R2.1/2F(1/5B1)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 | 12 | 2.4111×10-6 | 0.350915 + 0.581403i @ 0.004657 |
| 27 | R2.1/3F(2/3B1)S | 12 | 2.4052×10-6 | -0.234419 + 0.826435i @ 0.004654 |
| 28 | R2F(1/4B3)S | 7 | 2.2678×10-6 | 0.386393 + 0.569007i @ 0.004518 |
| 29 | R2F(1/4B1)FS[0]S | 9 | 2.0461×10-6 | 0.360121 + 0.615092i @ 0.004291 |
| 30 | R2F(2/7B4)S | 11 | 1.9481×10-6 | 0.123677 + 0.656892i @ 0.004187 |
| 31 | R2.1/3F(1/3B1)S | 12 | 1.9054×10-6 | -0.006958 + 0.806366i @ 0.004141 |
| 32 | R2F(5/11B2)S seahorse valley sequence | 13 | 1.7838×10-6 | -0.715175 + 0.298824i @ 0.004007 |
| 33 | R2F(1/3B2)FS[0]S | 8 | 1.6047×10-6 | -0.074191 + 0.970449i @ 0.003800 |
| 34 | R2F(2/7B3)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 | 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 | 13 | 1.0423×10-6 | -0.355706 + 0.657881i @ 0.003063 |
| 57 | R2F(1/9B1)S | 10 | 8.3044×10-7 | 0.356854 + 0.069659i @ 0.002734 |
| 87 | R2F(1/10B1)S | 11 | 4.7450×10-7 | 0.339454 + 0.050823i @ 0.002070 |
| 100 | R2F(1/2(1/2(1/3B1)B1)B1S | 7 | 4.2820×10-7 | -1.408358 + 0.136296i @ 0.001963 |
| 122 | R2F(1/11B1)S | 12 | 2.8280×10-7 | 0.325631 + 0.038164i @ 0.001596 |
| 143 | R2.1/5F(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 |
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 during my earlier work on the area. The search for the area was inspired by Jay Hill.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index
Robert Munafo's home pages on HostMDS (c) 1996-2010 Robert P. Munafo. about contact
This work is licensed under a Creative Commons Attribution 2.5
License. Details here
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