| Largest Islands |
Robert P. Munafo, 2008 Jan 27.
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.
This work is licensed under a
Creative Commons Attribution 2.5 License
.