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gussetCrimp
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« on: October 20, 2010, 01:27:18 AM » |
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Does anyone know of any study that counts the number of minibrots of at least a certain size in the Mandelbrot set?
Like there are n1 minibrots of radius greater than 1; n2 minibrots of radius greater than 1/10; n3 minibrots of radius greater than 1/100
("of radius greater than" needs to be defined properly, of course)
Is it clear enough what is and isn't a minbrot to be able to answer the question?
I would be interested to know how the series n1, n2, n3... grows. If someone can calculate it, they should submit it to the OEIS. (Or there might be a more "natural" sequence, following some other progression than powers of 1/10--this would grow quite slowly at first!)
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The Rev
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« Reply #1 on: October 20, 2010, 01:30:01 AM » |
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Interesting question. I hope someone here has an answer to it.
The Rev
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Tglad
Fractal Molossus
Posts: 703
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« Reply #2 on: October 20, 2010, 02:51:35 AM » |
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I don't know and interesting question.
Compared to the minibrots the tendrils are negligable in size, so we could actually say that the mandelbrot set approximates a cluster (a fractal distribution of separate solid parts), and the shape of the series will give it a fractal dimension.
So another way of putting it is- what is the fractal dimension of the mandelbrot set as a cluster?
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Prokofiev
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« Reply #3 on: October 20, 2010, 10:16:14 AM » |
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Interesting question, really.
Another related question would be : "Apart from size, is there a natural hierarchy among the minibrots ?".
I have no clue.
Periodicity could be a hint, but there seem to be an infinity of minibrots of periodicity 3 (all the minibrots of the main antenna ?).
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« Last Edit: October 20, 2010, 11:07:26 AM by Prokofiev »
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Sincerely,
Alexis
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mrob
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« Reply #4 on: October 21, 2010, 01:57:51 PM » |
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Does anyone know of any study that counts the number of minibrots of at least a certain size in the Mandelbrot set?
I have surveyed the whole Mandelbrot set down to sufficient detail to locate the top 4800 minibrots (which I call "islands"). Fairly extensive details, including coordinates of the top 37 islands and links to pictures for some of the more notable islands, can be found here: http://mrob.com/pub/muency/largestislands.htmlLike there are n1 minibrots of radius greater than 1; n2 minibrots of radius greater than 1/10; n3 minibrots of radius greater than 1/100
("of radius greater than" needs to be defined properly, of course) Radius is a bit difficult to define in an unambiguous and clear manner. For example, compare these two minibrots, which have been sized so as to have the same area as shown here. Imagining that they actually are the same area, which would you say has a larger radius? http://mrob.com/images/0-muency/r2f(1-15b1)s-sized.jpg http://mrob.com/images/0-muency/r2f(1-3b1)s-sized.jpgYou could try to measure across the "cardioid" part of each, but it's hard to find the edges accurately enough for it to be worth it. So ignoring radius for the moment and just using "square root of the area" as a definition of radius, the answer is pretty easy. I have measured each area to within +- 0.1 percent (that is, accurate to within one part in 1000 or better). I therefore know the "radius" to within one part in a million, for each of the largest 4800 islands (and most of the next few thousand, but there may be a few missing) based on my data: The Mandelbrot set has area 1.50659, The largest island has area 5.1023*10 -4, The 5th largest island has area 2.4560*10 -5, The 10th largest island has area 9.9227*10 -6, The 20th largest island has area 2.5839*10 -6, The 50th largest island has area 1.0423*10 -6, The 100th largest island has area 4.2820*10 -7, The 200th largest island has area 1.5332*10 -7, The 500th largest island has area 4.1960*10 -8, The 1000th largest island has area 1.5573*10 -8, The 2000th largest island has area 5.6897*10 -9, The 5000th largest island has area 1.5090*10 -9. - Robert Munafo http://mrob.com/pub/muency.html |
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Tglad
Fractal Molossus
Posts: 703
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« Reply #5 on: October 21, 2010, 03:12:29 PM » |
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Wow, this is excellent data  From that I calculate a fractal dimension of the cluster as about 1.387! I'm no expert but the dimension is ln(number of minibrots at radius)/ln(radius), so I did a ln of the data: x = ln(area): 0.40945, -7.58, -10.61, -11.52, -12.8, -13.77, -14.66, -15.69, -16.98, -17.97, -18.98, -20.31 y = ln(frequency): 0, 0.693, 1.791, 2.397, 3.044, 3.931, 4.615, 5.303, 6.216, 6.9087, 7.601, 8.517 Which gave a fairly straight line scatter plot with a gradient of about 0.693. But since we want to use radius, we scale by half the x axis, which means doubling the gradient. So it means that if you double the resolution you see 2.615 times as many minibrots. So the minibrots are 'denser' than a straight line, and sparser than an area. It is sparser than the cluster of craters on the moon/mars/venus which have dimension 2. |
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« Last Edit: October 21, 2010, 03:19:22 PM by Tglad »
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mrob
Forums Freshman
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« Reply #6 on: October 21, 2010, 04:29:42 PM » |
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So it means that if you double the resolution you see 2.615 times as many minibrots. Yes, that's about what I get. Using my data, here are some statistics going by powers of 2 in size. My program generates a "center" and a "view size" for each island. The center is the center of gravity of the island (which comes from the area measurement survey) and the size is large enough to see each island, scaled so they all appear the same size if plotted with that view size. (For example the largest island is at -1.75967 + 0.0 i and its "view size" is 0.06776; the largest period-12 island is at -0.916297 + 0.27718 i and its view size is 0.00469). Using the view size of the 4868th island (which is 0.000119) and going in steps of doubling the view size each time, here are the results: there are 0 islands larger than view size 0.122 (assuming we do not count the whole Mandelbrot set as an "island") there is 1 island larger than view size 0.0609 there are 2 islands larger than view size 0.0305 there are 4 islands larger than view size 0.0152 there are 12 islands larger than view size 0.00762 there are 32 islands larger than view size 0.00381 there are 103 islands larger than view size 0.00190 there are 260 islands larger than view size 0.000952 there are 707 islands larger than view size 0.000476 there are 1864 islands larger than view size 0.000238 there are 4868 islands larger than view size 0.000119 You can look at some of the data here: http://mrob.com/pub/mu-data/largest-islands.txt This is data from a level-10 search; each level scans a grid that is twice the resolution. Each scan identifies some islands at smaller sizes, but their size ranks are uncertain because there are always a few islands that fall through the cracks. That is why the file shows the top 1005, but the ranks numbers after number 707 are in parentheses because they are considered uncertain. I have full data for up to level 12 but the file is a bit big for my server. (Along with all the uncertain rankings, there are over 42,000 lines of data) |
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Prokofiev
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« Reply #7 on: October 21, 2010, 04:48:17 PM » |
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Welcome mrob ! Interesting results  I bookmark your website and have a look at all that when I have some time. |
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Sincerely,
Alexis
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Prokofiev
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« Reply #8 on: October 21, 2010, 05:09:33 PM » |
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Tom, What you have calculated is the exponent of the power law linking size to frequence, it is not the fractal dimension, I'm affraid. The fractal dimension of your cluster equals 2 since anyone of the minibrots has a positive area. Still, I removes nothing from the interest of that result. |
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Sincerely,
Alexis
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gussetCrimp
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« Reply #9 on: October 21, 2010, 09:53:32 PM » |
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mrob, thank you! Looking at your data at the link, and using successive powers of two to divide the area of the whole set, I find the following numbers... read the table so that n in the first column refers to the exponent of (1/2) for the area multiplier. The line that starts with 22 for example should be read to say: there are 108 islands whose area is as big as, or bigger than, 1/(2^22) times the area of the main island (R2) (and that area is 3.593408346176e-07). I kept all digits, way past the point of significance, to be sure rounding errors wouldn't accumulate.
0 1.5071847 1
1 0.75359235
2 0.376796175
3 0.1883980875
4 0.09419904375
5 0.047099521875
6 0.0235497609375
7 0.01177488046875
8 0.005887440234375
9 2.943720117188e-03
10 1.471860058594e-03
11 7.359300292969e-04
12 3.679650146484e-04 2
13 1.839825073242e-04
14 9.199125366211e-05 3
15 4.599562683105e-05
16 2.299781341553e-05 6
17 1.149890670776e-05 9
18 5.749453353882e-06 15
19 2.874726676941e-06 20
20 1.437363338470e-06 39
21 7.186816692352e-07 62
22 3.593408346176e-07 108
23 1.796704173088e-07 179
24 8.983520865440e-08 281
25 4.491760432720e-08 475
26 2.245880216360e-08 777
The OEIS sequence for this would be 1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,6,9,15,20,39,62,108,179,281,475,777...
(and this assumes mrob didn't miss any islands in his data).
The ratio between successive counts, starting from line 16, goes (rounded):
1.5
1.67..
1.33..
1.95..
1.59..
1.74..
1.66..
1.57..
1.69..
1.64..
Wouldn't it be neat if it stabilised at 1.618... ?
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« Last Edit: October 21, 2010, 09:57:12 PM by gussetCrimp »
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Tglad
Fractal Molossus
Posts: 703
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« Reply #10 on: October 22, 2010, 01:28:20 AM » |
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Prokofiev, I agree that the standard fractal dimension is and must be 2, what I calculated is the fractal dimension of minibrot size distribution.
To quote from the analogy of craters on the moon from this awesome google books link I found called 'fractals in the physical sciences'-
"N(r) proportional to r^-D. N(r) denotes the number of craters whose diameters are larger than r. This D may be regarded as the fractional dimension of crater size distribution."
The book looks really interesting, I'm going to read it after this post.
It will be interesting to know what the 1.387 actually converges to as my calculations aren't that accurate. Is it irrational? or an existing quantity?
GussetCrimp, you can get the size distribution fractal dimension D = 2*log2(your ratio). Interestingly, if your value was 1.618 then that gives D=1.388 which is very close to what I calculated and well inside the margin of error.
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« Last Edit: October 22, 2010, 02:14:33 AM by Tglad »
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mrob
Forums Freshman
Posts: 10
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« Reply #11 on: October 22, 2010, 03:09:04 AM » |
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I probably should have pointed out right away that the rank numbers only include islands on or above the real axis (because I don't think two mirror-image identical islands count as "different" for the purposes of this survey). So what I call the 2nd largest island is actually 2 islands tied for 2nd place, and the next one would be 4th place, and so on. This arrangement is shown in the table on my largest islands page (3rd link below) where I list two islands in the 2nd, 4th, 6th, 7th, 8th, etc. positions of the table (near the end I stopped doing this, out of laziness). If you're really concerned about figuring out this power law thing, then we need to adjust the rankings to count the islands below the real axis. Then the rankings in the table would read: 0, 1, 2, 4, 5, 7, 8, 10, 12, ... I can modify my software to do this if there is significant interest. Wouldn't it be neat if it stabilised at 1.618... ? Funny, I thought about the golden ratio too when I noticed that the ratio 4868/1864 = 2.6116... is close to 1+phi (which of course is phi squared). But I don't think there's anything special about "powers of 2" and so the phi thing is probably a coincidence. Back when they started trying to measure the area of the Mandelbrot set there was speculation (see reference below) that it was pi/2 = 1.5707963... which turned out to be wildly off from the value I later measured with statistical rigor (see second link), current best estimate 1.50659177+-0.0000002. The OEIS sequence for this would be 1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,6,9,15,20,39,62,108,179,281,475,777... The trouble with the OEIS sequence idea is that as we get to smaller and smaller sizes, it takes more precision to measure areas to resolve close ties. My software outputs a listing of "close pairs", pairs of islands that set a new record for having almost exactly the same size. Here is the start of that list, based on a level-9 survey, which takes about 45 minutes on an 8-core machine: rank 4 period 5 ratio 1.11 rank 9 period 8 ratio 1.11 rank 14 period 7 ratio 1.017 rank 22 period 7 ratio 1.012 rank 23 period 7 ratio 1.0066 rank 24 period 12 ratio 1.0044 rank 26 period 12 ratio 1.0015 rank 27 period 12 ratio 1.0012 rank 48 period 15 ratio 1.0012 rank 63 period 21 ratio 1.00059 rank 89 period 14 ratio 1.00038 rank 93 period 16 ratio 1.00023 rank 122 period 17 ratio 1.00011 rank 188 period 30 ratio 1.000086 This tells us that the 4th island is only 1.11 times smaller than the 3rd, and skipping down a bit, the 27th island is only 1.0012 times smaller than the 26th. Going back to the raw data, we see that #26 and #27 are at 0.350916 + 0.581401 i (area 2.4097(+-0.0023)e-6) and at -0.234417 + 0.826434 i (area 2.4067(+-0.0013)e-6). The errors are based on statistics, like the "margin of error" in a poll, so it's reasonable to expect that that the area of #26 might be 2.4097-0.0023 = 2.4074, and the other one might be 2.4067+0.0013 = 2.4080, which is enough to make them switch places. And that's just the 27th largest island. By number 93 there is a tie that is 5 times closer in ratio, and it keeps getting closer and closer. The trend of record-close ratios as a function of size-ranking is erratic, but it's faster than inverse square. Based on my Mandelbrot set area work (same link below) I found that by doing 8 times as much computation you can reduce the errors by about a factor of 2.3 (you need to double the grid resolution and also double the max number of iterations). As it stands presently, my islands surveying program takes 10 days to find and measure the top 4800 islands. - Robert References: Jay Hill, "Area of Mandelbrot Set Components and Clusters", Aug 18 1997 (republished on my web site with the author's permission): http://mrob.com/pub/math/jay-hill-2003-area-mandelbrot.html Robert Munafo, "Pixel Counting", web page (last updated Sep 23 2005) Explains the methodologies of using random sampling in a rigorous manner to measure the area of a solid region with a fractal boundary to produce estimates of the value as well as the experimental error in the measurement http://mrob.com/pub/muency/pixelcounting.html Robert Munafo, "Largest Islands", web page (last updated Jan 27 2008) http://mrob.com/pub/muency/largestislands.html |
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Prokofiev
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« Reply #12 on: October 22, 2010, 10:10:57 AM » |
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To Robert, I just had a look at your site. Very interesting ! I'll use your terminology to ask my question: You calculated Nm(n), the number of "Mu-molecules" in Mandelbrot set whose "seeds" have period n. ( http://mrob.com/pub/muency/enumerationoffeatures.html) Your result is: Sloane's sequence A006876 http://oeis.org/classic/A006876: 1, 0, 1, 3, 11, 20, 57, 108, 240, 472, 1013, 1959, 4083, 8052, 16315, 32496, 65519, 130464, 262125, 523209, 1048353, 2095084, 4194281, 8384100, 16777120, 33546216, 67108068, 134201223, 268435427, 536836484, 1073741793, ... Can you confirm these points ? 1) It represents the " number of minibrots of period n" + the "continent" (the first term), right ? . 2) You consider the entire Mandelbrot set and not the minibrots above the real axis ? 3) You consider all the minibrots and not just the ones you could survey numerically ? Thanks |
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Sincerely,
Alexis
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mrob
Forums Freshman
Posts: 10
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« Reply #13 on: October 22, 2010, 11:37:42 AM » |
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Prokofiev -- 1) It represents the "number of minibrots of period n" + the "continent" (the first term), right ? . Yes, and sorry the webpages get a bit formal, and hard to understand sometimes, but yes that's what I'm trying to count with the 1, 0, 1, 3, 11, 20, 57, ... sequence 2) You consider the entire Mandelbrot set and not the minibrots above the real axis ? Yes. If you count only the ones above the real axis it would be 0, 0, 0, 1, 4, 8, ... and if you count only the ones ON the real axis it's 1, 0, 1, 1, 3, 4, ... and if you count both on and above, but not below, it's 1, 0, 1, 2, 7, 12, ... I have never bothered to work out the formulas for those sequences and it might be a bit tricky. 3) You consider all the minibrots and not just the ones you could survey numerically ? Yes, as explained on that page, all of the sequences can be derived from abstract principles without looking too hard at any actual images. You do need to understand the Farey addition principle, multiplicity of roots, and a couple other algebra concepts. You definitely do not need to use root-finding algorithms (which would give you the coordinates of the minibrots and mu-atoms but no easy way to sort out which is which) or a pixel-counting survey (which would not even work, because many minibrots of a given period are far too small to find that way, for periods greater than 6 or 7). I will try to update the page to make that more clear. http://mrob.com/pub/muency/enumerationoffeatures.html |
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Tglad
Fractal Molossus
Posts: 703
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« Reply #14 on: October 23, 2010, 02:05:40 AM » |
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Hi Robert, great data "If you're really concerned about figuring out this power law thing, then we need to adjust the rankings to count the islands below the real axis" Of course the fractal dimension isn't affected by doubling the count, and maybe concerned isn't the right word, but I do think the fractal dimension of minibrot size distribution is an interesting property to know, it might even be the same value for all multibrots, who knows. With the 'fractal dimension', I almost forget to measure the size or coefficient (which does require using the whole plane), so I get roughly: frequency of minibrots larger than radius r = 0.00766*r -1.3871.387 being the dimension... actually... frequencies have negative dimension (per second, per metre etc), so possibly better to say the dimension is -1.387. |
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« Last Edit: October 23, 2010, 04:47:00 AM by Tglad »
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