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# Area History

Robert P. Munafo, 2012 Aug 18.

Following is a history of most of the events regarding the effort to determine the area of the Mandelbrot set:

### 1988

In 1988, Tord Malmgren in his high school thesis used a Commodore-64 to estimate the area of the Mandelbrot Set with pixel counting and got 1.57. He also used an approach similar to that of Jay Hill (see the Area page) and got 1.71. These are currently the earliest known estimates of the Mandelbrot Set's area.

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### 1990

In 1990, the Laurent series technique was discussed on alt.fractals by Gerald Edgar (edgar at math ohio-state edu) (who posted an area estimate of 2.089 based on 72 terms) and Yuval Fisher (who pointed out that 256 terms give a much lower value). Yuval Fisher also described how to compute coefficients in the area series.

On 1990 Dec 7, Scott Huddleston (scott at ferrari labs tek com) or (scott at crl labs tek com) posted a message to alt.fractals giving his results of carrying the Laurent Series out to 115,232 terms, and a best estimate of 1.73847. His table of estimates is given on the Laurent series page.

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### 1991

In 1991 Feb (23 or 24), John H. Ewing gave a talk at an MAA meeting describing a precise way to measure the area, the Laurent series technique. It takes an extraordinary number of terms to get even a few digits of accuracy. Using 240,000 terms, they got an estimate of 1.7274, and speculated that it was converging to a limit between 1.66 and 1.71. However, the rate of convergence is very slow, and "slows down" in such a way that traditional exponential or hyperbolic curve fits do not accurately model its shape.

Ewing also noted that pixel counting method gives a value of about 1.52, and stated that he was uncertain for the reason for the discrepancy. Ewing gave two possile explanations: either the pixel-counting method is missing the area of the fine detail features near the boundary of the Mandelbrot set, or the higher estimate (an estimate of the asymptote of the Laurent series calculation) is wrong.

Ewing's talk was reported to sci.fractals on 1991 Feb 25 by Stan Isaacs (isaacs at hpcc01 hp com).

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### 1992

Early in 1992, John H. Ewing and G. Schober published an article in the journal "Numer. Math." %% an abbreviation for what? (volume 61, pages 59-72) entitled "The area of the Mandelbrot set", presenting the results presented by Ewing in 1991 Feb. This article was mentioned on sci.fractals on 1992 Mar 10 by Jeffrey Shallit (shallit at graceland waterloo edu)

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On 1992 Sep 18, Keith Briggs (kbriggs at mundoe maths mu oz au) reported his estimate of the area using the conformal map technique, 1.499936. He conjectured that the area was exactly 3/2.

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### 1993

I (Robert Munafo) wrote the following in 1993, submitting it to sci.fractals (Yes, there is an error in the first sentence, the words "real" and "imaginary" should have been swapped)

The rectangle ranging over (-2.0,0.5) in the imaginary axis and (0.0, 1.125) in the real axis covers the top half of the Mandelbrot Set. This rectangle is divided up into WIDTH rows of HEIGHT rectangles each, and the midpoint of the left edge is used as a representative point. With each successive approximation, the width, height, and DWELL LIMIT are doubled:    dwell pixels area estimated width height limit in set estimate error 16 8 64 35 1.538 0.195 32 16 128 140 1.5380 0.0920 64 32 256 553 1.5188 0.0411 128 64 512 2195 1.5071 0.0187 256 128 1024 8804 1.51130 0.00855 512 256 2048 35144 1.50821 0.00382 1024 512 4096 140466 1.50703 0.00170 2048 1024 8192 561804 1.506875 0.000746 4096 2048 16384 2246985 1.506720 0.000327 8192 4096 32768 8987647 1.506671 0.000142 16384 8192 65536 35949947 1.5066449 0.0000616 32768 16384 131072 143796687 1.5066123 0.0000265 65536 32768 262144 575183466 1.5066037 0.0000114    The area appears to converge to a value somewhere near 1.50659. This limit has been confirmed by Jay R. Hill, who using a similar method arrived at an estimate of 1.5067 +- 0.00015

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Here is Jay Hill's pixel-counting estimate from 1993:

I have run a sequence of estimates on the positive half plane (-2,0)-(0.5,1.125), doubling the iteration limit and pixel count both in x and y. The results approach 1.5067.    nx ny iter area pixels 8 4 32 1.77369231916047618 10 16 8 64 1.57029576987507275 37 32 16 128 1.51345811319519949 138 64 32 256 1.50957245193378302 551 128 64 512 1.51248993398036579 2204 ( 256 128 1024 no data ) 512 256 2048 1.50804182611525017 35156 1024 512 4096 1.50711795556483235 140506 2048 1024 8192 1.50692766993201912 561901 4096 2048 16384 1.50683686137199402 2247158

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On 1993 Mar 15, Jay Hill posted his results of measuring the area by enumerating all mu-atoms and using an analytic numerical method to measure the area of each one indivdually, arriving at values of 1.504106 and 1.507818, neither of which he considered too reliable, because they were based on Milnor's formula r = sin(pi i/p)/(p2) for the radius of a mu-atom rather than actual measurements.

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On 1993 Apr 13, I posted to sci.fractals giving the estimate 1.506595 +- 0.000002, based on an average of 40 runs using a dwell limit of 524288 and period detection. It was the first result that measured the sampling error by taking many runs at the same resolution and dwell limit and calculating the standard error. Despite this, there was still an additional error of 0.0000056 due to the dwell limit (524288 is way too low)

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### 1994?

Later Jay Hill developed a significantly more reliable method, still based on the idea of enumerating the mu-atoms but actually measuring the size of each one once found, rather than using Milnor's formula.

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### 1996

In the book Fractal Horizons, Jay Hill reports the results of his calculations giving a lower bound on the area of 1.50585063, based on locating and measuring %% mu-atoms.

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On 1996 Dec 23, I posted new results to sci.fractals. It is an average of 20 runs on 65536 grid with a dwell limit of 16777216. The area estimate is 1.5065923 with a standard error of +- 0.0000006.

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### 1997

On 1997 Aug 18, Jay Hill placed updated area estimate results on his web page. His new lower bound is 1.506303622, based on measurement of 430809 mu-atoms.

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### 1998

On 1998 March 4th, I reported the estimates 1.50659173 +- 0.00000020 for the Mandelbrot Set's area and -0.28676832 +- 0.00000010 for its center-of-gravity, in the newsgroup sci.fractals. I also said that I thought I had found a formula for the center-of-gravity, -((ln(3) - 1/3)F), using the Inverse Symbolic Calculator (http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html).

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### 1999

On 1999 Oct 21, I obtained better estimates of the area and center-of-gravity: 1.50659177 +- 0.00000008 and -0.28676844 +- 0.000000025. I did not post to a newsgroup but updated these web pages. The margin of error for the center-of-gravity makes it appear that the formula -((ln(3) - 1/3)F) is not valid, so I removed the formula from the web page (later adding it back here in the history section).

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### 2000

On 2000 Feb 3, Cyril Soler reported the formula sqrt(6 . pi - 1) - e to me, and I added it to this web page on Feb 4.

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### 2001

Kerry Mitchell uses statistical analysis (by an independent approach and methodology from above) to arrive at a value of 1.506484±0.000004. His work is reported here: A Statistical Investigation of the Area of the Mandelbrot Set and mirrored here.

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### 2003

On Sep 25, I begin a measurement of 20 grids at gridsize 524288

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### 2004

On Sep 3, the area estimate begun almost a year earlier has completed, however the data are useless because the number of counted pixels in the 524288×1048576 grids overflowed the variable used to compute that total (which was a 32-bit integer).

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### 2010

As of Oct 20 2010, I have begun a new area survey after fixing the bugs from 2004. (This survey is suspended before reaching the 524288×1048576 grid size)

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### 2012

In early 2012 I continues the new area survey mentioned in 2010, and reproduced the calculation from 2004, this time without overflow:

gridsize itmax area center of gravity 25 12800 1.4895(254) -0.29136(717) 50 25600 1.50389(555) -0.28676(273) 100 51200 1.50404(283) -0.286789(849) 200 102400 1.506973(834) -0.286372(344) 400 204800 1.506685(353) -0.286790(119) 800 409600 1.506704(139) -0.2867344(577) 1600 819200 1.5066042(592) -0.2867648(244) 3200 1638400 1.5065813(195) -0.28676480(719) 6400 3276800 1.50658583(670) -0.28676542(320) 12800 6553600 1.50658975(254) -0.28676929(108) 25600 13107200 1.50659165(116) -0.286768307(532) 51200 26214400 1.506592210(521) -0.286768176(129) 102400 52428800 1.506591964(222) -0.2867683969(464) 204800 104857600 1.5065919785(936) -0.2867683743(266) 409600 209715200 1.5065918561(254) -0.2867684229(111)

The last estimate in the table, 1.5065918561(254), is:

1.506 591 856 1 ± 0.000 000 025 4

For details of my area estimate technique with illustrations of the grids, etc., see the pixel counting article.

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Separately, Thorsten Forstemann used twin Radeon HD  5970 GPUs to do a similar pixel counting survey up to a grid size of 2097152×2097152. The resulting area estimate is 1.5065918849(28), which is:

1.506 591 884 9 ± 0.000 000 002 8

There is a link to his webpage from the Thorsten Forstemann entry.

revisions: 20000228 oldest on record; 20120223 add recent results; 20120818 add FĂ¶rstemann results

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2013 Jan 21. s.27