🚀 go-pugleaf

In article <1993Dec13.194400.26337@ast.saic.com>,
   hilljr@jupiter.saic.com (Jay R. Hill) writes:
|>        A warning to DEEPZOOM users.
|>
|> Few can forecast the effects of technology on our futures as well
|> as Arthur C. Clarke.  Since he has written of at least one who
|> zoomed too deep once too often, we might want to be careful.  In
|> his 1990 novel _The Ghost from the Grand Banks_, he describes one
|> Edith Craig at the Torrington Clinic for psychological disorders,
|> who is mentaly locked in a 40 digit deep zoom into the Mandelbrot
|> Set.

In the novel Clarke also describes a small lake on the grounds of a 
wealthy Mandelmaniac which had been contoured into the shape of the 
Mandelbrot Set.  Now we don't know to what level of detail the lake
was created, but it was large enough for one to enjoy boating.  

Now let's suppose we want to build one of these lakes using bricks to
line the edge.  For starters, let's just design the contiguous interior
composed of the cardioid and tangentially attached buds.  We will not
consider the filements and detached cardioids with their buds.  Now
how many bricks will we need?  This is a question not unlike the one
Dr. Mandelbrot asked about the length of coastline around England.  

What is the fractal dimension of the contiguous Mandelbrot Set?  

Let P be the perimeter of this contiguous Mandelbrot Set lake and let
p denote the perimeter of the smallest bud to be constructed.  The 
size of p is some function of the brick size, noting that it will take
six or eight bricks minimum to line a nice looking bud.  Now let's
start by computing the perimeter of the cardioid and neighboring buds.
Then we shall sort this list by the perimeters from large to small, 
so we can ignore those smaller than whatever cutoff size we want.  We 
shall also compute the area of each component.  That will let us know
how much water we will need to fill this lake.  The list of components
will look something like this.

period    real(center)      imag(center)    perimeter        area
   1   0.00000000000000  0.00000000000000 4.0000000000 1.17809724509617
   2  -1.00000000000000  0.00000000000000 1.5707963268 0.19634954084936
   3  -0.12256116687665  0.74486176661974 0.5938968503 0.02806800174027
   3  -0.12256116687665 -0.74486176661974 0.5938968503 0.02806800174027
   4  -1.31070264133683  0.00000000000000 0.3708738371 0.01094567455674
   4   0.28227139076691  0.53006061757853 0.2764613682 0.00608217682639
   4   0.28227139076691 -0.53006061757853 0.2764613682 0.00608217682639
   5  -0.50434017544624  0.56276576145298 0.2437413687 0.00472768603336
   5  -0.50434017544624 -0.56276576145298 0.2437413687 0.00472768603336
   6  -1.13800066665096  0.24033240126210 0.1666690123 0.00221054754153
   6  -1.13800066665096 -0.24033240126210 0.1666690123 0.00221054754153
   5   0.37951358801592  0.33493230559750 0.1490826608 0.00176866021683
   5   0.37951358801592 -0.33493230559750 0.1490826608 0.00176866021683
   6  -0.11341865594944  0.86056947250157 0.1401668781 0.00156343898703
   6  -0.11341865594944 -0.86056947250157 0.1401668781 0.00156343898703
   7  -0.62243629504129  0.42487843647563 0.1307578033 0.00136058402674
   7  -0.62243629504129 -0.42487843647563 0.1307578033 0.00136058402674
   7   0.12119278610591  0.61061169221075 0.1052875813 0.00088215405359
   7   0.12119278610591 -0.61061169221075 0.1052875813 0.00088215405359
   8  -0.35910239011245  0.61735345339883 0.0963333110 0.00073848743594
   8  -0.35910239011245 -0.61735345339883 0.0963333110 0.00073848743594
   8  -0.99944238720657  0.26538753246841 0.0953172405 0.00072299127841
   8  -0.99944238720657 -0.26538753246841 0.0953172405 0.00072299127841
   6   0.38900684056977  0.21585065087082 0.0889638069 0.00062982058930
   6   0.38900684056977 -0.21585065087082 0.0889638069 0.00062982058930
   8  -1.38154748443206  0.00000000000000 0.0814869765 0.00052840454564
   8   0.32481970146546  0.56381562214033 0.0668384002 0.00035550214864
   8   0.32481970146546 -0.56381562214033 0.0668384002 0.00035550214864
   7   0.37600868184677  0.14474937132163 0.0571213948 0.00025964965170
   7   0.37600868184677 -0.14474937132163 0.0571213948 0.00025964965170
   8   0.35903106283661  0.10093487686430 0.0387604563 0.00011955504299
   8   0.35903106283661 -0.10093487686430 0.0387604563 0.00011955504299

In this table, perimeters of components whose periods greater than 3 are
approximate, good to about 6 digits.  The fractal dimension formula is

D = -ln(P/4)/ln(p/4)

Using the values in this table, when I sum perimeters up to p=0.03876046 
I get P=12.08072520.  So D = 0.23838700 and Area = 1.49206225.  Now, if 
the lake is to be about 200 meters across (unity scaled up to 100 meters) 
we will have about 1200 meters of boundary to build.  The smallest bud 
will be about 5/4 meters in diameter.

I got the impression that Clarke's lake is somewhat larger than this.
How about making it about 5 times larger or about a kilometer across.
Running the table on out to p = 0.0075 I get P = 18.4559, A = 1.50146 and
D = 0.2436.  In the limit of very small p I expect D approaches 1/4.

P = sqrt(8/sqrt(p)),  approximately.

My wife's favorite TV show is Wheel of Fortune (the one featuring Vanna).
Their theme song is something along the lines of "I'm a Wheel Watcher,
I'm a Wheel Watcher..." (funny, for the longest time I thought they were
singing "I will watch her").  Well, I prefer a different screen to watch.

Warmly,
Jay "I'm a cardioid watcher, I'm a cardioid watcher..." Hill
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}