area of the middle Riddle Mandelbrot set

[Roger Bagula]

The theory of dimension two sets of the Mandelbrot set type is new.

http://www.research.att.com/~njas/sequences/100K/riddle_Mand.jpg
h(z,c)=c*z2/8+7*z2/8+7*c/8+1/8
Which is made from the Bezier of the Mandelbrot:
m(z,c)=z2+c
and the Fiegenbaum set:
f(z,c)=1+c*z2
h(z,c)=(1-t)*m(z,c)+t*f(z,c)
at t=1/8
The science fractals newsgroup post is found at;
http://www.archivum.info/sci.fractals/2006-03/00134/a_third_quadratic_like_the_Mandelbrot_set_:_the_middle_Riddle_set
There are two questions connected with this:
1) is the area of the Feigenbaum set near or the same as the
Mandelbrot set.
2) Is the area of the middle Riddle set equal to: ( conjectured)
Area_MR=A_Mandelbrot+A_Fiegenbaum

Roger Bagula

[Roger Bagula]

Two new views:
http://www.flickr.com/photos/fractalmusic/4061950155/
http://www.flickr.com/photos/fractalmusic/4061950023/

--
Respectfully, Roger L. Bagula
11759 Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
http://www.google.com/profiles/Roger.Bagula
alternative email: roger....@gmail.com

[Roger Bagula]

The problem with this this middle Riddle Mandelbrot
it that the set isn't entirely connected
in any of the plots that I have done.
Here is the working area by Pixel counting in Mathematica:
Clear[f, x, nz, max0]
(*3D middle Riddle Mandelbrot*)
(*Mandelbrot with x^2+y^2 limited measure*)
(*by R. L. BAGULA 2 Nov 2009 © *)

Clear[n]
numberOfz2ToEscape[z_] := Block[
{escapeCount, nz = N[z],nzold=0},
For[
escapeCount = 0,
((Re[nz]^2+Im[nz]^2 < 128) && (escapeCount < 511)),
nzold=nz;
nz = (7/8)*(nz^2 + z) + (1/8)*(z*nz^2 + 1);
++escapeCount
];
escapeCount
]
FractalPureM[{{ReMin_, ReMax_, ReSteps_},
{ImMin_, ImMax_, ImSteps_}}] :=
Table[
numberOfz2ToEscape[x + y I],
{y, ImMin, ImMax, (ImMax - ImMin)/ImSteps},
{x, ReMin, ReMax, (ReMax - ReMin)/ReSteps}
]
d = Length[arraym]
area = Sum[Sum[If[arraym[[n,
m]] - max0 == 0, 1, 0], {n, 1, d}], {m, 1, d}]*9^2/d^2
N[%]
8.12302715420057
gr=ListPlot3D[arraym, Mesh -> False,AspectRatio -> Automatic,Boxed-
>False, Axes->False,ViewPoint->{-0.884, -1.543, 2.879}];

What I found was that a Bezier of three sets:
(Mandelbrot set , zero Julia in the middle and Feigenbaum set )
nz = (1 - t)^2*(nz^2 + z) + t^2*(z*nz^2 + 1)+2*t*(1-t)*nz^2
gives a better Ovals of Cassini type of kissing set of
two Mandelbrot like bulbs at:
t=3039/8192
with an area of:
11.8201 ( at 300 by 300 I get 11.6651)
In both cases the area is much larger than quoted on several web sites
on the web.
https://www.fractalus.com/kerry/articles/area/mandelbrot-area.html
1.506484193
The triple Bezier is just mot as pretty as a fractal.
http://www.flickr.com/photos/fractalmusic/4067935729/