FOTD -- June 30, 2009 (Rating 8)
Fractal visionaries and enthusiasts:
Today's image is a puzzle. Everyone knows that Mandelbrot-type
minibrots are impossible in Julia sets, so how do we explain the
obvious midget M-set in today's image, the parent of which is a
Julia set. To see this Julia set, reduce the magnitude to 0.5
and recalculate the image. The Seahorse-Valley Julia set that
soon fills the screen is obvious.
Well, two things need to be explained. To begin, the minibrot
is somewhat warped. Its two period-3 buds as well as its main
stem are missing. And in addition, the parent Julia set is not
quite a Julia set. It is double-rotated 1/1000 of one degree
from the true Julia orientation. This rotation is extremely
small -- the width of one centimeter as seen from a distance of
over 1/2 kilometer -- but it is enough to reveal the Mandelbrot
shape of the minibrot. To see the minibrot magically turn into
the Julia set of the center of the main Mandelbrot bay, change
real(p1) and real(p2) to 90. Yes, 1/1000 of one degree can make
this much of a difference.
Other things of interest are the four partially-filled peanut
holes and the breakdown of the 2,4,8... series of elements
surrounding the minibrot. This breakdown is no surprise, since
it often appears in warped minibrots such as the minibrot in
today's image. The unusual texture of the image was achieved by
rendering it with the outside set to 'summ'.
I rated the image at an 8. I think it has enough interest to
earn such a rating. The calculation time of just over 9 minutes
is rather slow, but nothing to get excited about, especially
since the image may be viewed on the FOTD web site without the
need to calculate it. The web site may be accessed at:
<
http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
Picture perfect weather prevailed here at Fractal Central on
Monday -- at least the fractal cats thought so. They thoroughly
enjoyed the blue skies, puffy white clouds, low humidity and
temperature of 84F 29C.
>From indoors I half enjoyed the outside conditions. Meanwhile,
FL, having asked a question on Sunday that I had no immediate
answer for, enjoyed an afternoon in her garden, keeping a sharp
lookout for japanese beetles in the roses.
The next FOTD, the first of a new month, will be posted in 24
hours. I wonder what the new theme, if any, will be. Until
then, take care, and the pleasantly cool summer we're having
this year should quiet those greenie fanatics clamoring about
the imaginary perils of equally imaginary man-made global
warming . . . at least until it gets hot again.
Jim Muth
jamth@???
jimmuth@???
START PARAMETER FILE=======================================
Seahorse_Valley-30 { ; time=0:09:05.19-SF5 on P4-2000
reset=2004 type=formula formulafile=basic.frm
formulaname=SliceJulibrot4 passes=1
center-mag=-0.000394521/0.010564/31.23732/1/72.5/0
params=89.999/0/89.999/0/-0.7500830511589915/0.007\
7264749796986/0/0/2/0 float=y maxiter=15000
inside=0 outside=summ periodicity=10
colors=000mO2zP5mP6la8mPAzPCmPDzPEmOKaMNYLSVKWSJ`P\
IfMHkKGpJLlJQiJVgJZcJcaJh`KnXKsVKxTQwXUv`Zvbctfhsi\
lslprpsssxsutprqlonhmkfjhbgeZebXb_T`YQYUNWRKTOGRME\
QPHTTKWWMY_Q`bSbfVeiYhm`kqcntfpxiszkuznxzluvksrjqn\
hpigmgflbdiYbhVagYihbskdzlczhczebzbazZ`zW`zSZzP_zL\
YzIYzESs8Mk2Gc0JgBLkOMn`OzzPezPzzPezQezzzzQzzQezRe\
zzvzRezRtSSuzSezSezSezzzzTxJzyIzzHTzGTzESzDTzDRzCQ\
zCOzBOzBNzALz9Lz8Kz8Jz7Iz7Hy6Fy6Fy6It8KpAMkCOgFQbH\
SYJTVLWRNYNP_KRXMQWNQTPPSQPQROQSOOVNNWNLYMK_MIaLHb\
LEeJDfKEeLGeMHeNIcOJcPKcQLbRObTPbUQaVRaWSaXT`YU`ZW\
`_X_`Y`aZ`b`````_b`Yc`WfaTg`RcfS_kUWqVTvVStUSsUSrT\
SqTSpSSpSSnSSmSRlQRlQRkPRiPRhORgORfOQeMQcMQbLQbLQa\
KQ`KQ_KQYJQXIPWHQVHPTGPTGPSGPRFPQFPPEPOEPNCPMCPLCP\
KBPJBPIAPHAPG9PF9PE9PD8PC8PC7zC8PA5P85m97P64P54m65\
P33bHHzHHaGG`EEmGG_EE_DDmEEmDDmDDzzDmCCzCCzzCmBBzB\
BzzAmAAzAAm88zz1zY0zQ0zz1 }
frm:SliceJulibrot4 {; draws most slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
r=u*sg+v*ca*sb*cg, s=v*sin(a),
c=p+flip(q)+p3, z=r+flip(s)+p4:
z=z^(p5)+c
|z|<=9 }
END PARAMETER FILE=========================================
