Hi. Once I had a front end utility for inputing formula and parameter files to fractint.I have xince lost it and forget the name.Could any one please help.
Brian.
FOTD -- December 11, 2002 (Rating 3)
Fractal visionaries and enthusiasts:
Today's fractal might show actual new material in the hypercom-
plex Mandelbrot set, or it might be an artifact of mathematical
imprecision. But regardless of what it is, it's an image with
spirals everywhere. There are single spirals, double spirals,
even signs of triple spirals. In fact, the image is filled with
spirals without end. And "Spirals Without End" is what I named
it.
The image is a zoom into a hidden valley that appears peeking
between the converging straight arms of yesterday's "Hyperbud"
image, where a new world appears to be only partly revealing
itself. In contrast to yesterday's image, which suffered from
resolution breakdown, today's image suffers from the near impos-
sibility of finding an acceptable coloring. I finally settled
on a very broad purplish palette, though the result still leaves
much to be desired and drops the rating to a below-average 3.
While fractaling, I sometimes wonder about the meaning of it
all. Is the existence of fractals filled with a deep, as-yet-
undiscovered meaning, perhaps about the nature of reality, or
are fractals mere candy for the eyes? And what of the concept
of meaning itself? Is it something invented by the highest
known form of life on the planet Earth, or is the universe
filled with it? Does meaning actually exist before a self-aware
entity asks the question 'why'? Does meaning exist on a two-
mile-wide chunk of ice orbiting in the cold loneliness of the
Oort cloud, where observers never have and never shall venture
to ask 'why'?
And what of this fourth dimension with its hyper-objects? We
can infer exactly how the various parts would fit together if it
existed, but even knowing these things, we cannot visualize some-
thing as simple as a tesseract. Does this mean that the fourth
dimension does not exist? Does it mean that it does exist but
we cannot visualize it? Does it mean that we have invented the
whole puzzle as a mental exercise? Or does it mean anything at
all?
Don't expect answers any time soon.
And don't expect an image soon if you try to run today's para-
meter file. It takes 1-1/4 hours on my less-than-supersonic
P-200 machine. The best way to see the puzzling image is to
download it from Paul's web site at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
or from Scott's site at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
Tuesday's fractal weather was cloudy and chilly here at F.C.,
with a high temperature of 34F 1C. The ex-dynamic duo passed
the unsatisfactory day lounging by the heat, except for when
they were asking for food.
This morning has brought freezing rain, which is even worse.
It's a good day to remain indoors and get ahead on some work.
Until tomorrow, when things will hopefully be more pleasant,
take care, and try to make the best of bad situations as they
arise.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
SpiralsWithoutEnd { ; time=1:15:40.15--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=HyperMandelbrot passes=1 center-mag=-0\
.04192332435179621/+0.08550882276013791/4021858
params=0/0/10/0/1e-014/0 float=y maxiter=20000
inside=255 logmap=-3100 periodicity=0
colors=0009ECAECBEDCEDDEDEEDFEDGGDHGFIGFJGFKGFLGFM\
GFNGHNGHMGHMGHMGHMGHMHHMHIKHIKHIKHIKHIKHIKHKIHKIHK\
IHKIHKIHKIHKKGMKGMKGMKGMKGNKGNMGNMGNMENMEPMEPMEPNE\
PNEPNERNERNDRNDRPDTPDTPDTPDTPDTPDVPDVRBVRBVRBVRBWR\
BWRBWTBWTBWTAYTAYTAYTAYVA_VA_VA_VA_V8_V8`V8`W8`W8`\
W8`W8bW8bW8bY6bY6dY6dY6dY6dY6d_6f_6f_4f_4f_4f_4g`4\
g`4g`4g`4g`4i`4i`4ib4ib4kb4kb4kb4kb4kd4md4md4md4md\
4md4nf4nf4nf4nf4pf4pf4pf4pg4pg4rg4rg4rg4rg4ri4ti4t\
i4ti4vi4vi4xk4xk4yk4yk4zk4zk4zm4zm4zm4zm4zm4zm4zm4\
zn4zn4zn4zn4zn4zn4zp4zp4zp4zp4zp4zp4zr4zr4zr4zr4zr\
4zr4zr4zt4zt4zt4zt4zt4zt4zv4zv4zv4zv4zv4zv4zx4zx4z\
x4zx4zx4zx4zy4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4\
zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx\
4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4z\
x4zx4zx6zx6zx6zx6zx6zx8zx8zx8zx8zxAzxAzxAzxAzxBzxB\
zxBzxBzxBzxDzxDzxDzxDzxEz }
frm:HyperMandelbrot {; periodicity must be turned off
a=(p1),b=(0,0):
q=sqr(a)-sqr(b)+pixel,
b=(p2+2)*a*b+p3,
a=q,
|a|+|b| <= 100 }
END 20.0 PAR-FORMULA FILE==================================
FOTD -- December 09, 2002 (Rating 6)
Fractal visionaries and enthusiasts:
Well, fractal persistence paid off, and I finally found an image
worthy of an above-average rating. True, it's only a slightly-
above-average rating of 6, but it's still an above-average
rating, and that's all that matters.
I might have rated the image higher if it had been something I
found rather than created with the < epsiloncross > inside fill
feature of Fractint. But let's not quibble -- an above-average
image is an above-average image, and I needed an above-average
image. (Having repeated the phrase 'above average' a sufficient
number of times, I shall mention it no more.)
The expression (-Z)^(1.5)+C was iterated to draw the fractal.
Iterating -Z instead of Z merely reveals a different part of the
infinite complex logarithmic spiral. The 'SliceJB-new-min'
formula is a slight reworking of a formula that was posted to
the list several years ago by John Goering.
Today's image is an oblique slice of the -Z^1.5+C Julibrot.
The slice is oriented rather close to the Julia direction, which
is revealed when (p1) and (p2) are set to 0.5,0.5. I named the
image "The Nest" when it reminded me of some crazy bird's nest.
With its effect of being illuminated from behind, the image
could just as easily been named something inspirational such as
"Crown of Thorns".
The 'SliceJB-new-min' formula draws more rotations than any
other formula in my vast and partly forgotten collection, but it
does not draw every possible rotation in four dimensions.
Finding a formula that does this is still one of my quests.
The render time of 18 seconds is blazingly fast, but the down-
load of the completed image is still available at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
and at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
The fractal weather Sunday was pleasant enough, with hazy sun,
gentle breezes, and a temperature of 48F 9C. The problem was
that the snow turned wet and slushy -- far too unpleasant for
the cats' tender paws. They once again passed the day keeping
warm and wishing they could get outside to play.
This morning the snow has re-frozen and is dry, but the tempera-
ture is well below freezing -- far too chilly for the intrepid
ones. Having a pile of work to think about, I shall give the
cold little notice -- that is unless some reason arises that I
must wander outdoors. Until the next FOTD appears in 24 hours,
keep warm or cool, whatever the case may be, and keep your
fractals dry.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
The_Nest { ; time=0:00:18.46--SF5 on a P200
reset=2002 type=formula formulafile=julibrot.frm
formulaname=SliceJB-new-min center-mag=0/0.0688935\
/0.6666667 params=0.49/0.56/0.4/0.39/1.5/0/-0.0\
416275/0.7559499/0/0 float=y maxiter=1200 passes=1
inside=epsiloncross proximity=0.05 periodicity=10
colors=0007I00008GO7LQ6NT5QW4TZ4Wa5Zd6`g6ak7bo8cs8\
ew9hx9kyAnzBqzBtzCwzDzzDzzEzzEzzJzzOajSYiXUiaQheMh\
000000mRj000nQk000nNm000mKnmIolGplFplDqkCrkAsk9sjE\
kiIchMXhQPgUIfYAfa3d_7bZB`XFZWIXUMVTQTRTRQXPO`NNdL\
LgJKkHIoFHrMLpSOnYSlcVjiZhoagm_dkYbjW`hUZfSWeRUcPS\
aNQ`LNZJLXHJWGHYIHZLK_NM`PO`RQaTSbVUcXWc_Zda`ecbfe\
dfgfgihhkjhmlT_aUXcVVeVTfWQhWOiXMkYJmYHnZFpZDqiu_j\
t`kt`lsamsamrbnrborcpqcqqdqpdrpesoetoftofniihdlb_o\
XVrRQuLLxONuROrUQoXRlZTiaUfdWcgX`iZYl_VoaSrbPtcMr`\
NqZOpXPoVQnSRmQSlOTkMUjJViHWhFXgDYfAZe8_d6`c4aTiUV\
jTXkTYkT_lT`lTbmTcmSenSfnShoSioSkpSlpSmlPmhMmdJzmH\
mYEnUBnR9nN6nJ3nG1hQLc_cZivHD3EQ9BaF9mLAlOAlRAlTAk\
WAkYBk`BkbBjeBjgBjjBjlAei9ag8Yd7Ub7Q_6MY5IV4ET3AQ3\
6OK8T_9YoAbhFebKgXPjRUlLZoFbqOZiXWbeTWbUV`UVYVUWVU\
UVTRWTPWSMXSKXRIXRKYPMYNNZMPZKQ_JS_HT`GV`EWaDYaBZa\
A`YFbVJcSNeOSgLWhI_WI`JIa }
frm:SliceJB-new-min {; thanx to J.R.H. Goering, July 1999
pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5):
z=(-z)^(p3)+c
|z|<=100 }
END 20.0 PAR-FORMULA FILE==================================
Does anyone know anything about these beasties? I haven't had a
chance to read this in depth yet, but they appear to be 6 dimensional
cousins of quaternions/hypercomplex numbers.
Could probably define trig functions for these, and get some fractals
out of them (there's a Mandelbrot example in the article.) The
question is, would there be more to discover that is not already
present with hypercomplex and quaternion slices?
http://fibonacci-arrays.com/Triternions.pdf
Tim
FOTD -- December 10, 2002 (Rating 5)
Fractal visionaries and enthusiasts:
I see that the existence of triternions has been noticed. I had
not seen them mentioned until Tim mentioned them in his recent
letter, though I had assumed that they did exist and were less
well-known mainly because they are less useful than quaternions.
Three-part numbers would draw a three-dimensional Mandelbrot set
and a six-dimensional Julibrot. But I wonder whether they would
actually create new Mandelbrot material. The Quaternions and
Hypercomplex numbers draw interesting Julia sets, but merely
expand and distort the Mandelbrot set, introducing little if any
actual new material. This might be due to the nature of the
M-sets or it might be due to the way they are implemented in the
program. I question the implementation because the Julia
versions of the hyper-sets have 6 parameters, while the Mandel
versions have only 2 working parameters.
The quaternions draw a figure resembling the 3-D figure that
would appear if the M-set were rotated around its X-axis. This
figure, which basically consists of circles, can then be sliced
through any of its parts, but the result is always a stretched
and distorted version of the same Mandelbrot set that we know
and love. The Hypercomplex M-set appears as two separate sets
that overlap. Some interesting things happen where the two sets
intersect, but once again, no new material is created.
Perhaps the triternions would create a M-set with new material
to explore, perhaps not. I rather suspect the latter, though it
would be most interesting to actually try it and find out.
Several years ago, bored with the sameness, I wrote a generaliza-
tion of the Hypercomplex formula that does create new material.
I have never been convinced that this new material is real
rather than an artifact of mathematical imprecision, but it
exists in a consistent way that would imply that it is indeed
real.
Today's image, which was created with this formula, and rather
resembles a skewed gothic window with converging bars before it,
has been named "Hypercomplex Bud". I gave it this name because
it is a bud in the Hypercomplex Mandelbrot set unlike anything
that appears in the familiar complex M-set. Not only is the
image unique on the surface, but its inner details are surpris-
ingly complex -- more intricate than anything that appears in
the traditional M-set. I will show some of this incredible
inner richness in tomorrow's FOTD.
I had to rate today's image at an average 5. Its mathematical
interest is above average, but its artistic merit is below. My
alibi, as always, is lack of time. If I had more time, I could
have found a better color palette.
With a render time of 2/3 hour on my machine, the parameter file
is a slow one. A far better way of viewing the scene is to
download the finished GIF file from the internet and Paul's site
at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
or from Scott's site at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
Monday was sunny but cold here at F.C. With snow on the ground
and a temperature that never reached freezing, (it stopped at
28F -2C), the dynamic duo of fractal cats spent the day huddled
by the heat. Late in the afternoon, a treat of turkey eased
their distress, and consequently my distress also.
As is usually the case, I find myself with several non-fractal
tasks to accomplish before setting out on my fractal journey of
the day. This is a certain sign that I must now get busy. So
until next time in 24 hours, take care, and when you walk with
fractals, don't expect much conversation.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
Hypercomplex Bud { ; time=0:39:51.09--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=HyperMandelbrot passes=1
center-mag=-0.04037558685446038/+0.084551148225468\
65/122.6667 params=0/0/10/0/1e-014/0 float=y
maxiter=12000 inside=255 logmap=177 periodicity=0
colors=000fRKgSKhSKiTKjTKkUKlUKmVKnVKoWKpWKqXLrXLs\
YLtYMsXNrWNrWOqVOqVPpUPoUQoTQnTQnSRmRRmRSlQSkQTkPT\
jPUjOUiOUiNVhMVgMWgLWfLXfKXeKYdJYdJYcIZcHZbH_bG_aG\
``F``Fa_Ea_EaZDbZCbYCcXBcXBdWAdWAeV9eU8fV9eV9eV9eV\
9eV9dV9dV9dV9dV9dV9cVAcVAcVAcWAcWAbWAbWAbWAbWAbWAa\
WAaWBaWBaWBaWB`WB`WB`XB`XB_XB_XB_XB_XC_XCZXCZXCZXC\
ZXCZXCYXCYXCYYCYYDYYDXYDXYDXYDXYDXYDWYDWYDWYDWYDVY\
EVZEVZEVZEVZEUZEUZEUZEUZEUZETZETZFTZFTZFTZFS_FS_FS\
_FS_FS_FR_FR_GR_GR_GQ_GQ_GQ_GQ_GQ_GP`GP`GP`GP`HP`H\
O`HO`HO`HO`HO`HN`HN`HN`HN`HNaIMaIMaJMaJMaKMaKLaKLb\
LLbLLbMLbMLbMKbNKbNKcOKcOKcPJcPJcPJcQJcQJdRJdRIdRI\
dSIdSIdTIdTIeUHeUHeUHeVHeVHeWGeWGeWGfXGfXGfYGfYFfZ\
FfZFfZFg_Fg_Eg`Eg`Eg`EgaEgaEhbDhbDhcDhcDhcDhdDhdCi\
eCieCieCifCifBigBigBjhBjhBjhBjiAjiAjjAjjAjjAkk9kk9\
kl9kl9km9km9km8ln8ln8lo8lo8lo8lp7lp7mq7mq7mr7mr6mr\
6ms6ms6nt6nt6nt5nu5nu5TVn }
frm:HyperMandelbrot {; periodicity must be turned off
a=(p1),b=(0,0):
q=sqr(a)-sqr(b)+pixel,
b=(p2+2)*a*b+p3,
a=q,
|a|+|b| <= 100 }
END 20.0 PAR-FORMULA FILE==================================
FOTD -- December 08, 2002 (Rating 4)
Fractal visionaries and enthusiasts:
I named today's image "Arcing". My excuse is that the bluish
streaks surrounding the central area of activity could be
interpreted as a series of electric arcs. The actual reason is
that I rather enjoy using the word, which regardless of how it
is spelled, never seems to be spelled correctly.
The presumed correct spelling is 'arcing', but spelled this way,
the word looks as if it should be pronounced 'arse-ing', which
more sensitive fractalists could take to mean something un-nice.
The spelling 'arking' looks as though it would be pronounced
correctly, but seems unrecognizable when seen on paper. The
participial form of the word 'picnic' is 'picnicking', and this
word gives no trouble. Why then can not the participial form of
the word 'arc' be spelled 'arcking'? . . . I guess it looks like
a mis-spelling of 'racking'.
This little side trip into the world of correct spelling is in
no way meant to divert attention from today's image, which is
rated well above yesterday's lowly 2, but still comes in at a
slightly-below-average 4. The slit-like hole at the center is
actually a very odd slice of a midget that appears in the east
valley area of a larger midget that lies on a filament extending
from the northwest shore of the main bud of the M-set.
At this angle, the 2,4,8... series of elements has basically
disintegrated into a disorganized chaos, though the overall
pattern is still rather well defined. A glance at the < tab >
screen will show that the image has been greatly stretched and
skewed to undo the inherent stretching and skewing of the scene.
Such stretching and skewing is an important part of examining
the odd slices of the Julibrot, where features in their natural
form are often stretched and twisted beyond recognition.
This natural stretching is an inevitable part of the odd slices.
It exists for mathematical reasons, one of which is that objects
appear far larger in the Julia aspect than in the Mandelbrot
aspect, and the odd slices include parts of both aspects. I
will have more to say about this stretching in a future FOTD.
Right now, it's time to reveal that, with a render time of 4
minutes, the image is barely worth the effort of rendering it
from the parameter file. Those who choose not to render may
find the pre-rendered image ready for download on Paul's site
at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
and on Scott's site at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
The fractal weather Saturday was sunny but chilly, with a
temperature barely above freezing. The snow still on the ground
kept the dynamic ones safely indoors by their radiators. But I
could not avoid a trip outdoors to do some things that needed to
be done.
This morning is sunny and chilly again. I expect a similar day.
Until the next FOTD appears in 24 hours, take care, and see you
then.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
Arcing { ; time=0:04:06.95--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=multirot-XZ-YW-new passes=1
center-mag=-0.00000000004526324/-0.000000000006343\
38/1.471077e+011/0.0132/172.255691302663337/-56.45\
35740250172111 params=-26.03289895321512/95.611438\
3373516/2/0/-1.256850515434251/0.3795596138199557/\
-1.256850515434251/0.3795596138199557 float=y
maxiter=2500 inside=0 periodicity=10
colors=000cdOXZPPTRHNSAHTELYHObLRfOUkSXoV_tYbxWapU\
`iT_aR_VQZNOYGNY9LZDJ_GH`JFaNEbQCcTAdX8e_7eb8c`8b_\
9`Y9_XAYVAXUBVSBUREVQGWQJXPLYPNYPQZOS_OV`NXaNZaN_d\
K`fI`iFakDanAbp8bs5cu3cw1_m3Wc5SU7OK8PWPPgePsvMrnK\
rgIq`GqUDqMBpF9p87p18l39i5Ae6Bb8C_ADWBETDEPFFMGGJI\
HFKICLJ8NK5PK2QN6OQAMSELVHJXLH_PGbTEdWCg_Bic9lg7nj\
6oa9pTBqKErCGoDHmEHjFHhGHfGHcHHaIH_JIXKIVKITLIQMIO\
NIMNIOSPQWVS`aUegWjn`tyZqvYntXlrVipUfmTdkRaiQZgPXd\
NUbMR`UVHPSQLPZGNgCKp8Iy7Hv7Ht7Gr7Gp7Fn7Fk7Ei7Eg6E\
e6Dc6D`6CZ6CX6BV6BT17Q49R6BR8DSAFSCHTEJTGLUINULPVN\
RVPTWRVWTXXVZXX`Y_`_ZaYZaWYaUYaTXbRXbPWbOWbMWcKVcJ\
VcHUcFUdETdCTdATd9R`9QY9PV9OS9NP9MM9LI9KF9JC9I99H6\
9G39J6CL8FOAIQCLTEOVGRYJU_LXbN_dPbgReiThmewj_lkVjk\
PilKglEfm9dp1bm4ck7chAdfDdcGdaJeZMe_HfYLeXPeWTeVWe\
U_dTcdSgdRjdQhcQgbPfbPdaPc`Ob`O`_N__NZZNXYMWYMVXFK\
`JPZMUXQYVTbTXgR_kPcpNftL }
frm:multirot-XZ-YW-new {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p3, c=e*imag(pixel)+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
FOTD -- December 07, 2002 (Rating 2)
Fractal visionaries and enthusiasts:
On some days you just can't find a good fractal. That's what I
told myself as I reluctantly designated today's abysmal image as
FOTD for Dec. 7. My alibi is that, though I had expected a slow
day on Friday, a pile of unexpected work arrived, and after snow-
removal duty, I was left with basically zero time to devote to
fractals. But even a much-below-average image is better than no
image at all.
Today's image is a zoom into the 'New View' image of Dec. 4. To
add a bit of life to an otherwise lifeless scene, I colored the
image with the inside areas set to < fmod >. I named the
resulting mess "FMOD Modules".
The result vaguely resembles a surreal landscape, with stretched
and tortured shapes hanging in the sky above the straight
horizon. In the foreground, an eroded, brilliantly yellow peak
juts upward from nowhere, while in the distance, mysterious
features lurk.
The preceding paragraph is an exercise in saying something about
nothing.
Speaking of the fourth dimension can also be considered saying
something about nothing. But at least the fourth dimension is a
real abstraction, in the sense that we know exactly how it would
appear if it did exist and we had the eyes to see it. And who
knows? These days, when physicists speak of 10, 11, or 26
dimensions as though they actually exist, who knows what else
might exist. Perhaps, hidden in some undiscovered and undiscov-
erable parallel world, a hypercubic crystal of four-dimensional
salt actually exists. At least it's worth thinking about for a
few seconds.
With its abysmal rating of 2 and its render time of 14 minutes,
the image is unlikely to be worth the effort of running the
parameter file. A download of the completed image is strongly
recommended. The download can be found at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
and at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
Friday was a partly cloudy, breezy and cold day here at F.C.
The snow cover and temperature that never got above freezing
kept the intrepid duo snug indoors all day. I was kept quite
busy removing snow and by an unexpected job that came in around
noon and had to be finished by today. But all went well, and
things are back on track. Hopefully, the next FOTD, which will
appear in 24 hours, will have a bit more class.
Until then, take care, and better things are to come.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
FMOD_Modules { ; time=0:14:38.04--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=multirot-XZ-YW-new center-mag=-0.41302\
339764054050/+0.00629609963586154/116.7456/0.9579
params=4.5/22/2/0/0/0/-1.25/0.03 float=y
maxiter=5000 inside=fmod logmap=16 periodicity=10
colors=000000ZWFQVHRVJSVLTUNUUPVURWTTXTVYTXZSZ_S`_\
SbaUacW`eY`f__ha_jcZleZmfYmgYmhXmiXmjXmkSnlNomIpnD\
qq6tt8wwAzzBwzDtzFqzGnzIkzKizLgzNezOcz4ay6_x7Yx8Xw\
9YwBZvC_vD`uEauF_tIZsKXrMWrOVqQTpSSoURoWPnZOm`MlbL\
ldKkfIjhHijGilLknPloTmpYnqapreqsjrtnsuuuxrtvpstmrs\
kqqhppfondnmamk_ljXkhVjgSieQhdOgbLfaJe_GdZEcXBbW9a\
U7`TLbZYcdjejwfoshloiikjfgkccl`_mYWnVSoSNqROpQOoQP\
nQPmPPlPQlPQkPQjORiORhORhOSgNSfNTeNTdMTcMUcMUbMUaL\
V`LV_LWYKV_LV`LUaMUbMTcMTdNSeNSfNRgORhOQiPQjPPkPPl\
QOmQOnQNoRNpRNqRLoPKmOJkMIiLGgKFeIEcHDaFC`EAZD9XB8\
VA7T85R74P63N42L31K23M75OC6QH8SM9TRBVWDX`EZeG`jHao\
HanHblHcjHdhHefHeeGdgFdhEdiEcjDckCcmBcnBboAbp9bq9b\
r8Yg7UX7UZ8V`8Wb8Wc8XegVTiVSlWRoZSnWRnUQmSPmPPmNOl\
LNlINkGMkELkBLj9Kj7Jj5Ji6Ki6Li6Li6Mi6Mi6Ni6Oi6Oi6P\
i6Pi6Qi6Ri6Ri6Si6Si6Ti6Ti6Ui6Vi6Vi7Wi7Wi7Xi7Yi7Yi7\
Zi7Zi7_i7_i7`i7ai7ai7bi7b }
frm:multirot-XZ-YW-new {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p3, c=e*imag(pixel)+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
FOTD -- December 06, 2002 (Rating 5)
Fractal visionaries and enthusiasts:
Today's image catches one of those Mandelbrot midgets at an
angle where it is most distorted. The actual midget can be seen
by checking the M-set at the coordinates of (p4) of today's
image. At the angle of today's slice, little is left of the
hole, which once was the midget and now lies invisibly at the
center of the image, or even of the basin itself, for that
matter.
At this angle, the basin appears to have split into two
branches, though actually it is the angle of the slice that
creates the split, cutting the single curved basin in two
places. The surrounding areas of chaos, which somewhat resemble
comets, are far better defined in the M-set.
I named the picture "Passion Flower". At one time I thought I
saw a flower blossom somewhere in the scene, though the bloom is
now far less apparent. I rated the image at an average 5, which
is all I could do for it. (The corresponding midget, which
appears in the M-set, would rate a 6 or 7.)
With a 3-1/2 minute render time, the parameter file is one way
of viewing the scene. A more efficient way is to download the
completed GIF image from Paul's web site at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
or from Scott's site at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
Today's image is a slice of the four-dimensional Julibrot. The
Julibrot cannot be accurately considered an object, because
objects are things that exist objectively in the objective
world, and to our specialized senses at least, the objective
world is three dimensional.
I have spent many years trying to visualize a four-dimensional
hyper-object, knowing all the while that my effort was futile.
The greatest problem is that the surface boundary of a 4-D
figure must be three-dimensional, and my imagination invariably
produces flat two-dimensional bounding surfaces for all my
mental figures.
A 4-D hyper-object must have on its bounding outer surface as
many dimensions as there are altogether in our space. On the
3-D surface of a 4-D planet, there would be six cardinal
directions. Three parameters would be needed to fix a location,
and two parameters would be necessary to define a direction.
The horizon would not be a circle, but a sphere that totally
surrounds and encloses a visitor from the third dimension, as
the heavens surround the planet Earth.
On such a world, the sunrise would first appear as a brightening
concentrated in a particular area. Then a brilliant star-like
dot would suddenly pop into existence at the brightest point,
grow to the familiar blinding disk, then shrink again to a dot
and vanish. At sunset the same thing would occur at another
point of the field of view.
At night, assuming perfect clarity of the atmosphere, the stars
would appear as twinkling fireflies as they rose above and sank
beneath the horizon, passing through the 3-D visitor's field of
view. The view on a 4-D world would be interesting indeed --
provided, that is, that the visitor could find a way to shield
his retinas from light from the side, and also a way to avoid
crumpling into a wrinkled heap as a 2-D sheet of plastic wrap
does in our 3-D world.
It snowed at a moderate rate all day Thursday here at F.C.,
piling up 8in 20cm by the time it ended at 11pm. Combined with
a temperature of 27F -3C, the conditions proved far too harsh
for the intrepid duo, who passed the day alternating between
the window and the radiator. All things considered, their moods
were surprisingly cheery.
Unable to get out yesterday, I passed the day getting ahead in
the work. As a result, after clearing the sidewalk, I will have
extra time this afternoon to devote to fractal hunting. Hope-
fully, I'll find something worthwhile. Check back tomorrow to
see whether I actually find it. Until then, take care, and
fractals can and do make life interesting.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
Passion_Flower { ; time=0:03:35.04--SF5 on a P200
reset=2002 type=formula formulafile=julibrot.frm
formulaname=SliceJB-new passes=1 center-mag=0/0/2.\
632523e+010/0.006156/-8.21933390591836854/89.48120\
55080059565 params=0.22/0.51/0.505/0.326/2/0/-0.15\
40762385267798/1.030994820300086/-0.15407623852677\
98/1.030994820300086 float=y maxiter=2500 inside=0
colors=000WIFXJGYKHZLHZMI_NJ`OJ`PKaQKbRLbSMcTMdUNd\
VOeWOfXPd`XeZTfXPgVLhUHiSDjQ9mR3jP6hN9eLBcKE`IGZGJ\
XEMUDOSBRP9TN7WJ2_L6YNAWPDURHSTKQVOOXRMZVL`ZJbaHde\
FfhDhlBjo9ls7ms6kqDimKgiRkkVmmWfeXbYW_YXYYYWYYUYZR\
Y_PY_NY`Laa9ceEcgJYaNRWSLQXEKd4Gb6F`8FZAFXCEVEETGE\
SIDQKDOMDMOCKQCISCHUBFWBDYBB_A9aA7cA3dD6gA9j7Cl4Fm\
2Mk8SiEZfKecMeaOd`PcZQbYRaXS`VT_UUZTVYRWXQXWPYWN_V\
M`ULaTJbSIcRHdQFePEfODgNBhMAiM9jPIlSQmVToYWp_Zrbas\
ecuhevjguiiuhkshlrglpfmofmmenlenjdnicogcoebpdbpbaq\
a`q_`rZ_rX_rWaoXbmYckYdiZfgZgd_hb_i``kZ`lXamUbnSbp\
QcqOcrMdtJdvHewFixDly8lxBiwEfvHcuJ_sJWqJQoJPmJOkJN\
iJMgJLeJKcJJaJJ_JIYJHWJGUJFSJEQJDOJCMJBMBDLFCKJBKN\
AJR9IV8IZ7Ha6He5Gi4Fm3Fq2Eu1Cz0Ex1Fw2Gu3Ht4Ir5Jq6K\
o7Ln8Ml9NkAOiBPhCQgDReESdFTbGUaHV_IWZJXXKYWLZUM_TN\
_RL`SNaTPbUQcVSdWTeXVfYWgYYhZZi_`j`akaclbemcfndhod\
ipekqflrgnshotiqujrvktzmw }
frm:SliceJB-new {; by John R. H. Goering, July 1999
pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5):
z=z^(p3)+c
|z|<=9 }
END 20.0 PAR-FORMULA FILE==================================
FOTD -- December 06, 2002 (Rating 5)
Fractal visionaries and enthusiasts:
Today's image catches one of those Mandelbrot midgets at an
angle where it is most distorted. The actual midget can be seen
by checking the M-set at the coordinates of (p4) of today's
image. At the angle of today's slice, little is left of the
hole, which once was the midget and now lies invisibly at the
center of the image, or even of the basin itself, for that
matter.
At this angle, the basin appears to have split into two
branches, though actually it is the angle of the slice that
creates the split, cutting the single curved basin in two
places. The surrounding areas of chaos, which somewhat resemble
comets, are far better defined in the M-set.
I named the picture "Passion Flower". At one time I thought I
saw a flower blossom somewhere in the scene, though the bloom is
now far less apparent. I rated the image at an average 5, which
is all I could do for it. (The corresponding midget, which
appears in the M-set, would rate a 6 or 7.)
With a 3-1/2 minute render time, the parameter file is one way
of viewing the scene. A more efficient way is to download the
completed GIF image from Paul's web site at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
or from Scott's site at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
Today's image is a slice of the four-dimensional Julibrot. The
Julibrot cannot be accurately considered an object, because
objects are things that exist objectively in the objective
world, and to our specialized senses at least, the objective
world is three dimensional.
I have spent many years trying to visualize a four-dimensional
hyper-object, knowing all the while that my effort was futile.
The greatest problem is that the surface boundary of a 4-D
figure must be three-dimensional, and my imagination invariably
produces flat two-dimensional bounding surfaces for all my
mental figures.
A 4-D hyper-object must have on its bounding outer surface as
many dimensions as there are altogether in our space. On the
3-D surface of a 4-D planet, there would be six cardinal
directions. Three parameters would be needed to fix a location,
and two parameters would be necessary to define a direction.
The horizon would not be a circle, but a sphere that totally
surrounds and encloses a visitor from the third dimension, as
the heavens surround the planet Earth.
On such a world, the sunrise would first appear as a brightening
concentrated in a particular area. Then a brilliant star-like
dot would suddenly pop into existence at the brightest point,
grow to the familiar blinding disk, then shrink again to a dot
and vanish. At sunset the same thing would occur at another
point of the field of view.
At night, assuming perfect clarity of the atmosphere, the stars
would appear as twinkling fireflies as they rose above and sank
beneath the horizon, passing through the 3-D visitor's field of
view. The view on a 4-D world would be interesting indeed --
provided, that is, that the visitor could find a way to shield
his retinas from light from the side, and also a way to avoid
crumpling into a wrinkled heap as a 2-D sheet of plastic wrap
does in our 3-D world.
It snowed at a moderate rate all day Thursday here at F.C.,
piling up 8in 20cm by the time it ended at 11pm. Combined with
a temperature of 27F -3C, the conditions proved far too harsh
for the intrepid duo, who passed the day alternating between
the window and the radiator. All things considered, their moods
were surprisingly cheery.
Unable to get out yesterday, I passed the day getting ahead in
the work. As a result, after clearing the sidewalk, I will have
extra time this afternoon to devote to fractal hunting. Hope-
fully, I'll find something worthwhile. Check back tomorrow to
see whether I actually find it. Until then, take care, and
fractals can and do make life interesting.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
Passion_Flower { ; time=0:03:35.04--SF5 on a P200
reset=2002 type=formula formulafile=julibrot.frm
formulaname=SliceJB-new passes=1 center-mag=0/0/2.\
632523e+010/0.006156/-8.21933390591836854/89.48120\
55080059565 params=0.22/0.51/0.505/0.326/2/0/-0.15\
40762385267798/1.030994820300086/-0.15407623852677\
98/1.030994820300086 float=y maxiter=2500 inside=0
colors=000WIFXJGYKHZLHZMI_NJ`OJ`PKaQKbRLbSMcTMdUNd\
VOeWOfXPd`XeZTfXPgVLhUHiSDjQ9mR3jP6hN9eLBcKE`IGZGJ\
XEMUDOSBRP9TN7WJ2_L6YNAWPDURHSTKQVOOXRMZVL`ZJbaHde\
FfhDhlBjo9ls7ms6kqDimKgiRkkVmmWfeXbYW_YXYYYWYYUYZR\
Y_PY_NY`Laa9ceEcgJYaNRWSLQXEKd4Gb6F`8FZAFXCEVEETGE\
SIDQKDOMDMOCKQCISCHUBFWBDYBB_A9aA7cA3dD6gA9j7Cl4Fm\
2Mk8SiEZfKecMeaOd`PcZQbYRaXS`VT_UUZTVYRWXQXWPYWN_V\
M`ULaTJbSIcRHdQFePEfODgNBhMAiM9jPIlSQmVToYWp_Zrbas\
ecuhevjguiiuhkshlrglpfmofmmenlenjdnicogcoebpdbpbaq\
a`q_`rZ_rX_rWaoXbmYckYdiZfgZgd_hb_i``kZ`lXamUbnSbp\
QcqOcrMdtJdvHewFixDly8lxBiwEfvHcuJ_sJWqJQoJPmJOkJN\
iJMgJLeJKcJJaJJ_JIYJHWJGUJFSJEQJDOJCMJBMBDLFCKJBKN\
AJR9IV8IZ7Ha6He5Gi4Fm3Fq2Eu1Cz0Ex1Fw2Gu3Ht4Ir5Jq6K\
o7Ln8Ml9NkAOiBPhCQgDReESdFTbGUaHV_IWZJXXKYWLZUM_TN\
_RL`SNaTPbUQcVSdWTeXVfYWgYYhZZi_`j`akaclbemcfndhod\
ipekqflrgnshotiqujrvktzmw }
frm:SliceJB-new {; by John R. H. Goering, July 1999
pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5):
z=z^(p3)+c
|z|<=9 }
END 20.0 PAR-FORMULA FILE==================================
FOTD -- December 05, 2002 (Rating 4)
Fractal visionaries and enthusiasts:
Today's image resembles a disembodied grin -- a rictus someone
who likes to use fancy words might call it. I simply call it
"The Laugh", a name which properly describes the effect the
picture creates. Turned upside-down, the image might be called
"The Cry", but let's keep things as pleasant as possible.
I'll try to be as pleasant as possible about the image, for
which I make no excuses and which I can rate no higher than a
somewhat inflated 4. But even a 4-rated image is worth the
effort when the time required to render it is only 44 seconds.
The hole at the center, shaped like a gaping grin, is actually
an odd slice of a midget located near the tip of the northern-
most filament of the M-set. At the hole's upper shoreline a bit
of one of those ubiquitous straight bridges appears. (In the
M-set a slice of this bridge appears as a valley that forms a
lip of a bud.)
The grin is actually part of a four-dimensional hyper-object.
In a space of four dimensions, things that seem absurd in three
dimensions become everyday facts. Consider rotation. In a
space of zero or one dimension, rotation is impossible. In two-
dimensional space, rotation takes place around an axis-point,
while in three-dimensional space, rotation takes place around an
axis-line. The general rule is that in n-dimensional space, a
simple rotation takes place around an n-minus-2 axis.
But a simple rotation requires only two dimensions. The remain-
ing dimensions simply remain fixed while turning in place. In
four-dimensional space, this leaves the two-dimensional axis of
rotation motionless, its points rotating on themselves. But
even while serving as an axis of rotation, this plane may itself
rotate. The resulting motion, which is impossible in three
dimensions, is known as a double rotation. When subject to a
double rotation, every point of a four-dimensional hyper-object
except the point where the two axis-planes intersect is in
motion. If the two rotations are equal, all points move in a
circle. If the rotations are not equal, the points trace out a
surface of double revolution, which is shaped like a four-dimen-
sional doughnut.
This double-rotation motion can perhaps be more fully realized
by considering the Mandelbrot set and Julibrot. With the M-set
on the screen, which of the Z dimensions is perpendicular to it,
real(z) or imag(z)? Of course, they both are. In fact, every
line in the Z, or Julia, plane is perpendicular to the M-set.
Keeping the M-set undisturbed on the screen, the Julibrot can be
rotated around the M-set, so that every line of the Z plane in
turn lies in our 3-space and at a right angle to the M-set.
Looking at only the screen, one would never know that the
Julibrot was rotating. Now, with the Julibrot still rotating,
if the M-set on the screen were to begin rotating on the screen
around its origin, carrying the rest of the Julibrot with it,
the Julibrot would be in a state of double rotation. Do not try
to mentally trace out the path the points would take, it is
impossible.
It is far easier to imagine the fun of rendering today's super-
fast image from the attached parameter file. And those not
particularly fond of rendering can download the completed image
from one of the FOTD web sites at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
and at:
<http://sdboyd.dyndns.org/~sdboyd/fotd/index.html>
I'll have much more popularized 4-D curiosities in the weeks to
come.
Wednesday was hazy and cold here at Fractal Central, but the day
was made interesting by a late morning apparition of two
brilliantly colored sun dogs. The temperature of 26F -3C was
far too chilly for the ears of the intrepid duo, who also
dislike dogs of any variety. They wisely decided to pass the
day stretched by the warmest radiator they could find.
Gazing from the F.C. window this morning, I see snow falling at
a moderate rate with 6in 15cm already on the ground and the snow
expected to continue until evening. The fractal cats will need
comforting. Regardless of the weather, the human's work still
needs to be done however. And the best way to finish work is to
start it. So until tomorrow, take care, and be of good cheer.
Jim Muth
jamth(a)mindspring.com
jimmuth(a)aol.com
START 20.0 PAR-FORMULA FILE================================
The_Laugh { ; time=0:00:44.70--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=multirot-XZ-YW-new passes=1
center-mag=-0.00000000000001003/-0.000000000000001\
47/3.099006e+012/1.4001/0.404762088721556723/-20.7\
50360169932005 params=-73.9/161.5/2/0/-0.207107799\
5585138/1.122756154909389/-0.2071077995585138/1.12\
2756154909389 float=y maxiter=12000 inside=0
logmap=79 periodicity=10
colors=000PKAQLARMASNATOAUPAVQAWRAXSAYTAZUA_VA`W6a\
X5bY5bZ5a_4``4`a4_b3Zc3Zd3Ye2Xf2Xg2Wh9ViFUiLTiRTiX\
SibRihQinPitPizSkyUmyWoyYqy`sybuydwyfxyetwdpvcmtbi\
safq`bp__nZWmYTjXTkXWmPZoFaq0ds1cq1bp2ao2`n3_m3_l4\
Zk4Yj5Xi5Wh5Wg4Yh4_i4ai4bj4dk4fk4hl4il9gjDehIcfMbd\
Q`bVZ`ZXZbWXZSRVOMRLHNHCKE7RCBXAEc8Ii7LQBeigqgclf_\
hdXdcT`aPX`MT_IOYEKXBGV7CU38T04Q4GO8RLBbJFmHIxaTpu\
bin_hhXgaVgWSfPPeJNeCKd6Id7Lg8Oj9QlAToBWrCYtD`wDby\
F_wHYuIVsKTrLQpNOnOLlQJkRGiTEgUCfch5qfDhdK_cSRaZI_\
fAZmKTnTNnbInkCnt7nkHkcQiWZfOhdGqa8z_7mZ6gz5Qz4Ez5\
Fz6Fz7Fz7Fz8Fz9FzAFzAFzBFzCFzDFzDFzDDzDCzGDzGDzFGz\
EJzCMzBPzASz8Vz7Yz6`z4cz3fz2iz5dz8_zAVzDQzFLzLRzQW\
zVaz_fzdkzcizbhzbfzaez`cz`bz_`z__zcbzfezihzlkzonzr\
qzutzxwzurzrnzojzleziazfYzcUz`PzYLzVHzSDz_KzgRznYz\
m_zl`zkazjbziczhdzgezfgzehzdizcjzbkzalzamz_lzZlzYl\
zWlzVlzUlzSlzRlzQlzPlzQfz }
frm:multirot-XZ-YW-new {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p3, c=e*imag(pixel)+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
