If you've seen
this page
in which my Commodore 64 version of this part of the Mandelbrot set
appears, you'll know that in the "Computer Recreations" column of the
February 1989 issue of Scientific American, A.K.Dewdney presented a graphic
sent in by a reader, Andrew laMance, entiltled "Love Canal". This featured a
black, winding shape somewhat resembling a river or estuary, surrounded by
beautiful but dark bands of colour with a somewhat threatening look about
them.
In his article, Dr. Dewdney stated that he presumed that the name referred
to a certain notorious polluted area in New York State near Niagara Falls.
He suggested that the black part was reminiscent of "a drainage ditch filled
with toxic chemicals where strange vegetation struggles to survive".
Rather gloomy, don't you think?
The graphic above was my first attempt to produce a PC-based graphic of the
area. The basic shape is correct, so clearly I'd found the right area; but
apart from that, the picture bears little resemblance to Mr. laMance's
original. Well, thereby lay a challenge!
You can download and save a copy of the BASIC source-code for the 640 x 480
version by right-clicking
here.
In the following graphic, the white rectangle gives an indication of the
whereabouts of the Love Canal within the M-set. Note, however, that this
rectangle is actually too big by a factor of 1,000. Our area of interest is
at its centre. We're doing a deep zoom here! (At the bottom of this page,
you can see some animations which show the zoom actually occurring, thus
enabling a much better appreciation of the Love Canal's relationship to the
overall M-set.)
The black rectangle to the east of the white one shows where the Love Canal
isn't. A misprint crept into Dr. Dewdney's article, so that the
coordinates of the central point were given as (.235125, .82722), rather
than (-.235125, .82722). As mentioned in the page whose link appears above,
I found by experiment that changing the sign of the x-coordinate
successfully pin-pointed the required area.
(There's another page on the web which mentions this same fact - click
here,
scroll about 7/8 of the way down to the bottom, and read on from there.)
As in several others of my initial attempts at particular areas, the first
alteration was to space the ribbons out so that there would be much less
"crowding" of colours near the M-set itself. The following two graphics
resulted. You'll notice that there's very little difference between them; the
reason for including both here is that I later did further adjustments using
each of them as a starting point, as detailed below.
Love Canal - Variation "A"
You can download and save a copy of the BASIC source-code for the 640 x 480
version by right-clicking
here.
Love Canal - Variation "B"
You can download and save a copy of the BASIC source-code for the 640 x 480
version by right-clicking
here.
To produce the next graphic, I used Neopaint to vary the colours in
Variation "B" above, just as I've done in previous pages:
The main thing to notice about all three graphics presented so far is that
the "Canal" part corresponds to what I normally refer to as the "white
buffer zone", with the black M-set appearing as small islands within it. (I
used an iteration limit of 10,000 to produce them.) The usual four-colour
"aura" appears as usual around these black bits and within the white area.
The effect is that of a clean waterway. No hint of pollution here;
everything is sweetness and light!
However, in Mr. laMance's original, that entire area was black, with a
narrow white "aura" which then gives way to bands of lurid colour - creating
a very different impression. The final graphic in this page is my attempt to
produce something similar (given that I only have a 16-colour palette),
using the original as a guide. This time, I started with variation "A". To
get the wide black area, I reduced the iteration limit to 250. The remaining
colours are simply the standard DOS colours. I haven't altered them in any
way; I've simply applied them in a different order.
I think I've achieved a nicely poisonous effect - well, I certainly wouldn't
want to swim in it!
You can download and save a copy of the BASIC source-code for the 640 x 480
version by right-clicking
here.
UPDATE, Wednesday, 8th November, 2006
Well, here's a surprise.
While researching my
"It's still not easy being green"
page, I've inadvertently run across something about the real Love
Canal in Niagara Falls.
Above is my model of a molecule of
2,3,7,8 tetrachlorodibenzo-p-dioxin (TCDD), which I've referred to in
that page. While "Google-hunting" for some information about this very toxic
substance, I found
this
web-page which mentions the fact that TCDD was found at Love Canal.
It really is a small world, isn't it...?
Now, as promised above: an animated GIF showing a zoom from the whole M-set
down to the Love Canal, giving a perspective of just where it is and how it
fits into the big picture. I've done this in black and white, basically for
two reasons: firstly, in the interests of clarity, as a stark two-tone
version emphasizes the structures involved; and secondly, to keep the size
of the file down to a sensible level (151Kb). (Actually, there's a third
reason: the truth is that I don't know how to do an effective job of this in
glorious technicolour, and I'm not disposed at the moment to take the
trouble to find out!
You may have seen sophisticated videos showing zooms of the Mandelbrot set
or other fractals. I have an ABC video from 1988 entitled "Chaos: the Theory
which Imposes Order Within Disorder" which includes some delicious footage
of this type. Perhaps you've seen rock videos which use M-set zooms as a
background, giving a kind of up-to-date hi-tech psychedelic effect. (As
mentioned in a 31-10-2010 update to my
...Region #8
page, the video of the 12" version of a 1988 hit,
"Stand Up for Your Love Rights"
by Yazz, is an excellent example.) These are always spectacular; the
surrealistic effect of the sinuous black areas surrounded by fizzing
colour is definitely something to behold.
Well, that's not what we have here - sorry!
Such impressive movies, if they are to happen in "real time", require
powerful supercomputers to generate the individual frames quickly enough.
That's not what you get with an old 1997 Windows 95 PC! So what we
do have here is basically a black-and-white slide show.
This involves 101 graphic frames, with 100 individual zooms, using the same
ratio of dimensions each time. The vertical height (y-direction) of the
first frame (showing the whole set) is 2.5; and that of the final frame
(showing the Love Canal itself) is 0.0001008. This corresponds to an overall
zoom factor of 24801.5873, very close to the value used by Andrew laMance
(24,800) in his original. Thus each of the 100 individual zooms involves a
factor of 24801.58730.01 = 1.106483125,
or approximately 10.6% each time.
The "flea" from which the Love Canal "sticks out" first becomes visible in
Frame 16. Just keep watching that spot indicated at the beginning to see it
grow and assume its true form. Also, do keep in mind that the Mandelbrot set
is connected - which means that the "flea" is part of the whole set,
and joined to the more obvious large parts of it, even though this is not
necessarily immediately obvious at this level of magnification. Similarly,
all apparently isolated dots in the later frames are, in reality, also parts
of a connected whole.
To me, Frame 100 looks more like a rampaging gnarly elephant than anything
else...
You can download and save a copy of the BASIC source-code by right-clicking
here.
I've also decided to include within this page the source code for the
program I used to generate the 101 frames:
REM *** MAD TEDDY'S M-SET ZOOM PROGRAM ***
REM Zooming in on the "Love Canal", starting from the entire M-set
SCREEN 12: REM graphics, 640x480
iterationlimit = 250
numberofframes = 100
REM coordinates for first frame
REM coordinates for final frame
REM dimensions of first frame
REM dimensions of first frame
k = LOG((yhigh1 - ylow1) / (yhigh2 - ylow2))
FOR frame = 0 TO numberofframes
REM print frame number at bottom - three digits
lambda = frame / numberofframes
ylow = ylow1 + z * ylowdelta
FOR i = 0 TO 239
REM don't bother plotting a black dot if point is in M-set
END IF
BEEP
a$ = ""
CLS
NEXT frame
xhigh2 = xlow1+(yhigh2 - ylow2) * 4 / 3
since I launched this website in early July 2006, has now been corrected to:
xhigh2 = xlow2+(yhigh2 - ylow2) * 4 / 3
(Note that this error was only in the transcript above - not
in the downloadable program itself.)
Apologies for any confusion this may have caused. (Please - if you spot any
other errors in any of my programs and/or transcripts of same within these
pages, do be good enough to
contact me
and let me know, so that I can fix them.)
The iteration limit used throughout is 250. This is the same figure I used
to generate the "toxic" version of the picture (above); it gives good
results without taking a ridiculously long time.
Note that this program can easily be modified to generate frames for a zoom
to any part of the M-set! All you have to do is put in appropriate
values for ylow2, yhigh2, and xlow2, corresponding to the location of your
"target area".
In fact, you can zoom towards that area from anywhere in the M-set;
you don't have to start from the whole set! To do this, simply change the
values for ylow1, yhigh1, and xlow1. One area may be totally enclosed in the
other (as in the current example), or they may overlap - or they may even be
in totally different parts of the plane, with no points in common at all! If
you wanted to, you could have a "journey" from any one of my examples
presented in these pages to any other, with or without a zoom (inward or
outward). Whether you get an interesting result will depend on the details
of the "journey", i.e. what lies between the two locations. (Obviously some
such "expeditions" will give better results than others.)
Finally, you can have more or less frames - all you need to do is change the
value of numberofframes - and, of course, you can also change the value of
iterationlimit to suit yourself (the bigger it is, the longer the process
will take).
To give a clear idea of how the zoom actually operates, here is an animated
GIF (102Kb) showing just every fifth frame of the above animation, together
with a graphic indication of where each frame is with respect to the entire
M-set:
Notice that the white box is only visible for the first half of the
animation (as far as frame 50); after that it becomes too small, and
disappears into the "vanishing point" at (-0.2351151, 0.8272533). The zoom
factor for this animation is
24801.58730.05 = 1.658523453, or
approximately 66% each time. The box dimensions therefore shrink by a factor
equal to the reciprocal of this figure, i.e. 0.602942678, with each zoom.
Hence the zoom factor for frame 50 - halfway through both this animation and
the previous one - is the square root of 24801.5873, i.e. 157.4851971. The
"flea" mentioned above first makes its appearance, in this version, in Frame
15.
The BASIC program used to generate the frames for this animation was
produced by modifying the program for the other animation, above. You can
download and save a copy by right-clicking
here.
(I used Neopaint to help in producing the opening frame, and also to make a
few colour modifications - in particular, the yellow for the boxes, which is
thus a bit darker than the standard DOS yellow.)
Feel free to modify either or both of these programs, and use the results to
take your own personalized tour around the Mandelbrot set. As a
starting-point, you may like to zoom up on some of my other examples within
these pages. Then you might get inspired to embark on some adventures of
your own.
Go ahead - experiment, and have some fun!
UPDATE, Saturday, 30th October 2010
A few weeks ago - and before I found on YouTube the video of Yazz's
"Stand Up for Your Love Rights" featuring fractal graphics, as mentioned
above - I discovered some other very good YouTube videos which do indeed
show the surrounds of various parts of the Mandelbrot Set in fizzing colour.
I've already made reference to this in an update at the bottom of my
Fractals #1...
page; and I've decided that it's highly appropriate to repeat the details
in this page also, in case you missed it on the way here.
Here
is a link to the first of six videos which together make up a program
hosted by the late Arthur C. Clarke which features some mathematicians -
including Dr. Mandelbrot himself - discussing aspects of fractals in
general and the M-set in particular, and which includes a delightful
soundtrack by
David Gilmour
of Pink Floyd. The other five videos are easily located within YouTube's
own indexing system, along with several other videos in similar vein. Well
worth a look - check 'em out!
My home page
Preliminaries (Copyright, Safety)
Mandelbrot set: Region #23
Click on the picture to see a 640 x 480 pixel version.
Click on the picture to see a 640 x 480 pixel version.
Click on the picture to see a 640 x 480 pixel version.
Click on the picture to see a 640 x 480 pixel version.
Click on the picture to see a 640 x 480 pixel version.
)
ylow1 = -1.25
yhigh1 = 1.25
xlow1 = -2.15
xhigh1 = xlow1+(yhigh1 - ylow1) * 4 / 3
ylow2 = .8271696
yhigh2 = .8272704
xlow2 = -.2351922
xhigh2 = xlow2+(yhigh2 - ylow2) * 4 / 3
xlowdelta = xlow2 - xlow1
ylowdelta = ylow2 - ylow1
xhighdelta = xhigh2 - xhigh1
yhighdelta = yhigh2 - yhigh1
LOCATE 16,16: PRINT "Frame ";
IF frame < 100 THEN PRINT "0";
IF frame < 10 THEN PRINT "0";
PRINT RIGHT$(STR$(frame), LEN(STR$(frame))-1)
z = 1 - EXP(-k * lambda)
yhigh = yhigh1 + z * yhighdelta
xlow = xlow1 + z * xlowdelta
xhigh = xhigh1 + z * xhighdelta
FOR j = 0 to 319
x = xlow + j * (xhigh - xlow) / 319
y = yhigh - i * (yhigh - ylow) / 239
xc = x: yc = y
iterations = 0
rsquared = 0
DO UNTIL rsquared >= 4 OR iterations > iterationlimit
u = x * x - y * y + xc
v = 2 * x * y + yc
rsquared = u * u + v * v
x = u: y = v
iterations = iterations + 1
LOOP
IF rsquared >= 4 THEN
PSET (j,i), 15: REM plots a white dot
REM - just leave it as it is, black, and save a bit of time
NEXT j
NEXT i
DO: a$ = inkey$: UNTIL a$ = " "
UPDATE, 5th November 2010:
The program simply generates the 101 frames one at a time, with a beep at
the end of each one. Pressing the space bar clears the screen and starts the
next one. I used Neopaint's "Neograb" feature to save the frames before
pressing the space bar each time, and then later used DISPLAY to crop the
results to 320×254 pixels (the top-left part of the screen). Finally, I used
Microsoft's GIF Animator to assemble them (along with a title frame and the
"Watch this spot" frame, both produced with the aid of Neopaint) into a
103-frame animated GIF.
Return to
Fractals #1: the Cantor and Mandelbrot sets