Hal,
I really enjoyed your four dimensional fractal animation! I am usng Mozilla 1.7 on an older 486 with braodband (cable) access. The .gif downloaded very quickly and runs very smoothly. It is really fascinating and a great improvement over the three dimensionals that I have seen previously.
Good work!
On Thu Jul 6 15:26 , 'Hal Lane' sent:
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About three years ago someone put a link to
the fractal program Quat (that can create
three dimensional fractals) on the Fractint
list serve. At that time I found some
interesting 3D fractals on the web and
created some of my own.
Quat is available at:
http://www.physcip.uni-stuttgart.de/phy11733/quat_e.html
Quat uses the quaternions (an extension of
complex numbers) to calculate its 3D fractals.
Note that Quat's 3D fractals are calculated
as three dimensional objects but only
displayed as shadowed, depth-cued 2D images
of the 3D fractal. This method of display,
however, does give quite a bit of feeling
for the third dimension without requiring
everyone who wants to view the three
dimensional fractal to buy a stereoscope or
cross their eyes.
(Note that Quat does allow you to calculate
a pair of 2D images from a 3D fractal --
with the viewpoints slightly rotated from
one another. These can be viewed as a true
stereoscopic 3D image with a stereoscope or
by crossing your eyes. There is a link:
'Stereo 3D Fractal' at the top of my web
page (see below) to view one of these.)
However, I have always been interested in
even higher spatial dimensions that fractals
can inhabit.
I wanted to calculate and view four
dimensional fractals -- but the real problem
is how to display them, not how to calculate
them. Calculating them just takes time.
After thinking about the problem of how to
display a four dimensional fractal for quite
a while, I finally hit upon a solution.
I would use Quat's 2D, shadowed, depth-cued
images to show three dimensions of the
4D fractal and use time to show the fourth
dimension. I would create an animation
where each *time slice* would be a different
fractal.
Each slice would be a 3D object (seen as a
2D, shadowed, depth-cued image). I would
assemble all of these 3D slices into an
animation.
Viewing the animation would then show the
fourth dimension of the fractal object over
the duration of the animation.
I set up a batch file to invoke Quat to
calculate the different 3D slices of the
four dimensional fractal and then put the
2D versions of these slices together into
a .gif animation. This works and makes
a very crisp and clean animation, but
creates a large file.
I realize that other animation encodings
(e.g. .wmv, .avi, etc.) are smaller but
I do not have access to these encoders.
I just put my animation of a 4D fractal
on the web for others to view:
http://home.earthlink.net/~hallane/hals_fractals_n_photos/
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Warning! Clicking on the still image
on my web page to view the animation will
download a 600 MB file. I do *not*
recommend using dial-up internet access to
view this large a file.
If you have dial-up access you should view
the image on a friend's computer that has
broad-band access. Just email them the link
and schedule a visit.
(If you really want to try to download the
file with dial-up internet access you could
use a 'download manager' program that can
restart the download from where it left off
because of a dropped phone line. This will
still take a very long time. The largest
file that I ever downloaded successfully
using this method was a 90 MB file.)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
I'm sorry that the animation is so large. I
originally intended it to only be viewed on
my own computer. With broad-band access
being more widely available now I felt
encouraged to put it on the web.
Sorry... /hals_fractals_n_photos/
has underscores in it that don't show up
obviously in the link above, although you
should be able to just click on the link
to get the web page to appear in your
default web browser. If that fails, the
easiest thing to do is copy and paste the
link into the address window of your web
browser.
I have successfully viewed the animation
with these web browsers:
- IE 6.0,
- Firefox 1.5.0.4
- Netscape 6.2
The smoothest rendering of the animation
(after the entire file is downloaded) seems
to be provided by Firefox and Netscape.
The animation loops continuously.
If you only get a jerky rendering of the
animation in your web browser you can right
click the fully downloaded animated image
and choose "Save image as..." and save
the animated .gif image to your hard drive
and try viewing it with a different image
viewing program.
I have discovered that the version of
Microsoft Paint that ships with my
Windows 98 does not display the animation.
Also, probably because the image is a
non-standard resolution and for other
reasons I have not been able to view it
successfully with Fractint.
The best I have been able to do using
Fractint is to use F3 - 320 x 200 x 256
to display the animated .gif file. It
shows a distorted version of the animation
once -- too quickly -- and then stops.
I would be interested in any comments you
may have about how successful I have been
in trying to view the fourth spatial
dimension.
- Hal Lane
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# hallane@earthlink.net #
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