############################################ 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 *spatial slice* of the four dimensional 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 ######################### # hallane@earthlink.net # ######################### ############################################ -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.1.394 / Virus Database: 268.9.9/382 - Release Date: 7/4/06