---------[ Received Mail Content ]----------
>Subject : Re: [RE]Re: [RE]Re: [Fractint] 3D pixel
>Date : Mon, 16 Oct 2006 19:44:57 -0500
>From : Jonathan Osuch
>To : Fractint and General Fractals Discussion

>Russell,

>> I'm now more familiar with the existing formulas, and it does appear
>> that most all I've suggested is already there. Indeed, to work from
>> this context would evidently be a quick and easy way to get started,
>> and that deserves consideration. There are, however, many possible
>> variants of these new formulas, and it seems that ultimately a 3D
>> pixel option will afford a lot more freedom to explore.

>Help me out with this. I am not seeing the possibilities of the 3D
>pixel to which you are referring. Thanks.


Jonathan,

Certainly. I'm happy to... but not in ten words or less I hope. It's useful, I believe, to review how I have come to advocate for this option. So indulge me for a moment, if you will.

Somewhere in the closing moments of '02, I saw that a system of ordered triplets could be fashioned on the cyclic group C6. The idea was that these "triternions" would generate fractals native to 3-space, not just 3D subsets of 4D fractals. As we kicked the concept around on this list, it led to a series of nice images and animations. However, we never quite realized the original goal.

After a while, the initial momentum dissipated and everyone went on to other adventures. It was nearly three years later that I finally saw clearly how to employ these numbers in a Fractint formula. A problem, though, was that my 2D formula offered just a glimpse of whatever it was that was lurking there in T-space. I was able to clarify this a bit by varying the Z-axis parameter and combining sequential images in a photo-stitching program. Then, by altering the formula, I could explore the three planes. In each of them, I found a stretched out M-set, but it was maddening to try to visualize the "solid" object that they must somehow combine to form.

Finally, after this prolonged gestation, everything came together. By virtue of Gerald K. Dobiasovsky's Rot3d_T-set formula, we were able to generate objects such as these:

http://ixitol.com/T-set.GIF
http://ixitol.com/C-set.GIF

This led to a brief flurry of elaborations and animations, but eventually it became clear that the Rot3d_T-set envelope has some major limitations. E.g., most of the free parameters that Fractint offers are spoken for in this formula, and it doesn't provide for any change of Z-axis angle. Moreover, if I understand the mechanisms here, there are potential problems in the trade-off of the depth of an image for its resolution. There is also the overhead of the relatively lengthly code.

It would be wonderful if we could start bringing more of Fractint's power to bear, directly, on this new exploration of 3-space. A 3D pixel would allow even more of the kind of free-wheeling investigations that have characterized this unique combination of mathematics, programming and artistry from its early days. The more basic, the better to start with, I suppose, leaving future enhancments to be inspired by the discoveries as we go along.

Again, your consideration of these suggestions is much appreciated. Thanks.

Russell