In regards to method #3, reconstruction, I'm looking for more methods.
In general my primary approach is converting the 3D space to a 1D space using a 3D space filling curve. Then I convert the 1D space to a 2D space using a 2D space filling curve. Right now my library of curves includes: 2D hilbert curve and 3D hilbert curve. That's it! They produce a very interesting result that might be optimal, but I don't want to blindly assume it is optimal.
Here is how these two curves together map 3D space to 2D space
(rgb values are xyz offsets)
There are a lot of good things about this output. Similar colors (neighboring spots in 3D) mostly end up close to each other. Here is a screenshot of this curve combo in action:
Alternate view using method #1
The blue tesseract is mostly kept together in the final image, and the red hypersphere is mostly kept together in the final image. I believe though that this is a mostly universal feature of my method in general. Different space filling curves should have similar "locality" but might be better overall.
So which other curves should I try?
Since the beginning, I've also been on the lookout for a good 3D space filling surface. The current method converts 3D to 1D and then 1D to 2D which "shreds" the image more than necessary. A better approach might be to go directly from 3D to 2D. The simplest method would be just a 2D hilbert curve that is extruded along the third dimension but I dislike how it makes 1 dimension act completely different than the other 2.
In general my primary approach is converting the 3D space to a 1D space using a 3D space filling curve. Then I convert the 1D space to a 2D space using a 2D space filling curve. Right now my library of curves includes: 2D hilbert curve and 3D hilbert curve. That's it! They produce a very interesting result that might be optimal, but I don't want to blindly assume it is optimal.
Here is how these two curves together map 3D space to 2D space
(rgb values are xyz offsets)
There are a lot of good things about this output. Similar colors (neighboring spots in 3D) mostly end up close to each other. Here is a screenshot of this curve combo in action:
Alternate view using method #1
The blue tesseract is mostly kept together in the final image, and the red hypersphere is mostly kept together in the final image. I believe though that this is a mostly universal feature of my method in general. Different space filling curves should have similar "locality" but might be better overall.
So which other curves should I try?
Since the beginning, I've also been on the lookout for a good 3D space filling surface. The current method converts 3D to 1D and then 1D to 2D which "shreds" the image more than necessary. A better approach might be to go directly from 3D to 2D. The simplest method would be just a 2D hilbert curve that is extruded along the third dimension but I dislike how it makes 1 dimension act completely different than the other 2.