Joining two things in a single tile

[t686]

You can see on the website, https://quantumfrontiers.com/, scrolling down the page, it mentions tiling the plane aperiodically.  If you can tile it correctly, you can make a turing machine out of it.  What I was noticing that when they make aerogel, the least dense commercial prouct, they put the gel into an autoclave, and the pressure combines a liquid and gas into one.  I was previously thinking that you need two things to make something interesting, when really it's just about the one thing.

It's a little bit complicated, but I made an X-shaped cellular automata, you can hit play on the first post in this thread: http://www.conwaylife.com/forums/viewtopic.php?f=11&t=3195&start=0

However, you see it's a solid square shape mass of white and black squares followed by 4 long diagonals.  It's the center part that's important.  I made a clockwise and flipping over that pattern, you have then a counterclockwise pattern.  However, splitting that center large square mass (remove diagonals), and joining the clockwise top half to a counterclockwise bottom half, makes a single tile that tiles the plane.  By rotating the tile 90 degrees or flipping it, you have essentially a different looking tile but it's the same tile, and they would join on the triangles edge to edge.  There 's a spacing in between the tiles, but like in the quantumfrontiers.net, if you lay out a row of the tiles, at some point there should only be one choice on how to add the next row.  This is a pattern for a cellular automata, but that's what it would be.  Also, http://mathworld.wolfram.com/SingularPoint.html, the reason why I decided to now update the center part is, look at the "ordinary double point" in that link.  Originally I made the pattern with a single tile with 4 long diagonals extending out from it.  They were originally clockwise or counterclockwise.  When you make the clockwise, for instance, the four long diagonals each curl in the clockwise direction.  By changing the center large mass of the X pattern, to clockwise on the upper half of the center mass, and then changing the bottom half of that center mass by only flipping the bottom half left-to-right, so you have a center tile of clockwise over counterclockwise in one.  That clockwise upper half, in the "ordinary double point" in the wolfram link, the upper two lines curve clockwise, and the lower two lines curve counterclockwise, same as in my new center tile (which I haven't posted or drawn but will show it later as an update).  You're joining the tiles connecting them edge to edge left-to right at the triangles, and then start another row, like in the turing machine tiling in quantumfrontiers.net.  The original tile actually spins in a circle at the center in the cellular automata.  Now the worlfram singular point, the new tile is like that, and notice the picture of the double point is basically "bound" on one half, and "diverges on the other half.

That's the same as joining jpeg compression with fractal compression.   On compressing a jpeg movie, the edges of the image are jagged or square and choppy, which causes ghost images in the lossy compression to fan out to a great distance from the image as ghostly remnants.  The fractal image compression leaves smooth edges, so it "bounds" any ghost images, the jpeg, "diverges" them.  I believe the tiling of the original center mass, you're removing the long diagonals, so you have an actual tiling on a cellular automata square grid in 2-dimensions.  That's a chaotic rule I used, and I'm not sure, but it may be that you make the cellular automata in a tiling like this, hit play, and since this rule looks like snow on a television screen, it may just be that you use the tiling as "background noise" pasted behind your image, so that the image will compress random images in a lossless form of the best compression method today, because random images or data cannot currently be compressed, if you tile the background with chaotic noise in a single image tiling aperiodically, it replicates a supercritical thing like when they make aerogel, so it would be able to compress random data.

Actually the original center mass, works by itself (the center area on the pattern, without the long diagonals, it then makes a square tile).  Flipping it left-to-rgiht, makes a mirror image of the original but doesn't change the tile.  You would use the mirror image and the original image somehow  (in some combination in the entire pattern) in tiling them in a long row left to right, joined at the "flags", and then start another row.  The reason the original image spins at the center is that rotating the original center piece through 90 degrees changes nothing about the appearance of the image at all, because it's invariant to 90 degree rotations, but you need the mirror image to make a tiling (flip left-to right or flip up-down, doesn't matter to make the mirror image)

Edited June 4, 2019 by t686