Realizing still life constraints as a planar tiling

For general discussion about Conway's Game of Life.
pcallahan
Posts: 410
Joined: April 26th, 2013, 1:04 pm

Re: Realizing still life constraints as a planar tiling

Post by pcallahan » January 14th, 2020, 2:57 am

Another idea is to forget about the octagons entirely and use these 6 domino-like tiles, assuming the user is able to count.
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These must be placed so colors of squares and red boundaries meet. Each corner square contains a number of dots equal to the number of black corner squares in the next two corners going clockwise. So when the tiles are placed, the dots give a quick visual indication of the number of neighbors.

This shows a block, a ship, a beehive, and and eater. It is easy to see that every black square contains two or three dots, and no white square contains exactly three dots. The tile without any dots is not really needed here but it can be placed around the boundaries.
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The tiles can be contoured to force the squares to line up by color, but the user is responsible for enforcing the cell counts.

I think if these were made with heavy enough material, like dominoes or mahjongg tiles, they might actually be fun to work with. I can only guess since I had to do it in Google draw. Another point is that they don't enforce any totalistic rule or the still-life property, so they could be used to set up and visualize the next step of any totalistic rule including life.

The disadvantage, of course, is that there is a brain involved in the process, if minimally. You would not be able to get a still life by some process of shaking the tiles together or translate it into a molecular computer. I wonder if these could be the basis of an actual game, e.g. pick some tiles randomly and build a still-life for points.

Update: we can make the counting automatic (e.g.) by clipping the corners off the above tiles and turning them into octagons. Then we can fill in the gaps with tiles that enforce the count. I am not sure if this results in a smaller tiling than the original one in this thread. At some point, the question is just whether the tiles vary too much in size to be usable.

Update 2: The same tile set with interlocking boundaries to force colors.
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Note that this tile set can be used for assembling non-stable Life patterns too. I think I've hit my attachment limit on this post, but by assembling three blocks in a row, you can see that two white squares above and below have a count of three (birth) and two black squares on the ends each have only one neighbor and will die.

pcallahan
Posts: 410
Joined: April 26th, 2013, 1:04 pm

Re: Realizing still life constraints as a planar tiling

Post by pcallahan » January 15th, 2020, 11:48 pm

Note that we can turn this into an automatic still life tiling by moving slightly into 3D. The idea is sketched below (and I emphasize it's just the idea, not a complete construction).
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The dots can be placed so that they have octagonal symmetry. "Color" the dots in a quadrant in clockwise order according to the two following quadrants, also in clockwise order. But instead of using colors, use bumps and dents. Now when the tiles are placed, the four quadrants incident to the same corner will form a symmetric octagon of bumps and dents representing neighbor counts. The semi-transparent blue octagons would have complementary bumps and dents so they can only be placed on a matching neighborhood, but may be rotated.

We could support flips I guess by having bumps and dents on both sides, but if one side is flat any we're limited to rotations, we have:

For live cells, counts 2 and 3, for 4+7 = 11 octagons total.
For dead cells, counts 0, 1, 2, 4, 5, 6 for 1+1+4+10+7+3 = 26 octagons total.

Note that there are only 3 neighborhoods of 6 cells, because one of them causes internal overcrowding and doesn't work for a still life.

So that gives us 37 octagons and 6 squares for 43 distinct pieces, ignoring the previous ways of splitting up octagon neighborhoods. This is not conceptually any better than the pure 2D tiling, but it could be easier to work with because the pieces are all about the same size.

The above is mainly for completeness, because I think that inspecting the dots visually may be adequate for using this as a puzzle and it really reduces the tile count by a lot, besides being more generalizable.

Update: A visual approach with octagonal symmetry, using "flower petals."
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Update 2: A mildly amusing aside (at least I find it so). After making these 6 tiles in Google draw, I decided to construct all rotations ahead of time to aid in making patterns out of them. 24 rotations I thought? Wait, no, some are symmetric and have duplicates. I went through the process of eliminating duplicates and was surprised that the result was a nice number like 16. It took me about a day to realize that obviously there are 16 2x2 bitmap patches before you start reducing the set with rotational symmetry. I completely forgot since I had been treating them as geometric objects so long.

Here's a link to the a drawing document with the tiles if anyone wants to copy it and try it out. https://docs.google.com/drawings/d/18aF ... lLdJs/view

Update 3 An alternative to matching dots to two chosen neighbors would be to match all three neighbors in each 2x2 grid by splitting the dots across boundaries. So one dot counts the diagonal, and each half dot counts the horizontal or vertical neighbor. This could be used to replicate the actual neighbors with correct spacing inside each assembled tile, sometimes talking a half dot each from adjoining quadrants. I might try to make some tiles like that, but I'm a little reluctant to split something that is already small. I think the petal design makes the neighbor connections very clear.

Update 4 Here's the split-dot solution if my comments above weren't clear. I did not try to add jigsaw contours, but this shows the idea. Now the dots in each compound cell form an exact image of the live neighbors around it. I think that's an interesting property and could help for non-totalistic rules, but I still prefer the flower petals.
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