Translucent Dominoes (Cooperation Games II)

Most tiling problems are strictly segregational, i.e. the tiles are not allowed to overlap. To change this, let’s consider tiles that are partially translucent, so that in order to really tile a part of the plane, one needs to cover it multiple times.

This is the first in a series of posts about partially translucent polyominoes, and we begin with translucent dominoes, of which there are three:

Dominoes 01

The purple one is a regular (non-translucent) domino, and to the right you can see my feeble attempt to tile a 4×4 square of which two opposite squares have been removed (This is of course a very classical puzzle). The other two dominoes have one or two translucent squares, which are shown as gray. This translucency means that in order to properly tile a square with dominoes, we need to cover it either with a single solid color square, or with two translucent (gray) squares, i.e. the gray portions of two dominoes must overlap, like so: Connect 01

The left image shows two fully translucent dominoes that overlap in the middle square, while the left and right squares are still only covered once. By counting the small connector squares you can see how many gray squares sit on top of each other. In the middle is a chain of four dominoes, all gray tiles are doubled. And to the right we have covered the middle gray square three times, which is illegal for now. 

If we use only the blue singly translucent domino to tile, two of them need to overlap to form a single classical (segregational) tromino, so tiling with these dominoes alone is equivalent to tiling with trominoes.

Domino tromino 01

You should try to tile the 4×4 square (with two opposite corners removed as above) with copies of either the singly or the doubly  translucent domino. In both cases, this is impossible (and impossible for larger squares as well, again with opposite corners removed. You will enjoy finding the arguments). Tiling becomes easier when you allow both types of translucent tiles, a simple solution to the 4×4 puzzle is shown below to the right. The left figure gives a hint what limitations you face when you try to tile with the doubly translucent domino alone.

Domino4x4 01

As a cooperative game, start with a 6×6 square, mark a few tiles as forbidden, and then take turns to place translucent dominoes on the board with the goal to tile the board completely, following the translucency rule.

Hic Sunt Dracones

Take a long strip of paper, and fold it left over right, then left over right again, and so on, a couple of times.
Even if your strip is very thin and long, you probably won’t be able to do that more than six or seven times.

Then carefully unfold the paper so that each bent makes a right angle. What you get will look like this:

Samples 01

Another method leads to the same curves. Start with a curve consisting of two segments, making a right angle. Think of it as being a track you want to walk along. Things being difficult, you happen to swerve slightly to the right on the first segment, and on the second slightly to the left, meaning that instead of following the blue path, you walk the red path:

Drunk1 01

Now try again, this time starting with the red path that is four segments long (and colored blue below). The same happens, you alternatingly swerve right and left, creating the next (red) path. The curves will be the same ones as above.

Drunk 01

Is there any sense to it? Things get more amusing if you replace each segment by a square that has that segment as a diagonal. This turns the curves into polyominoes, as you can see for the first few cases below.

Polyominization 01

You will also see that these shapes start resembling a common dragon. If you keep folding a little while, more details emerge.
Reddragon 01

But it gets better. All these polyomino-dragons tile the plane, interlocking perfectly. Both the young dragonlings

Dragontile1c 01
and also the older, wrinkly ones:

Dragontile2 01

Now imagine stacking these dragons on top of each other, generation by generation. If I had the money, my mansion would look like that.