More about Decorated Squares (Five Squares III)

In order to classify surfaces that have five coordinate squares around each vertex, we were led to consider planar tilings with six different colored squares. Today we will discuss a special case of this, namely tilings that use just two of these squares. The only rule to follow is that colors of tiles need to match along edges. Here is an example:


To classify all tilings by these two squares (and their rotations), we first simplify by solely focussing  on the gray color (making it dark green), and considering the blue, orange, green as a single color, namely light green. This way we get away with just one tile. Of course we hope that understanding how this single tile can fill the plane will help us with the two tiles above.


We first note that placing the tile determines three of its neighbors around the dark green square. So instead of tiling the plane with copies of this squares, we can as well place dark green squares on the intersections of a line grid so that for each cell of the grid, precisely one corner is covered by a dark green square, like so:


We first claim that if we do this to the complete grid, we must have a complete row of squares or a complete column of squares. Below is a complete row (given the limitations of images). The red dots indicate where we cannot place green squares anymore, because the grid squares have all their green needs covered.


If we do not have such a row, there must be a square without left or right neighbor. Let’s say a square is missing its right neighbor, as indicated in the left figure below by the rightmost red dot.

Notice how the two grid squares to the right of the right dark square have only one free corner. We are forced to fill these with dark squares, as shown in the middle. This argument repeats, and we are forced to place consecutively more squares above and below, completing eventually two columns.

As soon as we know that we have (say) a complete horizontal row, directly above and below that row we will need to have again complete rows of squares, as in the example above. These rows can be shifted against each other, but that’s it. So any tiling of the plane by the dark/light green tile consists of complete rows or columns with arbitrary horizontal or vertical shifts, respectively.

Finally we have to address the question whether this tells us everything about tilings with the two tiles above. This is easy: Each dark green square represents a light gray square that is necessarily either surrounded by blue or orange tiles. So we can just replace each dark green square by an arbitrary choice of such a blue or orange cluster. The final image shows such a choice for the example above.


It is now easy to stack several such tiled planes on top of each other, thus creating infinite polyhedral surfaces that have five coordinate squares at each corner.



We can play domino with identical copes of a single cube by insisting that the cubes have matching colors at the faces where they touch. This is hard to convey in a perspective image like the one above that shows a 4x4x1 cubomino tiling, so I will switch to a 2D representation that shows each cube in central perspective from above. Here is the flattened version:


As you can see, the single 3D cubomino can be represented by six 2D cubomino squares, which may be rotated:


These are a subset of the tiles I used for the Compass game a while ago.

This new Cubomino puzzle is a simple example that teaches to analyze tilings by understanding them as boundary value problems. To see how this works, we first notice that to some extent the color of the lower and the left edge of a tile determines that tile and its rotation.


It works for most color combinations, but there are exceptions. These occur precisely when the two chosen colors for bottom and left edge are antipodal, i.e. are either equal or are one of the three pairs of colors of two opposite faces of our cube:


This observation extends to larger rectangles: No antipodal colors can occur both in the left edge and the bottom edge of a tiled rectangle. To see this, assume the contrary. The vertical edge on the left of the rectangle that has one of the antipodal colors (pale yellow below) determines a horizontal strip of tiles that have the antipodal colors as vertical edges, and the horizontal edge on the bottom of the rectangle with the second color from an antipodal pair (dark purple below) determines a vertical strip of tiles that have the antipodal colors as horizontal edges. These two strips meet in a tile that must have a boundary consisting just of antipodal colors, which is impossible.


On the other hand, if the left and bottom edge have no antipodal colors in common, like in the example below,
there is always a unique tiling of a rectangle that has the two edges as its lower and left edge.


The reconstruction process is easy: Each choice of a left and bottom edge color determines a tile and its rotation uniquely. We begin by placing the only possible square into the bottom left corner of our boundary, and work our way to the right and up.