Dynamic Tilings

Abstract

In this paper I will investigate some properties of a special group of tilings. Namely those tiling’s that can be coloured with only two different colours in such a way that every set of two tiles which have a common edge have a different colour. In the works of M.C. Escher these tiling’s play an important role. In his book Regelmatige Vlakverdeling [1] M.C. Escher writes about the foreground-background phenomenon of this group of tilings. In fact these 2-colorable tilings can be seen as a compound of two groups of tiles. And so we can split a 2-colorable tiling in two sets of tiles and we can concentrate on only one of these sets.

1. Introduction

1.1. Foreground - Background. In M.C. Escher’s print Houtsnede II we see many figures, "flying fish", of which some are dark and some are light. It is hard to concentrate on all of them at the same time. You see either “white” flying fisches on a black background or “black” flying fishes on a white background. In many more of his prints Escher experimented with this phenomenon. In this example the foreground changes to background from left to right and vice versa. Especially in tiling patterns that can be coloured with only two colours we see this division in two groups of tiles.
1.2. Regular Tilings. In Escher’s regular tiling pattern No. 105 (Flying Horses) we can choose to concentrate on the red horses and to see the rest as just a background colour. When we do that as in Figure 1a we have left only half of the tiles. Because of the shape of the tiles we now can slide them together (Figure 1b) and they will nicely fit together again. The final result (Figure 1c) is exactly the same tiling but now with one colour and only half as much tiles.
Figure 1a: Red Horses
Figure 1b: Sliding together
Figure 1c: Original Tiling
2. Moving the Tiles

2.1. Black and White. Let us try to understand what is happening during this sliding transformation. In Figure 2 we start with a simple square pattern of which we slide the tiles away from each other.
Figure 2a: Squares
Figure 2b: Sliding
Figure 2c: Sliding
Figure 2d: Squares
After we have done this so far that we reach the situation of Figure 2d we again have a square pattern in which the shapes of the openings (which is in fact the background) are squares of the same size as the original tiles. So now we can fill in these holes with new square tiles. The result is again a simple square pattern but we have made it twice as big as the starting pattern.
Figure 3a: Squares
Figure 3b: Sliding
Figure 3c: Sliding
Figure 3d: Squares
The question now is: is it really twice as big? In principal a tiling pattern is an infinite pattern. So an infinite number of tiles. And two times infinity is again just infinity.