Examples of self-similar tilings

 

Both the tilings by square and equilateral triangles are self-similar: the figure below shows how to group the tiles to exhibit the self-similarity.

 
Figure 21: Self-similarity of square and equilateral triangle tilings: blue lines surround groups of four tiles each

In both cases, the similarity is simply the doubling map . The triangle tiling is slightly different in that the prototiles are placed in two different orientations. The tiling by regular hexagons is not self-similar; one cannot find larger hexagons tiled by smaller ones. However, there is a process by which one can find a self-similar tiling which is based on the hexagon tiling. We refer the interested reader to [Thu89].

To exhibit a self-similar tiling, we must follow the pattern of L-systems in the sense that we give ``production rules'' showing how to dissect each prototile into pieces that are smaller copies of the prototiles. Next, we give some slightly more complicated examples. The first is a tiling by ``triominoes'' (three squares glued together, rather than dominoes which are formed by composing two squares). The large triomino is shown in this figure partitioned into four triominoes half the size of the original one.

  
Figure 22: Grouping of triominoes

The tiling of the plane is obtained by expanding the smaller triominoes to be the same size as the original one (i.e. doubling the scale), and then subdividing the four resulting triominoes into half-size triominoes again. This process is called inflation; repeating the inflation without end, we obtain an infinite tiling. There is one small detail to be dealt with: where to position the enlarged triominoes; if we position the rescaled lower-left triomino to coincide with the original triomino, the tiling will extend over only the first quadrant of the plane. If we position the rescaled inner triomino over the original triomino, the pattern will extend over only quadrants II, III, and IV. To obtain a complete tiling of the plane, we take as our ``axiom'' a union of two nested triominoes, and keep the central vertex always at the origin of the plane. The sequence of generations is shown in this figure.

  
Figure 23: Generations of triomino tiling

The resemblance between the ``production rules'' for self-similar tilings and those for L-systems can be exploited to produce FRACTINT-style L-systems generating the tilings. In particular, FRACTINT has another example based on ``hexiamonds'' (unions of six equilateral triangles; a diamond is a union of two) called Sphinx.

Sphinx { ; by Herb Savage
; based on Martin Gardner's "Penrose Tiles to Trapdoor Ciphers
; This is an example of a "reptile"
Angle 6
Axiom X
X=+FF-YFF+FF-FFF|X|F-YFFFYFFF|
Y=-FF+XFF-FF++FFF|Y|F++XFFFXFFF|
F=GG
G=GG }

The production rule for the sphinx tiling is shown in this figure.

  
Figure 24: The production rule for the Sphinx tiling

Note that there are actually two different sphinxes, if we restrict ourselves to orientation-preserving similarities. Type X and Y sphinxes are mirror images of one another. This accounts for the two production rules for X and Y in the L-system. Type X is subdivided into three of type Y and one of type X, and the mirror image decomposition is used for Type Y sphinxes. The production rules F=GG and G=GG show that the scale factor from one generation to the next is 1/2. The lines drawn in the previous generation are erased, and only the new smaller sphinxes are drawn. In this figure, we trace out the production rule precisely for Type X.

  
Figure 25: Tracing out the X production rule

Several generations of the sphinx tiling appear in this figure. All the tilings in this section have production rules subdividing the prototile into 4 congruent smaller tiles, which are half the size of the original. The square and equilateral triangle tilings have translational symmetries as well as the self-similarity, while the sphinx and triomino tiling apparently do not. This difference is a particularly interesting distinction between dynamical systems, and our next section's main topic.

  
Figure 26: Generations of the Sphinx tiling