. Andrew Glassner’s Notebook http://www.research.microsoft.com/research/graphics/glassner Aperiodic Tiling ____________________________________ eople love patterns. We find recurring patterns tals” will be nothing but collections of polygons in the Andrew Peverywhere we look—in the structures of rocks, the plane. It’s well known that there are only three regular Glassner personalities of our friends, and the cycle of seasons in ploygons that can tile the plane, as shown in Figure 1. the natural world around us. Theme and variation have Here the verb “tile” means to cover the infinite plane Microsoft been a staple of creative invention since time immemo- with a set of polygons so that no gaps or overlaps exist Research rial, spanning every form of creative endeavor. I suggest among the polygons. Each polygon is called a “tile,” and that theme and variation represent a balance between the composite pattern is called a “tiling.” the extremes of regularity and sheer randomness. The tilings in Figure 1 may be theoretically interest- Imagine a million grains of sand, placed one next to the ing, but because they’re so perfectly repetitive, they look other in a straight line. Dullsville. Now imagine those boring. Usually when you use one of these patterns to grains of sand scattered at random over a rocky surface. cover a wall or floor, you decorate the tiles with colors That’s just as dull as before. But take those grains and stack or shapes to make the whole thing a little more inter- them up, let the wind play over them and shape them into esting—again balancing the tiling’s regularity with the dunes, and you’ve got something beautiful and interesting. variation in the tiles themselves. I did this in Figure 1 by Balancing between repetition and randomness can lead using two or three colors, but it didn’t help much. You to patterns that draw us in and keep our interest. can generate more interest by allowing more than one One of the most interesting ways of assembling small regular polygon in the game, resulting in the semi-reg- units is along one of the lattices that make up crystals. ular tilings in Figure 2. This looks better, but regularity In this column I’ll live entirely in a 2D world, so our “crys- still dominates the patterns. (a) (b) (c) (a) 2 Eleven semi- regular tilings 1 Three regular made of one or tilings made of more regular many copies of polygons. Every a single regular vertex contains (b) polygon. the same kinds of polygons (d) (e) meeting in the same order. (c) (f) (g) (h) IEEE Computer Graphics and Applications 83 . Andrew Glassner’s Notebook 3 Matching rules. (a) Vertex rules. Corre- sponding colors must overlap. (b) Edge rules. The deformed (a) edges must link together. (c) Face rules. The decoration on the faces (a) (b) (c) must be continuous. We can control our tiling patterns by adopting match- ing rules. These describe the permitted ways in which tiles can be placed next to each other. Figure 3 shows a few examples for assembling such a tiling from a single piece. All the examples in this figure share a matching rule that says tiles may only be placed so that the shared edge is of the same length (that is, we can’t push a short edge up against a long one). I also distinguished the two short ends. In Figure 3a, I used a matching rule that says adja- cent vertices must have the same color. Figure 3b enforces (b) the same result by modifying the common edge into a mountain, so the tiles can only interlock in the desired 4 (a) Isosceles triangles can tile the plane periodically. way. In Figure 3c the rule stipulates that colored bands (b) The same triangles can create a radial tiling, which must be continuous across tile edges. Whether described is not periodic. as vertex, edge, or face rules, we get the same pattern. The patterns in Figures 1, 2, and 3 are periodic. Essen- tially this means you find a group of tiles that you can definition is important. Figure 4a shows a periodic tiling pick up and use as a rubber stamp. In other words, the built from an isosceles triangle. Figure 4b shows a rota- pattern is created by a single patch—called the funda- tionally symmetric pattern built from the same triangle, mental cell—that repeats by translation to cover the but this one isn’t a periodic tiling, even though there’s plane. The distance from one copy to the next is called obviously a great deal of internal structure. the period of the pattern. The translational part of the You can create tilings in surprising ways. Figure 5 shows a spiral tiling (which is also nonperiodic), inspired by a con- struction by Heinz Voderberg. Although I used colors for decora- tion, this pattern is made up of many copies of only one tile. I found that the easiest way to construct a cell for this sort of tiling is to start with an acute isoceles triangle, as shown in Figure 6. The tiling works if the acute angle goes into 360 degrees an 5 A spiral tiling even number of times (I used 16 for made of a single this figure, so the acute angle is tile. The tiling 360/16 = 22.5 degrees). Let’s see can be extend- how this works. First, create a par- ed to cover the allelogram from the triangle. Draw entire plane. a curve, and then rotate that curve by 180 degrees to get it to match up with the opposite corner. Now take another copy of the triangle and move its acute point to the lower right corner of the parallelogram. Make a copy of the curve and rotate it around the lower right corner of the parallelogram by the acute angle. Now join the sides of the 84 May/June 1998 . (a) (d) (a) 7 (a) The tile made in Figure 6. (b) This tile fits into itself by (b) (e) rotation. (c) The tile also fits into itself by a verti- cal reflection, making a block that can tile the plane periodi- (b) cally. (c) (f) (g) (c) 6 Creating a tile for Figure 5. (a) Start with an isosceles triangle with an angle of 360/n degrees (here n = 16). (b) Make a parallelogram. (c) Find the midpoint. (d) Draw a curve from the upper right corner to the midpoint. (e) Rotate the curve 180 degrees. (f) Rotate around the lower right corner by an angle equal to the 8 (a) The rec- acute angle of the triangle. (g) Join the edges. tangle can tile the plane peri- odically. (b) The curves, and voilá. The only tricky part is ensuring the rectangle can curves don’t intersect each other. also tile nonpe- Figure 7 shows that a tile produced this way has two riodically. important properties: it fits into itself both by a small rotation equal to the acute angle of the original triangle and by a full 180-degree rotation. Notice that the spiral (a) (b) tiling in Figure 5 uses both properties: the two centers use slightly rotated copies of the tile, and they join each other with a half-rotated pair. From there it’s pretty since they can be coerced into a periodic pattern. If straightforward to add tiles and follow the spiral out- there’s a set of one or more tiles that fit together to cre- wards. Notice, though, the pattern changes slightly each ate exclusively nonperiodic tilings, that set of tiles—and time it wraps around 180 degrees. the resulting pattern—is called aperiodic. Typically in any periodic tiling an infinite number of To my eye, aperiodic tilings made of a few distinct tiles fundamental cells exist, but often there’s only one small- fit into that desirable class of patterns that balance reg- est such cell. For example, in Figure 1b the smallest fun- ularity (because of the recurring instances of just a few damental cell is a single square, but you could use two tile shapes) and randomness (because the pattern never squares adjacent horizontally or vertically, or a 2-by-2 repeats). This column is the first of two that talk about square of squares, and so on. aperiodic tiles, how to make them, and what you might It’s easy to build nonperiodic tilings—simply don’t do with them. The second column will focus on the create periodic patterns. Figure 8a shows a periodic details of a specific family of tiles. tiling of rectangles, and 8b shows the same rectangles in a nonperiodic tiling. Theoretically we could extend the From 20,000 to 2 pattern of Figure 8b, always mixing things up, so that The question of whether any aperiodic sets of tiles the final result is not periodic. existed at all went unanswered until recently. In 1961 Can you create tiles that only tile the plane nonperi- Hao Wang conjectured that there were no aperiodic tiles. odically? The rectangles of Figure 8 don’t fill the bill, That is, any set of tiles that could tile the plane aperiod- IEEE Computer Graphics and Applications 85 . Andrew Glassner’s Notebook Fa Fb Fa Fc Fa Fb Fc 9 The Robinson Fe tiles. Pieces can Fc Fd only go togeth- Fa Fe er so that the Fa tabs fit the slots and the corners are covered. 10 Building a 3-by-3 Robinson block. Fe Ff Fa Fe Ff Fe Fa Fa Fa Fe Fb Fe Fc Fc Fb Fe 11 A 3-by-3 Robinson block Fa Fc Fa Fa Fc (shaded grey) Ff 12 A decoration of Figure 11.