Contents:
The Tessera puzzleTM, created by John Osborn and sold by the DaMert Company, is inspired by the tessellations of M. C. Escher. It consists of 60 sculptured plastic "beetles" in eight different shapes and colors. They can be snapped together like a jigsaw puzzle, except that there is no one "correct" solution.
There are lots of ways to play with the beetles without getting mathematical -- at least not consciously! But eventually some questions arise in the curious mind. How do they work, first of all? How can I be sure I won't end up with odd-looking holes in my pattern? (Or, how can I make interesting holes?) Can I make symmetrical patterns? Can I make a repeating pattern that would go on forever if I only had enough beetles? How can I be sure?
What you are looking at is a first attempt to answer some basic questions. I explain how TesseraTM works by dissecting the insects. Then I start to explore the possibilities of infinite patterns by looking at a simpler problem. This much I understand pretty well, although I haven't proved everything with mathematical rigor.
When I return to looking at the original problem, my answers are very incomplete. I have seen some patterns, but I can't prove that they go on forever. If anyone sees this and has ideas about how to extend what I've done, or how to answer questions that I didn't touch, I would be interested in hearing them.
-- Rick Peterson, rmp605@aol.com
The eight shapes are based on hexagons, with each edge replaced with one of four curves to make an antenna or a leg. There is actually only one basic curve, which is reflected in two ways to yield the four curves. Calling the basic curve L, one reflection gives curve R; the other reflection gives curve L'; and both reflections together give curve R'. Curves L and L' only occur on the left side of a beetle; curves R and R' only occur on the right side of a beetle. Curve L interlocks only with curve L', that is, an edge on one beetle of type L fits together with an edge on a second beetle of type L'. Likewise types R and R' interlock. Edge types L and R make a long antenna or leg; types L' and R' make a short antenna or leg.
In the figure above you see the four edge types, drawn on wedges that can be fit together to form a beetle as in the example at right. This example can be identified as LL'L' (starting at the head). The right side always follows from the left side because all beetles have bilateral symmetry.
If we refer to edge L as 1 and to edge L' as 0, the eight beetle types can be identified by binary numbers, as seen in Table 1.
The example above, LL'L', is type 3, which is colored red (or reddish purple) in the TesseraTM set.
We are interested in finding regular tessellations of the plane by the eight types of beetles. That is, how can the beetles be interlocked so that (if we had an unlimited supply of beetles) we can be sure they would interlock with no empty spaces to cover any size surface?
We can start by simplifying the problem. We know that an edge on the left side of a beetle can only interlock with an edge on the left side of another beetle, and likewise right with right. So instead of beetles, consider hexagons with one side (the left side) painted white and the other side painted black. Plain hexagons tessellate in a honeycomb pattern. How can each hexagon in the honeycomb be oriented so that black meets black and white meets white?
Consider one vertex of the honeycomb. Apart from overall rotations and reflections, we find two patterns in which black sides meet at the vertex. In the first case a fourth hexagon (at the right) can be in one of two positions.
Starting with pattern 3 as a seed, we can work outward and find that this pattern is forced:
If, on the other hand, we start with pattern 2 as a seed, it does not force a single tiling for the entire plane. It forces a vertical half-strip above the seed, as in the following figure.
We can say furthermore that if the entire strip (downward as well as upward) is not tessellated in this "bent" configuration then the plane must be tessellated in the "triangular" configuration of figure 3. For if the hexagon just below the seed is not horizontal (white above black) then seed pattern 3 is forced, thereby forcing the entire triangular tessellation; but if the hexagon is horizontal, the "bent" tessellation is forced one hexagon downward; and this reasoning can be iterated infinitely. Thus any tessellation of the plane that contains seed pattern 2 will either be the "triangular" tessellation or contain a strip of the "bent" tessellation.
Seed pattern 1 (Figure 2) allows for more variations than patterns 2 and 3. It only forces a block of eight hexagons. If we do not allow seed pattern 3 to appear in the tessellation, we get four different (partial) tessellations. Three of these necessarily extend to infinity both ways in one dimension (shown as vertical in Figure 5). The fourth is a patch which can be extended in the same fashion, but it is not forced to do so. These four "straight" stripes can be "mixed and matched" with one another arbitrarily to tile the plane, just by setting them side by side.
The fourth tessellation, in Figure 6, can also be extended to make an infinite stripe in a different direction. It is shown rotated so that this second direction is vertical. In this way it can be seen to combine with the "bent" tessellation stripe of Figure 4: the side columns of the "bent" stripe and this "slant" stripe are identical, so stripes of either type can be overlapped by one column to make another "mix and match" tessellation of the plane.
I am not entirely sure I have found every tessellation of the plane using half-colored hexagons. Perhaps something else can be done with the patch tessellation of Figure 6. But what I have found, in summary, is one tessellation with threefold rotational symmetry (the "triangular" tessellation, Figure 3) and two sets of tessellations with translational symmetry in one dimension. In Figure 7 these two sets ("straight" and "bent/slant") are depicted in a simpler form than that in which they were derived.
Now that we have enumerated all the oriented-hexagon tessellations, it is only necessary to choose one of them, then choose which of two types to use for each edge (L or L' on the left side of a beetle; R or R' on the right side).
Suppose we choose an oriented-hexagon tessellation that is a regular lattice, that is, symmetric under two particular translations in different directions. Then we can choose a unit cell and assign a type to each edge in the unit cell, remembering that opposite sides of the unit cell are equivalent. For example, Figure 8 shows the smallest unit cell for a tessellation consisting of stripe 1a only, with all possible edge type assignments. The little triangle symbols point from the hexagon having an L or R ("long") edge to that having the corresponding L' or R' ("short") edge. The hexagons are colored to match the beetle color, found in Table 1.
Note that there are two colors of triangles. Choosing one red-triangle edge to be long or short determines all the other red-triangle edges, either by bilateral symmetry of one beetle or by translational symmetry, i. e. identifying an edge on one side of the unit cell (drawn as a dashed rectangle) with an edge on the opposite side. The same holds for the blue triangles, so that there are 2 x 2 = 4 choices for all edges, as shown. However, the two tan-and-black tessellations are identical apart from a 180 degree rotation, and likewise for the two red-and-brown tessellations, so there are in reality only two different tessellations with this particular symmetry.
[Note also that the two tessellations above both form stripes of alternating colors. The stripes are diagonal in one case and vertical in the other. Are there other ways to make stripes of alternating colors, with different orientation patterns?]
As another example, take the "bent" tessellation of Figure 4, extended infinitely. The smallest unit cell is as shown in Figure 9, with two of the 16 choices for the four sets of edges. Note that the edges on the right side of the left configuration match those on the left side of the right configuration. Thus we could choose a unit cell twice as wide by merging the two cells shown, obtaining an infinitely repeating tessellation that uses all eight types of beetle. (Furthermore, it turns out that a pattern six unit cells wide and including only the leftmost five columns of hexagons uses exactly the 60 beetles that come in the TesseraTM set, which includes twice as many blacks and tans as the other beetles.)
The background pattern for this Web page is based on the "straight" tessellation of Figure 6, extended both vertically and horizontally. It is very similar to Figure 9 with the two unit cells joined, but it has a unit cell six hexagons wide, two high, and non-rectangular. The next unit cell to the right is shifted up (or down) by one hexagon. Like Figure 9, it uses all six beetle types; it needs three times as many black and tan beetles as the other types.
The same approach can be used in situations that do not have translational symmetry, except that there are no unit cells so the process must be continued indefinitely. The "triangular" tessellation is an example. If we require that the threefold symmetry of the oriented-hexagon tessellation be preserved, two possible tessellations are shown in Figures 10 and 11. It appears that the pattern of concentric rings of black and tan beetles will continue infinitely, but I do not yet have the tools to prove this.
