After many blog posts inspired by dates, including the Queen’s Birthday, Christmas Day, the National Days of Argentina, Brazil and Taiwan and the date of my Hip Op, it’s time for some basic tuition – though June, the sixth month, is still an inspiration, leading me to revisit, Lattice Labyrinths on the square lattice when both separation parameters are even – a situation I’ve been ignoring, as the Chinese and Serpentine families on the square lattice are completely soluble and the families of tessellations on the triangular lattice are so pretty.
When the separation parameters (a,b) are one even or zero, one odd, we get a Chinese Lattice Labyrinth such as (5,0) or (7,0). The area of the supertile a²+b² is always of the form 1+4n which corresponds to a tetragonally symmetrical supertile having a single central square and four identically-shaped arms. (A reminder: The separation parameters tell you how far across (a) and up (b) the lattice you have to count to come to a point identical in its environment to the one you started from; they tell you the scale on which the pattern repeats itself.)
If both a and b are even, a²+b² is of form 4+4n, so if a tessellation of tetragonally symmetrical supertiles is to exist, each must have at a block of at least four square tiles at the centre of each supertile. Finding such tessellations presents a big problem to me, as the missing-links graph algorithm as used above is of little help. Lower-order cases can be found by trial and error – below I illustrate case (6,4) – two different constructions.
I call the family of which (6,4) is an example the Windmill lattice Labyrinths, a name suggested by the shape of the lowest-order member of the family. I constructed the left-hand, yellow and green tessellation by trial and error. Its tessellation graph is drawn in black. When one “reverse-engineers” this tessellation by constructing its missing-links graph , shown in red, one discovers that the result is the tessellation graph of another version of Windmill Lattice Labyrinth (6,4), shown on the right in yellow and blue.
This property, that a Windmill Labyrinths is “complementary” to another Windmill Labyrinth is shared by all members of the family. Some members are complementary to themselves, some are complementary to a mirror image of themselves, the rest to a differently-shaped tessellation, as in the case illustrated above.
Despite the above beautiful property, I find the Windmill family unsatisfactory because the lattice point at the centre of each supertile is unconnected by/to the tessellation graph. We CAN connect these points without upsetting the symmetry of the tessellation as a whole by constructing a cross at each supertile centre to divide each supertile into four identically-shaped arms. This does however mean that these tiles meet in edges of the tessellation graph rather than at vertex (lattice) points, corner to corner. This feels like a generalisation too far. However, this is not the end of the story for (even,even) separation parameters.
In the last few years a Japanese academic team (see reference at the end of the post) have been working their way through all the low-order polyominoes that tessellate, by an ingenious method quite different from mine and soon becoming impractical within the times of the order of the apparent age of the Universe for higher-order cases. Working through their results and correlating them with my lattice Labyrinth Families, I found one that didn’t fit! Here it is.
Well! Both separation parameters are even, the full symmetry of tetrad and dyad axes is there, the supertiles meet corner to corner AND all lattice points (corners of the squares of the square lattice) are connected in the tessellation graph. the above tessellation is indeed the lowest-order member of yet another infinite family, which I had to call the Japanese Lattice Labyrinths in honour of its source – and to balance nomenclaturely with the Chinese Lattice Labyrinth family.
(R, the repeat unit (or fundamental domain) is the number of squares in the pattern that are repeated to form the tessellation, S is the area of each supertile. In this case four supertiles in the four possible different orientations make up the repeat unit)
To return to June, month 6, here is another member of the Japanese family, (6,4).
Unlike the Windmill Labyrinths, the Japanese family can be constructed via a straightforward missing-links graph (shown here in red), though spotting general construction rules that cover all cases is not easy. However, my attempts to construct a Japanese Lattice Labyrinth for another “June” case, (6,2) met with persistent failure. It can’t be done! here is the best I can do:
I decided that Scarthin had to get an entry into my terminology; it’s where I live and work (and well worth a visit to the bookshop, Scarthin Books, the pub, The Boat Inn or just to relax on The Prom, overlooking the millpond, The Dam). The Scarthin Lattice Labyrinth family is undoubtedly also infinite in membership; its lowest-order “basic tessellation” is case (4,0), which you might like to try constructing. Alas, this family, like the Windmills has a block of four squares in each supertile with the central lattice point not connected to the tessellation graph.
We can characterise which (even,even) cases are drawable as Japanese Labyrinths and which as Scarthin labyrinths algebraically, but it’s perhaps easier to get the point from this little sample table, below.
At present I’ve forgotten my clear and simple explanation of why the (even,even) separation parameter cases fall into these two distinct families. Analogously to Arthur Sullivan’s musical “lost chord”; this is my “lost proof”. Seated one day at the laptop, I was fluent and free of fear…..
If the constructions above have left you baffled, befuddled and bewildered, then you could always purchase a copy of the explanatory workbook and be admitted to hours of recreational mathematics and artistic designing. The Lattice Labyrinths workbook is available from the publisher or you-know-who , or from a good independent bookshop or via Google.
The Japanese research paper I referred to above is:
 Fukuda, Hiroshi et al, Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry, Symmetry 2011, 3, pp.828-851