HURRAH! 2017 is the first tessellatable year since 2011, either on the square or on the triangular lattice. To start with the last but best image I’ve found:
This shows symmetry axes and six supertiles of Trefoil Lattice Labyrinth (41,7), each supertile comprised of 2017 equilateral triangles. Here’s just one of the supertiles that fit together to potentially cover the infinite plane:
This unexpectedly wayward shape is in fact generated by an elegantly simple missing-links graph which I arrived at only by a hit-and-miss evolutionary process – a metaphor of last year’s wayward results of evolved democratic processes?
We had better re-cap to sketch out why 2011 and 2017 yield tessellations, but 2012, 2013, 2014, 2015 and 2016 do not, but by all means skip or speed-read the next two paragraphs if you like and go straight to some images of tessellations on the square lattice.
To construct a tessellation on the square lattice we set out points by counting a along one axis and b along the other, forming squares of area a²+ b², so for a number, such as 2017, to be the area of the repeat unit (fundamental domain) of a tessellation on the square lattice it is necessary that it can be partitioned into the sum of the squares of two integers, a and b (one of which may be zero). This is possible only if the number in question is of the form 4m, 4m+1 or 4m+2, never of form 4m+3. You can get a feel for why this is the case by drawing dots in square arrays or sticking matchsticks in a pinboard. Less obvious intuitively, it is also necessary that all the number’s prime factors (raised to whatever power they occur) must also be of one of the allowed forms. (See Conway and Guy’s Book of Numbers (expensive in paper). For instance, if 3 is a factor, or 3³, no tessellation is possible because both these powers of a prime are of form 4m+3, but if 3² is the factor, this presents no problem, as we can see that 9 can be expressed as 2m+1 where m=4.
For tessellations on the triangular lattice we set out points counting e and f along axes at sixty degrees to one another, and the area of the triangles so constructed will be the Loeschian number e² + ef + f². There is an analogous rule for the form of Loeschian numbers. All are of form 3n or 3n+1, never of form 3n+2. Again, you can get a non-rigorous feel for this by counting points or matchsticks set out in triangular arrays. Once again, this rule applies not only to the number itself but also to all its prime factors, raised to a power if it occurs as such. So 2 or 2³ cannot be factors of a Loeschian number but 2² can be, as 4 can be expressed as 3n+1, where n=1.
Note that the “tessellatability test” is a member of that beguiling but frustrating family of theorems that tells you whether something exists but includes no method of deriving it. One has confidence that e and f exist (there may be more than one possible pair) but get no further help towards finding them.
2011 is a prime number and of the form 4m+3 so cannot be the sum of two squares, but it is of form 3n+1 so can be a Loeschian number e²+ef+f² and a Trefoil Lattice Labyrinth can be constructed (I believe – I’ve yet to try) with e=39, f=10.
2012 = 2² x 503 and 503 is of form 4m+3 and also of form 3n+2, so is tessellatable neither on the square nor the triangular lattice. The same applies to the next four years:
2013 = 3 x 11 x 61 and both 3 and 11 are of form 4m+3 while 11 is of form 3n+2.
2014 = 2 x 19 x 53 and 19 is of form 4m+3 while both 2 and 53 are of form 3n+2.
2015 = 5 x 13 x 31 and 31 is of form 4m+3 while 5 is of form 3n+2.
2016 = 25 x 3² x 7 and 7 is of form 4m+3 while 25 = 32 is of form 3n+2.
2016 has 36 factors: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. leading me to think that 2016 might be a “highly composite number”, the smallest number to have 36 factors. But no, according to Wikipedia ( donate!) 1260 is the relevant highly composite number, the lowest with 36 factors. One observes that 1260 is an anagram of 2016. The highly composite number that comes first to my mind is 24, the lowest number to have 8 factors. One observes that 42, an anagram of 24, also has 8 factors.
End of digression. At last we come to:
2017, a prime of the form 4m+1 and 3n+1, so we should be able to construct tessellations on both the square and the triangular lattice. 2017 = 44² + 9², corresponding to Chinese Lattice Labyrinth (9,44) (my convention is to put the odd number first) and 2017 = 41² + 41×7 + 7², corresponding to Trefoil Lattice Labyrinth (41,7) (oddness and evenness are not especially significant for the construction, so I put the largest number first), a realisation of which heads this post.
Many of us are anxious that 2017 will be a difficult year, and so it proved when searching for a satisfactory tessellation on the square lattice. The standard missing-links graph for a Chinese lattice Labyrinth, consisting simply of nest of squares, work as always but leads to a very swastika-like (“swastikoid”) supertile. Attempt after attempt to use non-standard missing-links graphs failed – what emerged would have been (if completed) a tessellation of tiles within tiles rather than a monohedral tessellation. Here is an example:
At this point (a dyad axis) I gave up, as the tessellation graph traced thus far, if copied, rotated through 180 degrees and joined up, would lead directly to the next superlattice point – far too quickly, leaving most lattice points unvisited. I suppose I’d better show you what transpires if I do keep right on to the end of the road … tho’ the way be long … keep right on round the bend as the rather appropriate old song goes. So here it is:
– very (left-leaning) swastikoid of course. I’ve used three colours to distinguish the (two) different shapes that make up the tessellation, but two colours would do, yellows being surrounded by blue and vice versa.
Eventually, I arrived at a successful tessellation, but still reminiscent of a shape disgraced in Western eyes, though there is no such problem in South East Asia.
and finally, employing a radically new type of missing-links graph, a pattern that I feel able to recommend as a suitably tortuous representation of a year through which there is no clear path for many communities and nations.
I’ve illustrated all these designs against skies that at this time of year represent a Shepherd’s warning.
Happy New Year!