I have been inspired by find-the-factors blogger Ivasallay to tessellate prime number 439, that is to find a tessellation with that number of triangles (or squares) in its supertile. The result was Trefoil Lattice Labyrinth (18,5), which is not unattractive. I can’t do a tessellation of squares for a reason that will be outlined, though not entirely explained, further below.
The tessellation is a Trefoil because 439 is a prime number of the form 3n+1 and so is a Loeschian Number, expressable as e²+ef+f², being 18² + 18×5 + 5², the separation parameterts being (18,5) and each trigonally symmetrical supertile consisting of 439 triangles. The fundamental domain, or repeat unit as I prefer to call it, consists of two supertiles at 180° to each other, so has area 878.
To get ahead of Ivasallay for a short while, here is a table of primes up to 1013, indicating which correspond to a tessellations with supertile of that area counted in squares on the square lattice or equilateral triangles on the triangular lattice. Some prime numbers are possible for one or the other lattice, some for both and some for neither. I also give the separation parameters of the lattice labyrinths corresponding to the first column of primes. I’ll then tell of the test for tessellatability.
To recap on a previous post (I think), the test for whether a number can be the sum of two squares, that is can be of the form a² + b², and can form the basis of a lattice labyrinth tessellation with separation parameters (a,b) is that none of the powers of its prime factors can be of the form 4n – 1, or put more “mathematically”, none can = 3 modulo 4 (leave a remainder of 3 when all the multiples of four have been swept away).
You can see the basis of this from the first squares, 4,9,16,25, which, and all their successors, are equal to 0 or 1 modulo 4, so adding two of them can never get you to 3 modulo 4 …only its not quite so simple, you must also check that none of the powers of the prime factors = 3 modulo 4. For instance 21, 33 and 57 don’t qualify, although they are themselves of the form 4n+1
To introduce something new, an analogous test can be used to see if a number is Loeschian, that is of the form e² + ef + f² and can therefore form the basis of a lattice labyrinth on the triangular lattice with separation parameters (e,f). In this case none of the powers of its prime factors can = 2 modulo 3, that is that they must either be multiples of 3 or of the form 3+1. You can see how this come about by looking at any (it has to be equilateral) triangle drawn using the lines on triangular (isometric) graph paper – either there is opne little triangle in the middle, so the area is of the form 3n+1 or there is a point in the middle, so the area is of the form 3n. But once again you have to check the powers of the prime factors too, so, for instance 82,85 and 88 don’t qualify although they are themselves of the form 3n+1.
With the prime numbers in the above table things are much simpler, as they are their own prime factors so we only have to check the numbers themselves. Furthermore, (I nearly said “In addition”) multiples of 2, 3 and 4 are automatically ruled out by the primality of the numbers, so we are left with only two possibilities. To be an a² + b² a prime number must be of the form 4n+1 ( so 439 is out) and to be an e² + ef + f² a prime number must be of the form 3n+1, so 439 is possible.
Nuff for tonight.
P.S. if this isn’t mathematical enough for you have a look at some really elegant stuff on a particular post on the Mathemagical blog.