## Great Triangular Tessellations!

Birthdate Years that may be Supertile areas of Trefoil or Honeycomb Labyrinths

Trefoil Labyrinths have supertitle area, S = (e2 + ef + f2), so we are looking for e and f that give  (e2 + ef + f2) equal to birthdates within the last 100 years or so. Honeycomb labyrinths have S =2(e2 + ef + f2), so, strictly, we are looking for  smaller value of e and f that give values of (e2 + ef + f2) that are half the values of possible birthdates.  The reasons for and meaning of this are too long to spell out here – see my very affordable workbook.  The end product is the table of possible birth-years below. These are the only years that correspond to a supertitle area of a Trefoil or Honeycomb Labyrinth, which could become the basis of your personal logo if you were (or will be) born in one of these years. Unless otherwise annotated, the separation parameter pairs below all correspond to Trefoil Labyrinths, but I have yet to draw any of them and have little evidence, in most cases, that they CAN ACTUALLY be drawn,

but here’s one that certainly can be, Honeycomb (38,11) for which (e2 + ef + f2) = 1983, an important year for me, though the area of the hexagonally symmetrical supertile actually equals twice 1983, as mentioned above.

Blow me to Bermuda!   At last, some birthyears of MY family members (sparse if not totally absent from the square lattice table posted below) , and I know how to construct the Lattice Labyrinth for at least one of them.

Does anyone know of a simply-applicable sieve that will eliminate all those numbers that cannot be an (e2 + ef + f2) sum, where e and f are positive integers or zero? All I’ve been able to come up with is the fact that such a sum is always either a multiple of 3 or 1 plus a multiple of 3, but not all numbers that abide by that rule can be such a sum. The sequence of these numbers, starting with 1,3,4,7,9,12,13,16,19,21,… is known as the Loeschian Numbers , the link indicates how the sequence relates to distances between points on the triangular lattice, and thus, it has now turned out, to Lattice Labyrinths.