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,

TrefoilableHoneycomableBirthyearsbut 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.

Honeycomb (38,11) rev

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.

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About davescarthin

After a brief academic and local government career, long an independent bookseller/publisher at Scarthin Books, Cromford, Derbyshire, UK. An antiquarian bookseller in two senses, now also has time to be an annuated independent post-doc, developing the long dormant topic of lattice labyrinth tessellations - both a mathematical recreation and a source of compelling practical tiling/paving and textile designs. Presenting a paper and experiencing so many others at Bridges Seoul 2014 Mathart conference was a great treat, as was the MathsJam Annual Conference in November 2016. I'm building up to a more academic journal paper and trying hard to find practical outlets in graphic design and landscape architecture. An 8 ft square tiling design was part of the Wirksworth Festival Art and Architecture Trail 2016. I love giving illustrated talks, tailored to the audience. Get in touch to commission or to collaborate.
This entry was posted in Birthyear Labyrinths, Honeycomb Labyrinths, Mathematics, Trefoil Labyrinths and tagged , , , , , , , , , , . Bookmark the permalink.

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