Birthyears Tesselatable as Star, Dart or Diamond Lattice Labyrinths

The majority of this post is likely to be “all Greek” to you; what matters is the two tables, which list years which are also supertile areas I can do a Star, Dart or Diamond Lattice Labyrinth Tessellation for.

The post more or less completes the enumeration of those years which correspond to a Lattice Labyrinth tessellation with that supertile area. SOME years divisible by 3 are sums of the form e2 + ef + f2 where e and f , the separation parameters (e,f) of the tessellation are positive integers, or one may be zero. All these sums are related to the area, S, measured in equilateral triangles, of the supertile and the repeat unit, R, of a Honeycomb Labyrinth by the expression : R = S = 2( e2 + ef + f2 ).

The supertile of a Honeycomb labyrinth can be dissected into six supertiles of identical shape but six different orientations, successively at 60 degrees to each other, to yield a Star Lattice Labyrinth with supertiles of area, S, given by: R = 6S =   2( e2 + ef + f2 ) and therefore S = ( e2 + ef + f2 )/3. It can be shown that members of another family, a Dart Lattice Labyrinth can be constructed when (e,f) is of the form (even, even or zero). The supertile area of a dart Labyrinth is the same as that of the Star Labyrinth with the same separation parameters but the shape and topology of the tessellation are completelly different. Here is the table of possible years for Star and Dart Labyrinths.

StarDartBirthyears

If the separation parameters are of the form (odd, odd) or (odd, even or zero) then a Diamond Lattice Labyrinth is possible, but there are only THREE, not six, supertiles to the repeat unit, so that the supertile area, S, is related to the repeat unit, R, and to e and f by the expression: R = 3S =   2( e2 + ef + f2 ), so for the diamond family we have the supertile area given by:S = 2( e2 + ef + f2 )/3.  Here is the table of possible birthyears tessellatable by a Diamond Lattice Labyrinth with that supertile area.

DiamondBirthyears

I’ll try to make this a bit clearer by showing some low-order Stars, Darts and Diamonds in a subsequent post, though it may be some time before I manage to produce any more actual Birthyear Tessellations, despite my apparent obsession with this self-imposed task.

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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, followed by a triangles design in 2017. I love giving illustrated talks, tailored to the audience. Get in touch to commission or to collaborate.
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