Star Lattice Labyrinth (60,27) for Oliver and all 1983’ers

Onward to the true 1983 tessellation: Star Lattice Labyrinth (60,27). This can be constructed by dividing the 11,898-triangled supertiles of Honeycomb Lattice Labyrinth (60,27) into six 1983-triangled  Star supertiles  in the six different orientations. Although the constructor of the Giant’s Causeway, lacking even Euclidean geometry, never got beyond an approximation to the tessellation of hexagons, Honeycomb (1,1), supertile area 6, I propose to call Honeycomb (60,27) the IRISH GIANT and illustrate a small portion of it in the next post.

Finding Honeycomb/Star (60,27) has occupied every free moment of the last three days. Those troublesome threes, I should have been warned by 3 being a common factor of the separation parameters.I began blithely plodding through a most elegant and simple construction – despite its beguiling beauty it didn’t work. The next two days were spent in a tweaking and three re-tweakings, each of which introduced progressively more inelegant complications and didn’t work either. Oliver can be a very expensive son. Almost defeated, I returned to the original elegancy and realised it could be stripped down into a procedure still more elegant. This must work! It didn’t. On the verge of giving up (MUST get on with rewriting the BIG book), the last hope was to reverse the handedness of a spiral, with no loss of simplicity. Success ? Yes! But no – the final construction didn’t join up.  The last chance; I tried cutting out a lattice link to join up two corridors. Yes, I THINK I’ve got it, but don’t quite know why the final despairing link-snipping tweak worked. I can’t face reverse-engineering it, here it is. I THINK it’s right. Twelve supertiles are shown, coloured to distinguish them, with one changed from the original green to black. Don’t be taken in by the at-first-glance symmetry of the individual supertiles. Star Lattice Labyrinths do display the full 632 rotational symmetry, but the individual supertiles are not themselves symmetrical.

Star(60,27)FinalestestFancy counting the triangles? Below, I show just one of the supertiles, struggling so hard to be diagonally symmetrical, with the individual triangles marked; maybe you can make them out.



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.
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