Tessellations for Her Majesty Queen Elizabeth the Second’s Ninetieth Birthday. Long may you reign.

I am old enough to remember the celebration of  the Coronation of Queen Elizabeth II, to whom we refer always as simply “The Queen”. I can even remember the date – 2nd. June 1953. In the village of Spratton, Northamptonshire, each of  the pubs hosted a party. The Fir Tree Inn (long since closed) put on a barn dance, which sounded magic to me, but we, the village schoolmaster’s family, were too respectable for that; we went to the party in the village hall (alcohol-free? I wonder). I can just about remember watching the procession and service on a tiny black-and-white television screen. Afterwards we had the classic village hall tea – triangular sandwiches, probably including salmon and cucumber (Mmmm!), butterfly buns. iced cake and jelly or trifle. There was a children’s obstacle race, which included the terrifying task of blowing up a balloon until it burst. In 1977, for the Silver Jubilee of the Queen’s actual accession (not of the  Coronation) I closed the young, but already full-time Scarthin Bookshop and joined in Cromford, Derbyshire’s big Street Party, the tables stretching the length of North Street. The Golden Jubilee in 2002 seemed to pass without a village celebration, or perhaps I was too busy to notice, but by 2012 I was heavily involved in the Celebrating Cromford organisation and helped put on a Diamond Jubilee Street Party on Scarthin Promenade, next to the Boat Inn. We mostly dressed up as queens or kings and there was music and an excellent children’s magician entertainer. The rain stopped and the sun came out just for that afternoon. Here’s a picture or two.

JubileeSreetPartyScarthin 027JubileeSreetPartyScarthin 017

JubileeSreetPartyScarthin 008

Anyway, now to the serious business of some Loyal Lattice Labyrinth Tessellations. The 21st. of April leads to the number pair (21,4), which specifies Trefoil Labyrinth (21,4). here it is.

Trefoil (21,4) ER2 Birthday

I think this has turned out very appropriately for the Queen, it’s a self-disciplined tessellation, tightly controlled, resisting temptations to go out on limbs.

It’s also possible to dedicate a Lattice Labyrinth to all those reaching ninety years of age. 90 factorises to powers of primes: 2 x 5 x 3². We can immediately tell that 90 cannot be a Loeschian number of the form a² + ab + b² because such numbers and their prime power factors cannot be of the form 3n+2, which both 2 and 5 are. This rules out a tessellation on the triangular lattice with tile area 90. However, this number CAN be the sum of two squares because none of it’s prime power factors are of the form 4n+3 so it can correspond to a tessellation on the square lattice. We can quickly spot that 90 = 9² + 3², yielding number pair (9,3) which as both are odd corresponds to Serpentine Lattice Labyrinth (9,3) and here it is, not in Union Jack colours this time. For once I’ve included some of the mechanics – the axes of symmetry and the missing-links graph employed in the construction.

Serpentine (9,3) 90th. BirthdayEach supertile contains 45 squares, but they come in two sets, orientated at 90° to each other, so the repeat unit (or fundamental domain) is 90 as desired.

If the construction above has aroused your curiosity, then you could always purchase a copy of the explanatory workbook and be admitted to hours of recreational mathematics and artistic designing. The Lattice Labyrinths workbook  is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

Time to go off to choir.

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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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