Tricky Common Factor Case

To begin at a beginning.

The figure below shows just four interlocking supertiles of a typical member of the Chinese Lattice Labyrinth family, member (9,6) a case with separation parameters (a,b) where a and b share a common factor, 3 in this case. Such common-factor cases require a little more finesse in the construction process but tend to be attractive.

To embolden the effect of the illustration, I’ve hidden the boundaries of the 117 square tiles that make up each supertile. You can see how this pattern could be extended by repetition in all directions until the infinite plane bumps up against the edge of the Universe, or comes round to meet itself again.

Chinese(9,6)And shall be ever further. (A Googlewhack) – and while we’re contemplating infinity, suppose we wind up the order a little, to Chinese Labyrinth (27,18), also a member of the (3a,2a) common-factor sub-family. In this case it’s the highest common factor, 9, rather than 3, that determines the construction.

Chinese (27,18)

Notice the similarity. For an even higher-order member of this sub-family, see the Chinese Labyrinth (39,26) post.

About davescarthin

After terminating academic and local government careers, long an independent bookseller/publisher at Scarthin Books, Cromford, Derbyshire, UK. An antiquarian bookseller in two senses, I now have more 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 and textile designs. Presenting a paper and experiencing so many others at Bridges Seoul 2014 Mathart conference was a great treat, as were the spirited MathsJam Annual Conferences in November 2016 and 2017. I'm building up to a more academic journal paper and trying hard to find practical outlets in graphic design and landscape architecture. I submitted 8 ft square tiling designs to the Wirksworth Festival Art and Architecture Trail in 2016 and 2017. I love giving illustrated talks, tailored to the audience. Get in touch to commission or to collaborate.
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