## May the Fourth be with You … Lattice Labyrinth Tessellations for Star Wars Day 5/4 … May Fourth

My aged ears recently overheard  “May the Force be with You” as “May the Fourth be with You”. A quick Google© confirms that this is not an original discovery but is disparaged as an “obvious” pun. Nevertheless and therefore 4th. May = May 4th. has long been appropriated as Star Wars Day and from the Star Wars archives I have awoken an unearthly confirmation.

The gorgeous carpet which the warriors bestraddle is Trefoil Lattice Labyrinth (5,4).  They preferred this option to Chinese Lattice Labyrinth (5,4).

Or even this version of Chinese (5,4), strewn with Capital F for Force

Each supertile of Chinese Lattice Labyrinth (5,4) contains 5² + 4² = 41 squares. Prime number 41 is a Sophie Germain prime.

Here again is a portion of Trefoil Lattice Labyrinth (5,4).

Trefoil Lattice Labyrinth (5,4) is a typical member of the Trefoil (a,a-1) sub-family, with its Isle-of-Man-ish jester of a supertile and its overall 3D effect …..steps made of a pile of boxes? The 3D interpretation doesn’t survive close scrutiny; the central triangle of each supertile is fought over by  three apparently mutually perpendicular planes. In each supertile there are  5² + 5×4 + 4² = 61 triangles and prime number 61 is a Keith Number.

Inspired by (5,4) I’ve compiled a list of  number pairs (a,b) where both (a² + b²) and (a² + ab + b²)  are prime, checking all cases up to a² + b² = 2100. The highest-order case in this range is (42,17), where the primes are 2053 and 2767.

The corresponding  tessellations can only be Chinese and Trefoil as prime values of (a² + b²) can only be of form 4n+1 (congruent to 1 modulo 4)and prime values of  (a² + ab + b²) can only be of form 3n+1 (congruent to 1 modulo 3).

In the case of primes congruent to 1 modulo 12, both a Chinese AND a Trefoil Lattice Labyrinth can be constructed. The lowest order example is 13, the supertile area of Chinese (3,2) and Trefoil (3,1). A higher order example is 2089, Chinese (45,8) and Trefoil (43,5).

Let me know if you find al this emboldening irritating.

To finish with, here is the Tessellated Star Wars Set rotated and distended.

As always, here are some links to the inexpensive how-to-do-it  Lattice Labyrinths workbook which is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

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.

### 2 Responses to May the Fourth be with You … Lattice Labyrinth Tessellations for Star Wars Day 5/4 … May Fourth

1. ivasallay says:

I love the carpet they are standing on!

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• davescarthin says:

The colours are tuned to matched the light sabres; carpet-fitting service by my architectural student son.

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