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.
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.