Pythagorean Pairs of Pairs…and the Occasional Triplet of Pairs

LatticeLabyrinths

Here are small parts two Serpentine Lattice Labyrinth tessellations.

Serpentine (7,1)Tess

Serpentine (5,5 ) TessThough these two patterns look very different and the arrays of superlattice points (which we can take to be marked by the black squares) and symmetry axes are at different inclinations, the supertiles of each tessellation are of the same area – 25 squares.

This equality arises simply because 7² + 1² = 5² + 5² = 50. two supertiles each of area 25 and oriented at right angles to each other constitute the fundamental domain of the tessellation.

I am yet to discover the term for pairs of number pairs such as (7,1) and (5,5) so for the time being will refer to them as Pythagorean pairs of pairs. This is because two right angles triangles of integral lengths 7 and 1 and 5 and 5 respectively share a hypotenuse of equal, though not integral.   (7,1) and (5,5)…

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