# David Mitchell

Lattice Labyrinths are tessellations of animals - polyominoes laid out on the square lattice or polyiamonds laid out on the triangular lattice. They are required to be slim, one triangle or square wide, enclosing no lattice points. This rule implies elegant algebraic and geometrical properties and leads to powerful construction algorithms. Polyiamond Lattice Labyrinths repeat at intervals specified by separation parameters (e,f), measured along axes at 60 degrees to each other. Lattice Labyrinth animals, which I call supertiles, interpenetrate to the maximum degree. High-order cases bewilder the eye and suggest technical applications. Low-order cases can inspire Escherization, logos, fabric designs or pavings. Unexplored avenues abound.

Star Lattice Labyrinths are tessellations of six identically-shaped supertiles of area S oriented at six symmetrically-disposed angles. The area of the repeat unit, R = 6S = 2(e^2+ef+f^2), where the Loeschian number (e^2+ef+f^2) is necessarily of the form 3n. I like constructing Lattice Labyrinths with S equal to the birth-years of family and friends. The supertile of Star Labyrinth (60,27) consists of a convolution of 1983 equilateral triangles and is dedicated to my son Oliver and others born in 1983. Without losing the tessellation's 632 symmetry we can assemble 6 supertiles into a hexagonally-symmetrical animal, but such hexagonal-pot-bellied supertiles, generating a seductive Honeycomb Lattice Labyrinth family, are not truly slim.

This Trefoil Lattice Labyrinth, with separation parameters (44,1), is in honor of my daughter Rosie, born in 1981.Each trigonally-symmetrical supertile contains (44^2+44x1+1^2) =1981 equilateral triangles. In the Trefoil family, the Loeschian number (e^2 +ef+f^2) is necessarily of form 3n+1. Two supertiles oriented at 180 degrees to each other make up the repeat unit of the tessellation, just as two triangles generate the isometric graph-paper pattern. The latter is the basic tessellation of which the 632 -symmetrical Trefoil Lattice Labyrinth family is a generalization. I can draw any Trefoil Labyrinth with (e,f) of the form (e,1) but am still far from discovering a general procedure for constructing any member of this beautiful family.

The Diamond Lattice Labyrinths are a prolific source of striking images; I've dared to order 500 sheets of (7,1) and (13,1) gift-wrap. Like the Trefoil and Star families, Diamond Labyrinths are laid out on the triangular lattice, with the repeat unit specified by separation parameters (e,f). Try counting these out along lattice axes. For all Diamond Labyrinths (e^2 + ef + f^2) must be of the form 3n, with the additional constraint that (e,f) must be (odd,odd) or (odd,even or zero). A different family, the Dart Labyrinths, correspond to the remaining (even,even or zero) cases. I have worked out an algorithm for constructing all (e,1) Diamond Labyrinths and some other sub-families, but a grand unifying theory eludes me. You can do better.