in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape. See Quilts of the Underground Railroad. In these decorative applications, the rhombi may appear in multiple colors, but are typically given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms. Other applications The rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field. The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers. In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice. It is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals, and it has also been studied in percolation theory. Related polyhedra and tilings The rhombille tiling is the dual of the trihexagonal tiling. It is one of many different ways of tiling the plane by congruent rhombi. Others include a diagonally flattened variation of the square tiling (with translational symmetry on all four sides of the rhombi), the tiling used by the Miura-ori folding pattern (alternating between translational and reflectional symmetry), and the Penrose tiling which uses two kinds of rhombi with 36° and 72° acute angles aperiodically. When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry. Tilings combinatorially equivalent to the rhombille tiling can also be realized by parallelograms, and interpreted as axonometric projections of three dimensional cubic steps. There are only eight edge tessellations, tilings of the plane with the property that reflecting any tile across any one of its edges produces another tile; one of them is the rhombille tiling. See also Tiling by regular polygons References Further reading Keith Critchlow, Order in Space: A design source book, 1970, pp.77–76, pattern 1 Category:Euclidean tilings Category:Isohedral tilings Category:Isotoxal tilings Category:Quasiregular polyhedra