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        2. Background and Notation

The Archimedean tessellations of the plane are tessellations by regular polygons (not all congruent), edge to edge, with the property that the cyclic arrangement of polygons at each vertex is the same. (If the polygons are all congruent the tessellation is called regular. The three regular tessellations, by equilateral triangles, squares, or regular hexagons, are well known.) A convenient notation for an Archimedean tessellation records the sequence of polygons around each vertex, in parentheses, separated by commas. Thus the tessellation by dodecagons and triangles is (3,12,12). The two Archimedean tessellations which have three triangles and two squares at each vertex are (3,3,3,4,4) and (3,3,4,3,4). The other five Archimedean tessellations are (4,6,12), (4,8,8), (3,4,6,4), (3,6,3,6), and (3,3,3,3,6).

John Rigby introduced the notion of precise coloring in connection with his studies of colorings of regular tessellations of the hyperbolic plane. (See, for example, Precise colorings of regular triangular tilings, The Mathematical Intelligencer 20(1), 4-11 (1998) and Perfect precise colourings of triangular tilings, and hyperbolic patchwork, Symmetry: Culture and Science, to appear). Rigby was primarily interested in regular tessellations by equilateral triangles, as indicated by his answer to a question posed by David Gale and Raphael Robinson concerning the existence of a precise coloring of the regular hyperbolic tessellation (3,3,3,3,3,3,3). However, the terminology is equally applicable to any tessellation, or indeed any planar graph or convex polyhedron, in which the same number of polygons appear at each vertex. ( If this number is n, the tessellation is called n-valent.) In the case of the Archimedean tessellations of the Euclidean plane, it turns out that in some of the 5-valent cases there are infinitely many precise colorings. To produce a manageable number of precise colorings it is natural to further restrict consideration to those which are also perfect, or chirally perfect. The results are summarized in the next section.