The Theory of Quality Translations with Applications to Tilings

In this paper, we consider a broad class of simply connected complexes that includes, for example, the face-to-face tilings ofEnandSn.Along with a complex (or tiling) we consider a set Q ofqualitiesthat individually can be assigned to the various cells or tiles of the complex, and a group G which permutes the elements of Q. These qualities might be colours, a set of positive real numbers or a set of affine functions. If each ordered pair of adjacent cells is associated with a group elementg∈G,and a qualityq∈ Ais assigned to any particular cell of the complex, qualities can be assigned to adjacent cells using the action of the group G on Q. In fact, since the complex is assumed to be simply connected, a quality can be assigned to any cell of the complex bytranslatingqualities along paths of adjacent cells using the group action. Our main theorem gives sufficient conditions that the quality assigned to a cell in this manner is independent of the particular path used for thetranslation,and is therefore uniquely determined. As applications of this theorem we establish ann-dimensional generalization of the converse of Maxwell ’s theorem on frameworks [10, 11], we obtain some theorems on colouring the cells of a complex, and we generalize the main part of Voronoi’s famous theorem on primitive parallelohedra [16]. See [7] and the survey [1].