Duality, sections and projections of certain euclidean tilings
The author introduces the notion of a pegged tiling, that is, a tiling (of part of Euclidean \(n\)-space) by polyhedra, with a peg (i.e., a point) assigned to each tile in such a way that for every two tiles which meet on a facet the difference of the pegs (the connecting line) is an outer normal vector of that facet. In the first part of the paper fundamental properties of pegged tilings are established. In particular, they admit a natural duality (pegs become vertices, and vice versa). The main motivation for the study of pegged tilings is the fact that they are a natural framework to describe the projection method used in the construction of quasi-periodic tilings: Dualizing a pegged tiling, taking a general section of the result, and then dualizing again, yields the projection. In the second part an interesting connection to convex functions is investigated, namely, that a pegged tiling can be lifted to (the graph of) a strongly locally polyhedral convex function, i.e., there is a convex function such that the maximal regions on which it is affine coincide with the tiles. As a consequence, pegged tilings are (slightly) generalized Laguerre (and therefore generalized Voronovi) tilings. Some fundamental operations on tilings (including the projection method) have natural counterparts in terms of operations on convex functions applied to their lifts.
scientific article; zbMATH DE number 569281
scientific article