Letf be a harmonic function on a finite planar Markov chainM whose boundary consists of two vertices on the same face. We construct a geometric realization of (M, f) as a tiling of a rectangle with trapezoids, each trapezoid having two horizontal edges. Conversely, each such tiling arises in this way. Similar results hold for harmonic functions with more general boundary conditions.Certain prescriptions of transition probabilities on edges inM give rise to tilings with prescribed shapes. This allows us to give necessary conditions for the existence of a tiling of an arbitrary polygon with squares, equilateral triangles, and so on. Using this method, we classify all polygons with at most one non-convex vertex which can be tiled with squares. A similar classification holds for tiling with equilateral triangles. We determine the Euclidean tori which can be square-tiled.
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