We investigate tilings of cubiculated regions with two simply connected floors by 2 × 1 × 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that " almost " characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local… (More)
We construct Peano curves γ : r0, 8q Ñ R 2 whose " footprints " γpr0, tsq, t ą 0, have C 8 boundaries and are tangent to a common continuous line field on the punctured plane R 2 tγp0qu. Moreover, these boundaries can be taken C 8-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.
In this paper, we consider domino tilings of regions of the form D×[0, n], where D is a simply connected planar region and n ∈ N. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic… (More)