Ambiguous Tiling

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We propose a new concept“ambiguous tiling”，which is the combination of tiling art and ambiguous cylinders. Tiling art is an artistic pattern consisting of copies of one or a few kinds of tiles which cover the whole 2D plane without gaps or overlaps. Escher ’s regular-division artworks are typical examples. There is a strong condition for figures to be used as the tiles for tiling. On the other hand, ambiguous cylinders are 3D solid objects that give two quite different appearances when they are seen from two specific viewpoints. There is also another strong condition for a pair of figures to be satisfied to realize the ambiguous cylinder. In this paper we find figures that satisfy both of the conditions and so we can create 3D objects which give two different tiling patterns when we see them from two specific viewpoints. We first give a necessary and sufficient condition for a pair of tiling patterns to be created as the ambiguous cylinder. This condition is theoretically correct but not practically useful. So we next consider special type of tiling for which we can design the ambiguous tiling relatively easily.

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