| Properties of Tilings by Convex Pentagons (2005) | |||||||||||||||
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| Abstract. Let us consider an edge-to-edge and strongly balanced tiling of plane by pentagons. A node of valence s (≥3) in an edge-to-edge tiling is a point that is the common vertex of s tiles. Let W 1 be a finite closed disk satisfying the property that the average valence of nodes in W 1 is nearly equal to 10/3. Then, let T denote the union of the set of pentagons meeting the boundary of W 1 but not contained in W 1 and the set of pentagons contained in W 1, and let V s denote the number of s-valent nodes in T. If the tiling in T is formed of only 3- and k-valent nodes, then V 3: V k ≈ 3k – 10: 1 where k ≥ 4. On the other hand, if the tiles in edge-to-edge tiling are congruent convex pentagons, then at least two of the edges (of this congruent convex pentagon) are of equal length. 1. | |||||||||||||||
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