Properties of Tilings by Convex Pentagons

Teruhisa Sugimoto¹* and Tohru Ogawa^2,3

¹The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan
²(Emeritus Professor of University of Tsukuba), 1-1-1 Tennodai, Tsukuba-shi, Ibaraki 305-8577, Japan
³The Interdisciplinary Institute of Science, Technology and Art, GanadorB-102, 2-10-6 Kitahara, Asaka-shi, Saitama 351-0036, Japan
*E-mail address: sugimoto@ism.ac.jp

(Received September 10, 2005; Accepted April 6, 2006)

Keywords: Pentagon, Tile, Tiling, Pattern, Tessellation

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 W₁ be a finite closed disk satisfying the property that the average valence of nodes in W₁ is nearly equal to 10/3. Then, let T denote the union of the set of pentagons meeting the boundary of W₁ but not contained in W₁ and the set of pentagons contained in W₁, 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 V₃ : 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.