Forma, Vol. 21 (No. 2), pp. 113-128, 2006

Properties of Tilings by Convex Pentagons

Teruhisa Sugimoto1* and Tohru Ogawa2,3

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:

(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 Vs 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 : Vk ≈ 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.

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