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

*Review*

## Properties of Tilings by Convex Pentagons

Teruhisa Sugimoto^{1}* and Tohru Ogawa^{2,3}

^{1}The 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

^{3}The 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 *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} ≈ 3*k* - 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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