Systematic Study of Convex Pentagonal Tilings, I: Case of Convex Pentagons with Four Equal-length Edges

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 May 10, 2005; Accepted September 26, 2005)

Keywords: Convex Pentagon, Tiling, Tile, Pentagon, Pattern, Tessellation

Abstract. At the beginning of the series of papers we present systematic approach to exhaust the convex pentagonal tiles of edge-to-edge (EE) tilings. Our procedure is to solve the problem systematically step by step by restricting the candidates to some class. The first task is to classify both of convex pentagons and pentagonal tiling patterns. The classification of the latter is based on the analysis of vertex patterns of pentagonal tiling. As the first step of the procedure, the candidates are restricted to the pentagons with four edges of the equal length. Furthermore, the analysis is restricted to the simplest category of node conditions. As a result, we obtained the result that in the above restricted tilings, 14 patterns are possible by the combinatorial analysis, the topological judgment and the geometric judgment.