Properties of Nodes in Pentagonal Tilings

Teruhisa Sugimoto¹* and Tohru Ogawa^2.3

¹The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, 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, Suzukidaini-building 211, 2-5-28 Kitahara, Asaka-shi, Saitama 351-0036, Japan
*E-mail address: ismsugi@gmail.com

(Received November 17, 2008; Accepted March 13, 2010)

Abstract. A node of valence k in an edge-to-edge tiling is a point that is the common vertex of k tiles. We show that an edge-to-edge tiling of plane by pentagons each of which has m nodes of valence 3 and 5 - m nodes of valence k has properties of (m, k) = (3, 4) or (m, k) = (4, 6) if it is normal. Then we discuss tilings by congruent convex pentagons using the properties.

Keywords: Convex Pentagon, Tiling, Tile, Pentagonal Tiling, Node