No. 1056

Title:

Properties of Nodes in Pentagonal Tilings

Author(s):

Sugimoto, Teruhisa (Institute of Statistical Mathematics)
Ogawa, Tohru (Emeritus Professor of University of Tsukuba)

Key words:

Convex pentagon; Tiling; Tile; Pentagonal tiling; Node.

Abstract:  

A tiling by polygons is called normal if it is edge-to-edge and there are positive numbers $r$ and $R$ such that each polygon contains a certain disk of radius $r$ and is contained in a certain disk of radius $R$. A node of valence $k$ in an edge-to-edge tiling is a point that is the common vertex of $k$ tiles. Let $\Im$ be an edge-to-edge tiling of plane by pentagons each of which has $m$ nodes of valence 3 and $5-m$ nodes of valence $k \quad (0 < m < 5, k \ge 4)$. If $\Im$ is a normal tiling, then $(m, k) = (3, 4)$ or $(m, k) = (4, 6)$.