No. 943

Title:

Properties of Tilings by Convex Pentagons

Author(s):

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

Key words:

Convex Pentagon, Tile, Tiling, Pentagon, Tessellation, Pattern.

Abstract:  

A node of valence $k$ in an edge-to-edge tiling is a point that is the common vertex of $k$ tiles, and let $V_{t}$ ($t \ge 3)$ be the number of $t$-valent nodes in the finite tiling. If a very large finite strongly balanced tiling by pentagons is formed of only 3- and $k$-valent nodes, then $V_3 :V_k \approx 3k - 10$ where $k \ge 4$. On the other hand, the pentagons can be classified into twelve kinds depending on the varieties and arrangements of five edge-lengths. Then, 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.