TILING THE PLANE WITH CONVEX PENTAGONS
Abstract
We consider the problem of classifying the convex pentagons that tile the plane edge-to-edge. It is proved that in each such tiling of the plane by pentagons, there exists a tile whose set of vertex degrees can be one of the following: (3,3,3,3,3), (3,3,3,3,4), (3,3,3,3,5), (3,3,3,3,6), (3,3,3,4,4). This provides the possibility of searching which finally leads to exhaustive classification of such pentagons.
References
1. Agaoka, Y. An example of convex heptagon with Heesch number one / Y. Agaoka // Mem. Fac. Integrated Arts and Sci. Hiroshima Univ. Ser. IV, Science Report. - 2005, December. - 31. - P. 117 - 123.
2. Bagina, O. Tiling the Plane with Congruent Equilateral Convex Pentagons / O. Bagina // J. Combin. Theory. Ser. A. - 2004. - 105 (2). - P. 221 - 232.
3. Grunbaum, B. Tilings and Patterns / W. H. Freeman and Company / B. Grunbaum, G. C. Shephard. - 1987. - P. 472 - 518.
4. Hirschhorn, M. D. Equilateral Convex Pentagons Which Tile the Plane / M. D. Hirschhorn and D. C. Hunt // J. Combin. Theory. Ser. A. - 1985. - 39. - P. 1 - 18.
5. Kershner, R. B. On Paving the Plane / R. B. Kershner // American Mathematical Monthly. - 1968. - 75. - P. 839 - 844.
6. Reinhardt, K. Uber die Zerlegung der Ebene in Polygone: Dissertation / K. Reinhardt. - Universit дt Frankfurt, 1918.
7. Schattschneider, D. Tiling the Plane with Congruent Pentagons / D. Schattschneider // Math. Magazine. - 1978. - 51. - P. 29 - 44.
8. Делоне, Б. Н. Комбинаторная и метрическая теория планигонов / Б. Н. Делоне, Н. П. Долбилин, М. И. Штогрин // Труды математического института. - 1978. - 148. - С. 109 - 140.
Review
For citations:
Bagina O.G. TILING THE PLANE WITH CONVEX PENTAGONS. The Bulletin of Kemerovo State University. 2011;(4):63-73. (In Russ.)
Views:
214
Email this article 