Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-18T11:32:55.853Z Has data issue: false hasContentIssue false
  • English
  • Français

Cell Growth Problems

Published online by Cambridge University Press:  20 November 2018

David A. Klarner
David A. Klarner*
Affiliation:
McMaster University, Hamilton, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.

The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn, which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn, or (ii) in Tn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 851 - 863
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Eden, M., A two-dimensional growth process, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. IV (Berkeley, California, 1961), pp. 223–239.Google Scholar
2. Golomb, S. W., Checkerboards and polyominoes, Amer. Math. Monthly, 61 (1954), 275–282.Google Scholar
3. Harary, F., Unsolved problems in the enumeration of graphs, (Magyar Tud. Akad. Mat. Kutato Int Kozl.) Publ. Math. Inst. Hungar. Acad. Sci., 5 (1960), 63–95.Google Scholar
4. Harary, F., “Combinatorial problems in graphical enumeration,” Chap. 6, Applied combinatorial analysis, ed. by Beckenbach, F. (New York, 1964), pp. 185–217.Google Scholar
5. Klarner, D. A., Some results concerning polyominoes, Fibonacci Quarterly 8 (1965), 9–20.Google Scholar
6. Klarner, D. A., Combinatorial problems involving the Fredholm integral equation, to appear.Google Scholar
7. Pólya, G. and Szego, G., Aufgaben und Lehrsätze aus der Analysis, Vol. 1 (Berlin, 1925).Google Scholar
8. Read, R. C., Contributions to the cell growth problem, Can. J. Math., 14 (1962), 1–20.Google Scholar
9. Riesz, F. and Sz-Nagy, B., Functional analysis (New York, 1955).Google Scholar
10. Titchmarsh, E. C., The theory of functions, 2nd ed. (Oxford, 1939).Google Scholar