Introduction
We consider the complexity problem of plane square tiling with colored corners. The set of all local patterns with \(p\) colors on \(m\times n\) lattices is denoted by \(\Sigma_{m\times n}(p)\). A set, \(\mathcal{B}\subset\Sigma_{2\times 2}(p)\) of local patterns, is called basic set. Let \(\Sigma(\mathcal{B})\) be the set of admissible global patterns on plane, which is generated by \(\mathcal{B}\). On the other hands, \(\mathcal{P}(\mathcal{B})\) be the set of admissible global periodic patterns on plane, which is generated by \(\mathcal{B}\). In this study, we only consider \(p=2\) corner coloring in plane. Let \(\Gamma_{m\times n}(\mathcal{B})\) is the cardinal number of \(\Sigma_{m\times n}(\mathcal{B})\). Define the spatial entropy generated by \(\mathcal{B}\) is \begin{equation} h(\mathcal{B})=\lim_{m,n\to \infty}\frac{\log\Gamma_{m\times n}(\mathcal{B})}{m\times n}\,. \end{equation} Hence, we want to know determine that given the basic set \(\mathcal{B}\), whether the spatial entropy \(h(\mathcal{B})\) is positive or zero.
Goal
In fact, there are some rules. THerefore, this project includes two parts
- The first part of work is to decide whether the spatial entropy of any union of minimal cycle generators equals to zero or not.
- The second part of work is to study when the union of minimal cycle generators which is of zero entropy and some tiles are added into union without producing new minimal cycle generators, then whether the spatial entropy turns from zero to positive.
Results
There are 56 basic sets which is undetermined. (total: 3338)
References
- J.C. Ban and S.S. Lin, Patterns generation and transition matrices in multi-dimensional lattice models, (2005).
- J.C. Ban, S.S. Lin and Y.H. Lin, Patterns generation and spatial entropy in two dimensional lattice models, (2007).
Acknowledgements
I sincerely thank my advisor Dr. Songsun Lin for the guidance and encouragement. Also, I would like to thank Dr. Wenguei Hu and Dr. Hunghsun Chen for the guidance and assistance.