Christoffel and Fibonacci Tiles

Among the polyominoes that tile the plane by translation, the so-called squares have been conjectured to tile the plane in at most two distinct ways (these are called double squares). In this paper, we study two families of tiles : one is directly linked to Christoffel words while the other stems from the Fibonacci sequence. We show that these polyominoes are double squares, revealing strong connections between discrete geometry and other areas by means of combinatorics on words.

[1]  Tero Harju,et al.  Combinatorics on Words , 2004 .

[2]  Xavier Provençal.  Combinatoire des mots, géométrie discrète et pavages , 2008 .

[3]  Przemyslaw Prusinkiewicz,et al.  The Algorithmic Beauty of Plants , 1990, The Virtual Laboratory.

[4]  Christophe Reutenauer,et al.  On Christoffel classes , 2006, RAIRO Theor. Informatics Appl..

[5]  Gilbert Labelle,et al.  Properties of the Contour Path of Discrete Sets , 2006, Int. J. Found. Comput. Sci..

[6]  Herbert Freeman,et al.  On the Encoding of Arbitrary Geometric Configurations , 1961, IRE Trans. Electron. Comput..

[7]  Azriel Rosenfeld,et al.  Picture Processing and Psychopictorics , 1970 .

[8]  M. Lothaire,et al.  Combinatorics on words: Frontmatter , 1997 .

[9]  Grzegorz Rozenberg,et al.  The mathematical theory of L systems , 1980 .

[10]  Srecko Brlek,et al.  An Optimal Algorithm for Detecting Pseudo-squares , 2006, DGCI.

[11]  Achille J.-P. Braquelaire,et al.  Euclidean Paths: A New Representation of Boundary of Discrete Regions , 1999, Graph. Model. Image Process..

[12]  Danièle Beauquier,et al.  On translating one polyomino to tile the plane , 1991, Discret. Comput. Geom..

[13]  G. Rozenberg,et al.  Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology , 2001 .

[14]  Jan van Leeuwen,et al.  Arbitrary versus Periodic Storage Schemes and Tessellations of the Plane Using One Type of Polyomino , 1984, Inf. Control..

[15]  Srecko Brlek,et al.  On the tiling by translation problem , 2009, Discret. Appl. Math..

[16]  C. Reutenauer,et al.  Combinatorics on Words: Christoffel Words and Repetitions in Words , 2008 .