โ Scribed by Hikaru Kawamura
No coin nor oath required. For personal study only.
The random triangle-square tiling with twelvefold quasicrystalline order is studied by using the transfer-matrix method. Based on a systematic finite-size analysis for L ร oo lattices up to L = 9, the maximum entropy per vertex for an infinite system is estimated to be S= 0.119 _+ 0.00 I. The ratio of the number of triangles to that of squares in the highest-entropy state is estimated to be r= 0.433 +0.001, in excellent agreement with the value x/~/4 given by Leung, Henley and Chester.
Problems of tilings of the plane have attracted interest for many years. In the field of condensed matter physics, renewed attention was recently paid to the so-called Penrose tiling [ 1 ] and related tilings, stimulated by the experimental discovery of quasicrystals in 1984 [2]. When the tiling consists of more than one unit, it often happens that the tiling of the plane can be done in many different ways such that they give rise to finite entropy. By contrast, the two-dimensional Penrose lattice, which is known to be a quasicrystal with decagonal symmetry with quasiperiodic translational order [ 1 ], has zero entropy.
One example of random tiling may be seen in the so-called triangle-square tiling, which is obtained by closely packing the plane with regular triangles and squares without overlaps or gaps as shown in fig. I. As can easily be deduced from the figure, such a tiling can be done in many different ways and the resulting configuration looks spatially "random". Then, one can address some interesting questions: (i) In how many different ways can one tile the plane? Or equivalently, what is the maximum entropy associated with the random triangle-square tiling? (ii) In the highest-entropy state, what is the ratio between the number of triangles and that of squares? In order to avoid the complications due to boundaries, the thermodynamic limit is assumed to be taken in which the number of vertices N tends to infinity.
These problems were first systematically investigated by the present author in 1983 based on a transfer-matrix method, making use of a mapping property of the trianglesquare tiling to the correlated bond dilution on the triangular lattice [ 3 ]. In ref.
[ 3 ], the author formulated the statistical mechanics of triangle-square tilings by assigning proper Boltzmann weights to triangles and to squares, and investigated its thermodynamical properties. By studying the properties of semi-infinite lattices with the sizes Lรov, L being up to L=6, the maximum entropy per vertex, S~o, as well as the ratio
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