J. Propp, \Dimers and dominoes." Manuscript available at http://www-math.mit.edu/~propp/articles.html  Home/Search   Document Not in Database   Summary   Related Articles

....a lower bound from the solution of the Ising model in two dimensions. 1 Introduction Tilings of the plane are of interest both to statistical physicists and recreational mathematicians. While the number of ways to tile a lattice with dominoes can be calculated by expressing it as a determinant [1, 2], telling whether a nite collection of shapes can tile the plane at all is undecidable [3, 4] In the 1994 edition of his wonderful book Polyominoes [5] Solomon W. Golomb states that the problem of determining how many ways a 4 n rectangle can be tiled by right trominoes appears to be a ....

.... Ising = ln 2 1 2 1 (2 ) 2 Z 2 0 Z 2 0 d 1 d 2 ln cosh 2 (sinh ) cos 1 cos 2 ) For = 1 2 ln 2, where cosh 2 = 25 16 and sinh = 3 4 , we have Ising = 0:8270269567 and so 0:09501088358 For upper bounds, we can generalize an argument given in [2] as follows. If we construct a tiling of the plane with right trominoes by scanning from top to bottom and left to right, at each step the rst unoccupied site can be given a tromino with only four di erent orientations. Since we have at most n=3 such choices, the entropy is at most 1 3 ln 4 ....

[Article contains additional citation context not shown here]

J. Propp, \Dimers and dominoes." Manuscript available at http://www-math.mit.edu/~propp/articles.html

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC