Joakim Linde, Cristopher Moore and Mats G. Nordahl (2001), An n-Dimensional Generalization of the Rhombus Tiling, Discrete Mathematics and Theoretical Computer Science Proceedings AA, pp. 23-42.
Examples of random rhombus tilings
Click on the thumbnails to see larger versions of the images.- 2D
A random lozenge tiling created with CFTP.
The same tiling, but with the frozen tiles removed. An example of an "Arctic circle" phenomenon. - 3D
The outer shell of a tiling.
The unfrozen tiles in a tiling created with CFTP.
Wireframe of a tiling created with CFTP.
Shape of the average height function
After the DMTCS paper some measurements have been done regarding the shape of the average height function for rhombus tilings.
2d: The height function has been rescaled so that it is independent of system size. The measured results almost coincides for all examined system sizes. The measured results are also in close agreement with the analytical results for tilings of infinite size (H. Cohn, M. Larsen and J. Propp (1998), The Shape of a Typical Boxed Plane Partition, New York J. of Math. 4, 137).
3d: The height function has been rescaled so that it is independent of system size. It has been conjectured that the height function is flat in the unfrozen region. From the measurements shown above it looks like the height function is almost, but not quite, flat. If that is due to a thus far undiscovered measurement bug, or that the height function actually isn't quite flat, is unknown.
