For a description of the randomization process, refer to the Propp-Wilson Algorithm page.
A hexagon is unique in that it has a very apparent height function. While it takes a trained eye to visualize the height function of an Aztec Diamond, raising and lowering moves of a hexagon can be viewed as adding or removing a cube from a box. The minimum tiling (with height zero) corresponds to an empty box, while the maximum tiling has a height equal to the number of cubes inside it. To see this, pause the randomization by clicking on the applet and compare the number of apparent cubes with the height. Like the Aztec Diamond, brickwork will form in the corners while the circular region will seem fairly random.
Note: clicking the mouse within the applet will pause or resume the randomization process.