to download all images click here

It is easy to see that one can tile the plane with comets There are many ways to do this, some solutions are quite intricate and beautiful.

Some examples are shown in figures 1 and 2:

figure 1

figure 2

Now look at figure 3.figure 2

Copies of figure 2 tile the whole plane.

figure 3

Here we started with a flower-like arrangement of six tiles (yellow tiles
in the center). It seems that with this starting "seed" of six tiles in the
center no regular structure shows up.We still have six-fold symmetry, but apart from this the arrangement of tiles is quite chaotic, even after using 850 tiles.

Also, it is not clear whether we can extend this tesselation to a coverage of the whole plane.

Finally, the comet seems to me the only tile that produces a chaotic tiling from a tiny seed, and with only one single prototile used?

Do you know of anything like this?

If yes, I am very much interested to hear from you.

History.

I came across the comet tiling problem while writing the code for the Wolfram Demonstration "Tiling Constructor - Tile Dragging Variant",

which can be found here:

http://demonstrations.wolfram.com/TilingConstructorTileDraggingVariant/

You can run it inside your browser or download it and run it as a cdf-file, using a free cdf-player.

------------------------------------------------------------------------------------------------------------------

Here is another interesting tiling:

figure 4

The central "aperiodic seed" again forces any tiling that contains
it to be aperiodic, but the tiling of the plane is not chaotic.