Wang tiles (1st of January 2012)
I started last year with the post Moire pattern. I feel as it was only yesterday.
I wasn't sure that I will be able to fill up the whole year of the "Construction of Reality" and I thought that all that effort may
end in the last year. Still, it didn't and I open another year with another "surface science" post.
This time it is about Wang tiles (the image above). This is an (generally) aperiodic tiling of plane by using several predefined
"tiles" which are set up on a two-dimensional net (although I can also easily imagine Wang tiles in 3D - I may write about it
some other time). A detail of the above shown Wang tiling of a plane is shown in the image below.
From the image above one can see that the plane is tiled with only two Wang tiles, and one of those two tiles can be obtained by
the 90 degree rotation of the other one. Therefore, the tiles are totally useful - such, completely equal, tiles could be mass
produced, and a creative worker could set them up in an aperiodic interesting structure by deciding each time on the rotation of
the tile he/she sets up.
My Wang tiles were designed to form curved shapes and they are made of pieces of a circle (more exactly, a flattened torus) arranged
in such a way that they smoothly extend when lay out. It is of course possible to imagine many other solutions. If the story is interesting
to you, you can start your investigation with >> the Wikipedia paper on Wang tiles.
I also played a bit with different visual variants of my "snake-like" Wang solution, and one of them, on a square net is shown in
the blue-black version in the image below.
Wang tiles are used in the computer graphics to make textures of large objects, and even the whole plane. Convincing
aperiodic textures can be made from several cleverly chosen patterns (tiles) which are then assembled in a periodic
2D network.
Of course, 2D network does not need to be square. Hexagonal Wang tiles are shown in the image below.
In this case, one needs 5 Wang tiles, but there are only two basic tiles from which the other ones can be obtained by rotation (2 + 3). A detail of this tiling is shown in the image below.
Hexagonal Wang tiles in an orange-black version is shown in the image below. Here the toroidal and cylindrical pieces are flattened just a little bit so that one can easily see their three-dimensionality.
In the end, it is of interest to mention an earlier post which can probably also be set up in the category of hexagonal Wang tiling. The post in question is >> Periodic - aperiodic.
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Last updated on 1st of January 2012.