Site hosted by Build your free website today!

Tilings Explainations Page

Covering up the plane (think of it like an infinite sheet of paper.) with shapes called tiles. The tiles must not overlap or leave any gaps. We will consider for this entire web site that the tiles must be the same size and shape. (That is they will be congruent)(These tilings can be called monohedral tilings.)
Isohedral tiling:
A tiling where each tile acts like the other tiles. If you copied the tiling of the whole plane on an infinite sheet of clear plastic you could choose any tile on the plastic and line it up on top of any tile on the plane with the whole tiling matching on both the plastic and the plane.
Anisohedral tiling:
A tiling that is not isohedral; the tiles will play different roles in the tiling. For example, the tiling in figure 1 is anisohedral. You will notice that the tiles are grouped three per square. Think of them as a sandwich -- two pieces of bread outside and one layer of peanut butter inside. If this tiling of the whole plane were copied on a piece of clear plastic you would be able to line any slice of bread up with any other slice and make the whole tiling match. But if you tried to cover a slice of bread with a layer of peanut butter, the rest of the tiling would not line up. See figure 2. (It's a good thing that our real sandwiches aren't done this way.) You can also see this in a flash animation.
Anisohedral tile:
A tile that doesn't allow any isohedral tilings -- only anisohedral tilings. For example, the tiling in the definition of anisohedral tilings above uses a rectangle. The same rectangle could be used in this isohedral tiling. (figure 3) The tiling shown in figure 4 uses an anisohedral tile. We know that no tiling of it can be isohedral because to fill in the gap between the nose and chin, you have to use either a nose or a chin. So you will always have a pair of tiles that are playing different roles. One will be a chin biter; one will be a nose biter.
2-isohedral, 3-isohedral, k-isohedral tiles:
The number in front of the word isohedral means the tiles can have this many different roles in the tiling. (Mathematicians would say there is a tiling with this number of transitivity classes.) Note that in Figures 1 and 3 we see that this shape is both isohedral and 2-isohedral.
2-anisohedral, 3-anisohedral, k-anisohedral tiles:
The number in front of the word anisohedral means the tiles must play at least this many different roles in the tiling. (Mathematicians would say this is the minimum number of transitivity classes allowed for any tiling.) So 2-anisohedral tiles require at least two different classes of tiles. 3-anisohedral tiles would require any tiling to have at least three different roles for the tiles. etc.
classification of 2-anisohedral tiles
You will notice that we have numbers like 4353-11a2 with each 2-anisohedral tile. Since these are 2-anisohedral we have two classes of tiles: let tile A be from one class and Tile B from the other class. The classification number 4353-11a2 would tell us that Tile A has 4 neighbors and Tile B has 5 neighbors. The little threes tell us that Tile A has three neighbors who are Bs, and Tile B has three neighbors who are As. Note that the neighbor numbers show the ratio between the types of tiles. Most of the time these are in a one to one ratio with the only currently known exception being the unbalanced tiling which was discovered in 2002 which has the neighbor numbers 3 and 6 for a ratio of 1:2. The number and letter after the dash are part of the full topological classification explained by Delgado, Huson, and Zamorzaeva. The final number (or letter) is merely so that I can name the different tilings. I tried to add them in chronological order of when the tiles were published.
Peanut butter:
A paste made from grinding the seeds of the peanut plant to a smooth consistancy. Not previously published in the tiling literature. See anisohedral tiling.

This page designed by John Berglund.
You can reach me by email.