Introduction
Tiling, also known as tessellation, is the simple act of using a variety of geometric shapes to fill in a larger defined area. The word tessellation comes from the Latin word “tessellatus,” which referred to something that was made of small stone cubes or tiles. This paper will discuss the qualities of tilings and how to determine which surfaces can be tiled by a particular set of tiles.
Proofs of Possibility
One of the main things that mathematics has to do with tilings is whether or not an area can be tiled with specific units. There are many different ways to determine whether a tiling is possible in a given area. The most obvious way to determine whether a tiling can exist is to figure out if the total area of the surface being tiled is divisible by the area of the tiles. For example, if the area of a square is 60 square inches, and someone is trying to tile it with 7 square inch tiles, it is clear that 60/7 is not an integer and therefore 7 square inch tiles will not be able to tile a 60 square inch surface.
However, this is not a proof of existence for a tiling. If there was a 60 square inch rectangle and one was attempting to tile it with four square inch circles, it is obvious that this tiling will not exist. A regular tessellation is a tessellation that is made up entirely of a single regular polygon. These only exist with three shapes, the regular triangle, square, and hexagon. All other regular polygons cannot form tessellations on their own. Strictly when determining if a rectangle can be tiled by smaller rectangles is that the dimensions of the smaller rectangle must be able to be added up to equal the dimensions of the larger rectangle. Also, both of the larger dimensions must be divisible by either of the smaller dimensions. Ex: 3×5 rectangle to tile 9×15 board. 3+3+3=12, and 5+5+5=15, and 15 is divisible by 5 and 9 is divisible by 3 so this tiling can exist.
Another, more complicated way to determine whether a tiling can exist is to color the tiles in a pattern and determine how many times a tile will touch each color. Consider an 8×8 chess board. There is 32 white tiles and 32 black tiles on any regular chess board. If you place a 2×1 tile anywhere on the board, it will cover one black tile and one white tile. This can be used to prove that 32 2×1 tiles will cover 32 white tiles, and 32 black tiles. However, if you take away or add a different amount of black and white tiles, the tiling will not exist. For example, if you take away any two white color tiles, there will be 62 tiles, so there should be 31 2×1 to fill the chess board. However, there would be only 30 white tiles and 32 black tiles, whereas 31 2×1 tiles should cover 31 black tiles and 31 white tiles.
Something a little more complicated would be attempting to show that a 6×6 square could not be covered by 4×1 rectangles. The coloring shown in Table 1 shows that there are 9 of the red, blue, black, and white tiles. Also, it could be shown that a 4×1 rectangle will always cover either two tiles of no tiles of one color. This would go to prove that with any number of tiles, the number of tiles of each color that is covered will be even and could never be nine.
Counting Number of Possibilities
Another thing that relates tiling to mathematics is attempting to determine the amount of different tilings that can be done by particular tiles in a certain area. Mathematicians have spent years trying to determine how to figure out how many different arrangements of tilings can be made with specific tiles in a certain area. In 1961, Michael Fisher and Neville Temperley found an equation that shows the exact number of tilings that can be made from 2×1 dominoes in a rectangle that has dimensions of 2m x 2n. The equation is shown in Figure 2. For example, by plugging the number 8 in for n and 8 in for m, which is the dimensions of a standard chessboard, you get that a chessboard can be tiled using dominoes in 12,988,816 ways.
Another shape that has a definite equation for the amount of different domino tilings that can be done inside of it is the Aztec diamond. The Aztec diamond looks like a diamond that starts with two squares, increases by two each row until it reaches its max, has the max row twice, and then decreases by two every row until it is back at two. An Aztec diamond of size n=1 and n=2 are shown in Figure 3. If the largest row is represented by 2n, where n is how far from the top the largest row is, then the number of tilings with a 2×1 tile inside the Aztec diamond can be represented as 2n(n+1)/2.(1) For example, if one plugs in the Aztec diamond of n=1, it shows there are two different possible 2×1 tilings, which are shown in Figure 4. Also, if n=2 is plugged in, it turns out to be 2^3=8 different combinations of 2×1 tilings in the Aztec diamond of n=2. These are some of the few shapes that have been given an equation for how many different tilings can be done in a certain area. However, there are many more imprecise ways to determine how many tilings can exist in an area, such as comparing it to similar shapes.
17 Wallpaper Groups
One thing, that has been discovered due to mathematicians studying tiling and symmetry is the 17 wallpaper symmetry groups. MC Escher contributed immensely to these findings. Wallpaper groups consist of either translations, reflections, glide-reflections, or rotations. Glide reflections is where the shape is translated and then reflected. P1 is made up only of translations. Pm contains translation and reflection. Pg has glide reflections and translations. Cm has reflections and glide reflections. P2 has translations and 180-degree rotations. Pgg has glide reflections and 180 degrees rotations. Pmg has reflections, glide reflections, and 180-degree rotations. Pmm has translation, reflection and 180 degrees rotation. Cmm is the same as pmm with another 180-degree rotation. P3 has 120-degree rotations and translation. P31m has reflections, 120-degree rotations, and translation. P3m1 has reflections, 120-degree rotations, and translation, and also glide reflections. P4 has 90-degree rotation and then 180-degree rotations. P4g contains reflection and rotations of 180 degrees and 90 degrees, P4m contains two reflections each with 180-degree rotation, then a 90-degree rotation, and four directions of glide reflection. P6 contains 60-degree rotations, and also 90-degree rotation and then 180-degree rotations. P6m has 60-degree rotations, and also 90-degree rotation and then 180-degree rotations, but all axes of reflection are at the center of rotation.(3) Somehow, it is impossible to create a wallpaper pattern with symmetry that does not fall into one of these categories. A table for determining which category a pattern can go into is shown by Figure 5.
Tiling with the Knight
Another way that mathematicians have put tiling is to use is to study the relationship of the knight and the chessboard. The knight and its properties have been studied by mathematicians for a very long time. The knight’s tour is the problem of having the knight start off on one space of an 8×8 chess board and having it touch every spot on the board exactly once, moving in an L shape (two spots either vertically or horizontally, and one spot the opposite way). It seems difficult but taking how many different methods and starting spots there are, but, there are many ways to do it. People have found both very random ways to do it, and there are strategies. The most common strategy is to split the board into four quarters and look at four shapes that are in each of the quarters. These shapes are the left diamond (blue), right diamond (red), left square (white), and right square(black). In this strategy, you complete one shape for all four quarters, and then move to the next shape. In 1997, the mathematician Brendan Mckay discovered that there are over 13,000,000,000,000 closed knight’s tour, which is where the knight ends up one move away from its starting point. It is impossible to make a closed knight’s tour out of any 4 x n, with n being any integer, board. Also, if you start with 32 knights on every white piece, it is possible for each of them to move to a different black piece of the board. (2)
Conclusion
Although tiling does not seem to be very mathematical, the proofs of possibility, impossibility, and amount of tiling possibilities is able to be proven by extensive mathematical reasoning and equations. The most common use of tiling is shown mathematically in the 17 different wallpaper symmetry possibilities. You have to consider many things such as area, and shape of tiles in determining whether a tile can exist. Using the knight in chess gives you a very unique view on tiling and many different problems have come from it.
