Scenario Two

Using Matrices to Find Tile Proportions

In order to find the correct proportions for each tile type used, we used a simple system of equations. Unhatted ends are assigned positive values and hatted ends are assigned negative values. Since the number of hatted and unhatted cohesive ends has to be equal in every complete complex, we can write the equations such that the sum of the ends equals zero. These values are then added up within each individual tile and used as coefficients for the unknown tile proportions. For example, in a 3-regular complete complex with three bond edge types and four tile types, the equations would be set up as follows:

a₁r₁ + a₂r₂ + a₃r₃ = 0

b₁r₁ + b₂r₂ + b₃r₃ = 0

c₁r₁ + c₂r₂ + c₃r₃ = 0

Since we also know that the sum of all the proportions is equal to one, we can also use the following equation:

r₁ + r₂ + r₃ = 1

In this system of equations, a_i is the number of a type cohesive ends in the i^th tile, and r_i is the proportion of the i^th tile. b_i and c_i represent the same concept as a_i but for b and c bond edge types.

To solve the system of equations, we first input them into a matrix. We then converted the matrix to reduced row echelon form, giving us a quick and easy way to solve for the proportions. Since this process is easily done by a computer, it gave us the freedom to write a variety of programs to automate the tiling process.

Triangular Prism Sample

Starting with the pot P = {a²b̂,âb²,bb̂²}, we would first look at the distribution of a type bond edges in the pot. The first, second, and third tiles have 2, -1, and 0 cohesive ends, respectively. For b type bond edges, the three tiles would have -1, 2, and -1 cohesive ends. Using these numbers as coefficients of the proportions, we can generate the following system of equations:

2r₁ - 1r₂ = 0

-r₁ + 2r₂ - 1r₃ = 0

r₁ + r₂ + r₃ = 1

These equations can then be represented by the following matrix

which can be expressed in reduced row echelon form in the following way:

Knowing that the figure we are attempting to tile has six vertices, we can take the proportions for each tile from the reduced matrix. Tile a²b̂ needs to be used once, tile âb² needs to be used twice, and tile bb̂² needs to be used three times.