45-degree tiling with an infinite set of pieces

The following transform shows how an infinite set of 45-degree pieces can be used to create a design.

The first diagram shows the transformation of squares. Each square is transformed into a design that uses a larger square. Thus, as the design grows, the squares get larger and larger.

The diagram also shows how the edges of each transform are recursively developed. It is important that the edge of the larger square match with other pieces, including multiple smaller pieces. The transform of the larger square has an edge made of blue diamonds and smaller squares. This edge is created from two edges of the smaller square, plus a larger set of the same diamonds and squares as the smaller square.

The second diagram shows the transformation of the stars and the diamonds. It also shows how the edge of the square is truncated and combined to create the edges of the stars and diamonds.

The following shows the diamonds D₀ to D₃ and larger diamonds D2₀ and D2₁.