The diagram below shows one way how pattern block shapes can be used to create larger shapes.

In the first row, a beige diamond is "grown" to a larger diamond
with the same proportions. The new diamond uses beige half-diamonds and a blue
diamond (which is the same shape as two green triangles). The first diamond is
called D_{0} and the second is D_{1}.

In the second row, a green equilateral triangle is "grown" to a
larger green equilateral triangle. The larger triangle consists of beige
half-diamonds and a Y-shaped star. ). The first triangle is called T_{0}
and the second is T_{1}.

In the third row, a Y-shaped star is "grown" to a larger Y-shaped
star with the same proportions. The new Y uses beige half-diamonds and green
triangles. ). The first Y-shaped star is called Y_{0} and the second is
Y_{1}.

In all of the transforms the following are important:

- The edges of all transformed pieces are the same length as each other, i.e. the long diagonal of a beige half-diamond.
- The set is closed, i.e. the transformed pieces consist of only the original shapes.

The following diagram shows how these new pieces can be grown into larger pieces.

The first row shows a beige diamond (with the half-diamond) grown into the
larger beige diamond previously shown. Then it shows how these larger pieces
are put together to create an even larger diamond. This new diamond is called D_{2}.

Notice that D_{1} and T_{1} are put together to create T_{2}
in the same way that the original pieces D_{0} and T_{0}
created D_{1}.

The second row shows a larger triangle called T_{2}, and the third
row shows a larger Y-shaped star called Y_{2}.

Again, for these new shapes, the following is important:

- The edges of all transformed pieces are the same length as each other, i.e. two edges of the beige diamond plus the diagonal of a blue diamond.
- The set is closed, i.e. the transformed pieces consist of only the original shapes.

In the grown pieces above notice that all the beige half-diamonds are adjacent to other beige half-diamonds. So the only half-pieces really needed are at the edges of the designs. Internally whole pieces can be used.

When I use pattern blocks to create the designs I don't use half pieces.
Therefore the edges are not straight as shown above. The following diagram
shows how whole pattern block pieces can be used to create the next bigger
beige diamond, called D_{3}.

The diagram shows the diamond D_{0} in the first row, creating
D_{1} in the second row and D_{2} in the third row.

The diagram shows two ways to think about how D_{3} is created.

In the fourth row, pieces D_{2} and T_{2} are put together
the way the shapes in D_{1} are put together. So it is easy to see that
this process can be continued to create larger diamonds.

In the fifth row, pieces D_{1}, T_{1} and Y_{1} are
put together the way the shapes in D_{2} are put together. So you can
think about the operation of taking any pattern and creating a larger shape as
transforming the pattern by replacing all the shapes D_{0}, T_{0}
and Y_{0} by grown shapes D_{1}, Y_{1} and T_{2}.

The next two pictures show using whole pattern block pieces to create T_{3}
and Y_{3}.

The previous diagrams showed how a closed system can be created. This closed system has a property that, between each iteration, pieces are scaled by a certain factor.

Other pieces besides those shown above can be scaled. For example, in the same system, the following diagram shows how the square can be scaled.

The diagram shows how square S_{0} can be scaled to create squares S_{1},
S_{2} and S_{3}. This is done by creating a transformation for
the square, and then using the square's transformation and previous
transformations of the diamond, Y-shaped star and triangle to make larger
squares.

*Copyright 1998-2004 by Jim
Millar *