# Square Block Pattern Exploration

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Objective: Create wallpaper patterns using repeated square blocks.

Graph paper

## Exploration

This exploration uses Escher's line designs[1]:

1 2 3 4

which could be produced from a single stamp, rotated into four positions.

1. Draw the pattern corresponding to  $\begin{matrix}2 & 4 \\ 2 & 4 \end{matrix}$. Use at least four copies of the basic 2x2 shape.
2.
3. Draw the pattern corresponding to  $\begin{matrix}4 & 3 \\ 2 & 1 \end{matrix}$
4. What numerical pattern would generate this design:
5. Can you design a pattern that has reflection symmetry using these blocks?

Escher designed four square blocks which, together with their mirror images, make a large number of patterns that look like overlapping ribbons. Experiment with the EscherTiles applet to see these blocks in action.

Now consider a square block which looks like one line crossing over another. There are only two choices for rotating this block:

1 2
1. How many two-by-two arrays are there using only 1's and 2's
2. Find all possible patterns for these cross blocks, using only two-by-two arrays.

Handin: A sheet with answers to all questions.

1. Visions of Symmetry Page 44-52.