Square Block Pattern Exploration

Objective: Create wallpaper patterns using repeated square blocks.

Materials

Graph paper

Exploration

This exploration uses Escher's line designs:

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. Draw the pattern corresponding to  ${\begin{matrix}4&3\\2&1\end{matrix}}$
3. What numerical pattern would generate this design:
4. 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

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

Handin: A sheet with answers to all questions.