# Square Block Pattern Exploration

From EscherMath

Jump to navigationJump to search

**Objective:**
Create wallpaper patterns using repeated square blocks.

## Materials

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.

- Draw the pattern corresponding to <math>\begin{matrix}2 & 4 \\ 2 & 4 \end{matrix}</math>. Use at least four copies of the basic 2x2 shape.
- Draw the pattern corresponding to <math>\begin{matrix}4 & 3 \\ 2 & 1 \end{matrix}</math>
- What numerical pattern would generate this design:
- 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 |
---|---|

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

**Handin:**
A sheet with answers to all questions.

- ↑ Visions of Symmetry Page 44-52.