Square Block Pattern Exploration

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Time-25.svg

Objective: Create wallpaper patterns using repeated square blocks.

Materials

Graph paper

Exploration

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

1 2 3 4
Square-block-1.svg Square-block-2.svg Square-block-3.svg Square-block-4.svg

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

  1. 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.
  2.  
  3. Draw the pattern corresponding to  <math>\begin{matrix}4 & 3 \\ 2 & 1 \end{matrix}</math>
  4. What numerical pattern would generate this design: Square-block-4224-pattern.svg
  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
Square-cross-1.svg Square-cross-2.svg
  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.