Substitution Tessellation Exploration

Objective: Produce tilings using rep-tiles and substitution methods.

Materials

• Graph paper
• Triangular graph paper
• Two printed pinwheel tessellations

Procedure

1. Using triangular graph paper, make the substitution tessellation for the half-hexagon rep-tile:
2. Is the substitution tessellation that you get from the trapezoid a periodic tessellation or a non-periodic tessellation?

3. 

   The dissection for the pinwheel tessellation.

   On graph paper, draw a right triangle with legs of length 25 squares and 50 squares. Dissect it into smaller (similar) triangles using the pinwheel pattern. Every line you draw will be on a grid point. Now use the same pattern to dissect each new triangle into five still smaller triangles, which will have legs of length 5 and 10 squares. The resulting tessellation has 25 triangles and is stage two of the pinwheel tessellation. Feel free to dissect each triangle again to see stage three.
4. The pinwheel tessellation is not periodic. In fact, the triangular tile is rotated by new and different angles at each stage of the construction.
   • Learn more about the pinwheel tiling at http://paulbourke.net/texture_colour/tilingplane/ (scroll to bottom of page)
   • See an animated version at commons:File:Pinwheel_2.gif
   • Look for it in pictures of Australia's Federation Square
5. On your printed copy of the large pinwheel tessellation, the large triangle is made of five smaller triangles. Find them and highlight their edges. Can you identify the next level of smaller triangles?
6. Find rectangles in the pinwheel tessellation. Do you observe any patterns in their size or position? Shade in rectangles to emphasize the pattern (or lack of pattern).

Handin: Your trapezoid tessellation, your pinwheel tessellation, and the printed pinwheel tessellation with shaded rectangles.