Paving patterns - mathematics activities

Description

The photograph shows paving laid in a pattern that 'grows' rather than repeats - it has just two repeating elements. Within its border of small square tiles, it uses tiles of three different colours, and two different sizes. Working out how many of each tile is needed for a given large area requires the identification of the rule for growth pattern for each type of tile.

Activities

1. Describe in words the paving pattern in the photograph so that another person, without having the image to follow, could build it.
2. The border of this paving pattern is made up of grey tiles only. The width of the border is not clear from the photograph. What do you think are the relationships between these grey border tiles and the other tiles in the tessellation? What assumptions, if any, have you made?
3. Supposing that the border's smaller square tiles have side length of 100 mm, how many of each tile will be needed to tile a square courtyard of side length 50 m in this pattern, assuming the pattern 'grows' as the photograph suggests?
4. If the pattern is laid without the border, how many tiles of each type will be needed to cover a square courtyard of 50 m side length in this pattern, assuming the pattern 'grows' as the photograph suggests?
5. Assume the pattern 'grows' as the photograph suggests in a square courtyard. What will the corner opposite to where the photograph was taken look like? Use square grid graph paper to illustrate this opposite corner for an equivalent section to that in the photograph.
6. The photograph only shows the start of this pattern from one corner. What if the pattern grows as suggested by the photograph from each of the other three corners, until it reaches the mid-point. What would the resulting overall courtyard tessellation look like?
7. Identify the regularities in this periodic paving. There are pavings that fill the two-dimensional plane, but in an irregular, or non-periodic, way. Some have been known about for hundreds of years, others only recently. Find out what you can about non-periodic pavings and prepare a report for your classmates.