For Exercises 49–51, we define inflation and deflation of a sequence of “tiles” consisting of s (“shorts”) and s (“longs”).

  • Inflation: Replace each by and each by . For example, the inflation of ()()() would be , where we have inserted parentheses for clarity.
  • Deflation: Replace each by and each lone by . For example, the deflation of ()()() would be , where we have inserted parentheses for clarity.

Question 20.80

50. (a) Apply deflation repeatedly to your answer to Exercise 49a, which has 21 letters, until you can deflate no further. What do you end up with?

(b) Apply deflation repeatedly instead to the periodic sequence …. What do you end up with?

(c) Devise your own periodic sequence of and tiles. Apply the deflation process to it repeatedly. What do you end up with? What do you conjecture?

image (d) Explain how the Fibonacci tiling is quasiperiodic.