Question 20.74

44. A line can be tiled quasiperiodically with a pair of tiles but only if their lengths are in the right proportion, so that the pattern can be scaled up. We determine what that proportion must be. Let the two lengths be (for “short”) and (for “long”), with and , where the scaling factor is . The key requirement for the scaling factor is that scaling up a tile of length produces a total length of , that is, .

  1. Using substitution, reduce the two equations and to a single equation, eliminating and leaving just and involved.
  2. Use the quadratic formula to solve for two possible values of . Since we want , we choose the larger value.

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