MONT 106Q – Mathematical Thinking Problem Set 4 – Symmetric Strip Patterns November 8, 2016

Background

To practice with our ideas about the mathematical description of symmetric strip patterns, let’s look at the following questions.

Questions

I. The figure below shows a strip pattern of the type p111 (with only translation symmetries). For each of the six other types, complete this pattern to a pattern of that type by adding as little as possible and identify them by name (see the PowerPoint slides or Ascher’s book for the “official” names). Show at least 4 basic units of the pattern in all cases.

II. Name the type of the strip patterns for each of the examples on the back of this sheet. (These are taken from a book called Japanese Border Designs by Theodore Menten.)

III. Using coordinate equations for transformations.

A) If T (x, y) = (x + a, y) is a translation in Symm(S) for a strip pattern S then show or

explain why T^{−1}(x, y) = (x − a, y) B) If R(x, y) = (−x, −y), the 180 degree rotation about (0, 0) is also in Symm(S), then

the composition T RT^{−1} must also be in Symm(S). Compute the coordinate equations of T RT^{−1}(x, y) and identify this mapping. (Is it another translation, a rotation, a reflection, or a glide reflection?)