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\centerline{MONT 106Q -- Mathematical Thinking}
\centerline{Problem Set 4 -- Symmetric Strip Patterns}
\centerline{November 8, 2016}
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{\it Background}
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To practice with our ideas about the mathematical description of symmetric strip 
patterns, let's look at the following questions.
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{\it Questions}
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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.  
\vglue 1.0truein
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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 {\it Japanese Border Designs} by Theodore Menten.)
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III.  Using coordinate equations for transformations.
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\item{A)} If $T(x,y) = (x + a,y)$ is a translation in ${\it Symm}(S)$ for a strip 
pattern $S$ then show or explain why 
$T^{-1}(x,y) = (x - a,y)$
\item{B)}  If $R(x,y) = (-x,-y)$, the 180 degree rotation about $(0,0)$ is also 
in ${\it Symm}(S)$, then the composition $TRT^{-1}$ must also be in ${\it Symm}(S)$.
Compute the coordinate equations of $TRT^{-1}(x,y)$ and identify this mapping.  (Is it
another translation, a rotation, a reflection, or a glide reflection?)
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