CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...
CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...
CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...
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Once we know that a translation vector parallel to the reflection<br />
axis exists, then it is possible to pick the minimal such vector --<br />
recall that wallpaper patterns do not have arbitrarily small<br />
translations (4.0.4) -- which is easily shown to be the minimal<br />
gliding vector of a (hidden) glide reflection along the reflection axis.<br />
You should verify all these ideas for the examples in 6.3.1 and 6.3.3;<br />
you may in particular verify that the vertical translation guaranteed<br />
by the process in figure 6.23 is actually twice as long as the<br />
pattern’s minimal vertical translation.<br />
6.3.5 Symmetry plans. Even though we classified the pm types<br />
looking at their reflections and hidden glide reflections, we prefer<br />
to provide their symmetry plans based on reflections and parallel to<br />
them translations. It is of course easy to see that there exists a<br />
color-reversing translation parallel to the reflection axis if and<br />
only if the reflection and the corresponding hidden glide reflection<br />
have opposite effect on color.<br />
pm pm′ p b<br />
′ 1m<br />
p′ m p b<br />
′ g c′ m<br />
Fig. 6.24