Wallpaper groups (#23)

Reviewed-on: #23

src/Advanced/Wallpaper/parts/02 rotate.typ

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+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+= Rotational Symmetry
+
+
+#definition()
+A wallpaper may also have $n$-fold rotational symmetry about a point.
+#v(2mm)
+This means there are no more than $n$ rotations around that point that map the wallpaper to itself.
+#v(2mm)
+As before, two points of rotational symmetry are identical if we can perform a translation and rotation that maps one to the other without changing the wallpaper.
+
+#definition()
+In orbifold notation, rotation is specified similarly to reflection, but uses the prefix #sym.diamond.stroked.small. \
+For example:
+- #sym.diamond.stroked.small`333` denotes a pattern with three distinct centers of rotation of order 3.
+- #sym.diamond.stroked.small`4`#sym.convolve`2` denotes a pattern with one rotation center of order 4 and one mirror node of order 2.
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 50mm,
+  image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
+)
+
+
+#problem()
+Find the three rotation centers in the left wallpaper. \
+What are their orders?
+
+#solution([This is #sym.diamond.stroked.small`333`])
+
+#v(1fr)
+
+#problem()
+Find the signature of the pattern on the right.
+
+#solution([This is #sym.diamond.stroked.small`3`#sym.convolve`3`])
+
+#v(1fr)
+
+
+#remark()
+You may have noticed that we could have an ambiguous classification, since two reflections are equivalent to a translation and a rotation.
+We thus make the following distinction: _rotational symmetry that can be explained by reflection is not rotational symmetry._
+
+#v(2mm)
+
+In other words, when classifying a pattern...
+- we first find all mirror symmetries,
+- then all rotational symmetries that are not accounted for by reflection.
+
+#pagebreak()
+
+
+// MARK: glide
+
+= Glide Reflections
+
+#definition()
+Another type of symmetry is the _glide reflection_, denoted #sym.times.
+
+A glide reflection is the result of a translation along a line followed by reflection about that line.
+
+For example, consider the following pattern:
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 60mm,
+  image("../res/*x-a.png", height: 100%),
+)
+
+#problem()
+Convince yourself that all mirror lines in this pattern are _not_ distinct. /
+In other words, this pattern has only one mirror symmetry.
+
+#solution([
+  There may seem to be two, but they are identical. \
+  We can translate one onto the other.
+])
+
+#v(1fr)
+
+
+
+#problem()
+Use the following picture to find the glide reflection in the above pattern.
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 70mm,
+  image("../res/*x-b.png", height: 100%),
+)
+
+#v(1fr)
+
+#remark()
+The signature of this wallpaper is #sym.convolve#sym.times.
+
+#pagebreak()
+
+
+#definition()
+If none of the above symmetries appear in a pattern, then we only have simple translational symmetry. We denote this with the signature #sym.circle.small.
+
+#remark()
+In summary, to find the signature of a pattern:
+- find the mirror lines (#sym.convolve) and the distinct intersections;
+- then find the rotation centers (#sym.diamond.stroked.small) not explained by reflection;
+- then find all glide reflections (#sym.times) that do not cross a mirror line.
+- If we have none of the above, our pattern must be #sym.circle.small.
+
+
+
+#problem()
+Find the signature of the following pattern:
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 50mm,
+  image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
+)
+
+#solution([
+  This is #sym.convolve#sym.times.
+])
+
+#v(1fr)
+
+
+#problem()
+Find the signature of the following pattern:
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 60mm,
+  image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
+)
+
+#solution([
+  This is #sym.diamond.stroked.small`4`#sym.convolve`2`
+])
+
+#v(1fr)
+
+
+
+#pagebreak()
+
+#problem()
+Find two glide reflections in the following pattern.\
+#note[(and thus show that its signature is #sym.times#sym.times.)]
+
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 70mm,
+  image("../res/xx-b.png", height: 100%),
+)
+
+#solution([
+  #table(
+    stroke: none,
+    align: center,
+    columns: 1fr,
+    rows: 40mm,
+    image("../res/xx-a.png", height: 100%),
+  )
+])
+
+#v(1fr)