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src/Advanced/Wallpaper/main.typ

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+#import "@local/handout:0.1.0": *
+
+#show: handout.with(
+  title: [Wallpaper Symmetries],
+  by: "Mark",
+)
+
+#include "parts/00 arithmetic.typ"
+#pagebreak()

src/Advanced/Wallpaper/meta.toml

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+[metadata]
+title = "Wallpaper Symmetries"
+
+
+[publish]
+handout = true
+solutions = true

src/Advanced/Wallpaper/parts/00 arithmetic.typ

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+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+= Wallpaper Symmetries
+
+#definition()
+A _Euclidean isometry_ is a transformation of the plane that preserves distances. \
+Intuituvely, an isometry moves objects on the plane without deforming them.
+
+There are four classes of _Euclidean isometries_:
+- Translation
+- Reflection
+- Rotation
+- Glide reflection
+#note([We can prove that there are no others, but this is beyond the scope of this handout.]) \
+A simple example of each isometry is shown below:
+
+#let demo(c) = {
+  let s = 0.75
+  cetz.draw.line(
+    (0, 0),
+    (3 * s, 0),
+    (3 * s, 1 * s),
+    (1 * s, 1 * s),
+    (1 * s, 2 * s),
+    (0, 2 * s),
+    close: true,
+    fill: c,
+    stroke: black + 0mm * s,
+  )
+}
+
+#box(
+  height: 10cm,
+  align(
+    center,
+    table(
+      stroke: none,
+      align: center,
+      columns: (1fr, 1fr),
+      rows: (1fr, 1fr),
+      cetz.canvas({
+        import cetz.draw: *
+
+        demo(ored)
+        translate(x: 0, y: -2.5)
+        demo(oblue)
+      })
+        + [Translation],
+      cetz.canvas({
+        import cetz.draw: *
+
+        circle((-2, 0), radius: 0.2, stroke: none, fill: black)
+
+        demo(ored)
+        rotate(z: -45deg, origin: (-2, 0))
+        demo(oblue)
+      })
+        + [Rotation],
+
+      cetz.canvas({
+        import cetz.draw: *
+
+        line((-2, 0), (4, 0))
+
+        translate(x: 0, y: 0.25)
+        demo(ored)
+        set-transform(none)
+
+        set-transform((
+          (1, 0, 0, 0),
+          (0, 1, 0, 0),
+          (0, 0, 1, 0),
+          (0, 0, 0, 1),
+        ))
+
+        translate(x: 0, y: 0.25)
+        demo(oblue)
+      })
+        + [Reflection],
+      cetz.canvas({
+        import cetz.draw: *
+
+
+        demo(ored)
+
+        set-transform((
+          (1, 0, 0, 0),
+          (0, 1, 0, 0),
+          (0, 0, 0, 0),
+          (0, 0, 0, 0),
+        ))
+        translate(x: 2.5, y: 0)
+
+        demo(oblue)
+
+        set-transform(none)
+        line((-1, 0), (5, 0))
+      })
+        + [Glide reflection],
+    ),
+  ),
+)
+
+
+#definition()
+A _wallpaper_ is a two-dimensional pattern that...
+- has translational symmetry in at least two directions
+  #note([(and therefore fills the plane)])
+- has countably many reflection, rotation, or glide symmetries. \
+
+#problem()
+Is the empty plane a valid wallpaper?
+
+#solution([
+  No, since it has an uncountable number of symmetries.
+])
+
+#pagebreak()
+
+#definition()
+A _mirror symmetry_ is a reflection about a line.
+Its signature is `*`.
+An integer $n$ following `*` denotes $n$-fold mirror symmetry, the intersection of $n$ mirror lines.
+
+Two intersections of mirror lines are considered the same if we can perform a translation and rotation that sends one to the other, while leaving the pattern the same. There are various possible combinations of mirror symmetries.
+
+
+This flower pattern has signature `*632`: there are
+three distinct point of intersecting mirror lines with 6, 3, and 2 mirror lines respectively.
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 50mm,
+  image("../res/*632-a.png", height: 100%),
+  image("../res/*632-b.png", height: 100%),
+)
+
+#problem()
+Design a wallpaper pattern with signature `*2222`
+
+
+#pagebreak()
+Another symmetry is n-fold rotational symmetry about a point, whose signature is written n. Multiple bold numbers means multiple points of rotational symmetry.
+
+Two points of rotational symmetry are considered the same if we can perform a translation + rotation sending one to the other, while leaving the pattern the same.
+
+There are also patterns with both kinds of symmetries. To classify such patterns, first find all the mirror symmetries, then all the rotational symmetries that are not accounted
+for by the mirror symmetries.
+
+By convention we write the rotational symmetries before
+the `*`.
+
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 50mm,
+  image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
+)
+
+
+#problem()
+Mark the three rotation points in Figure 1.
+
+#problem()
+Find the signature of the pattern in Figure 2.
+
+#solution([`3 *3`])
+
+#pagebreak()
+
+
+Some exceptional cases: It is possible to have two different parallel mirror lines. In
+this situation the signature is ∗ ∗
+
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 60mm,
+  image("../res/**.png", height: 100%),
+)
+
+#problem()
+Draw another wallpaper pattern with signature `**`
+
+
+#pagebreak()
+
+There are two other types of symmetries. The first called a miracle whose signature is
+written ×. It is the result of a glide reflection, which is translation along a line followed
+by reflection about that line.
+This occurs when there is orientation-reversing symmetry not accounted for by a mirror.
+For example, if we modify Figure 3 slightly we get a signature of ∗×
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 60mm,
+  image("../res/*x-b.png", height: 100%),
+  image("../res/*x-a.png", height: 100%),
+)
+
+Signature ∗×. There is a glide reflection (shown by the by the dotted line)
+taking the clockwise spiral to the counter-clockwise spiral, reversing orientation
+
+
+#pagebreak()
+
+
+#problem()
+Find the signatures of the following patterns:
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 60mm,
+  image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
+  image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
+)
+
+
+#pagebreak()
+
+There is another exceptional case with two miracles, where there are two glide reflection
+symmetries along distinct lines. There are other glide reflections, but they can be obtained
+by composing the two marked in the diagram.
+
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 60mm,
+  image("../res/xx-b.png", height: 100%),
+  image("../res/xx-a.png", height: 100%),
+)
+
+Figure 7: There are two distinct mirrorless crossings, so the signature is `xx`.
+Lastly, if none of the above symmetries appear in the pattern, then there is only regular
+translational symmetry, which we denote by O.
+
+In summary, to find the signature of a pattern:
+- Find the mirror lines (∗) and the distinct intersections
+- Find the rotational points of symmetry not account for by reflections.
+- Look for any miracles (×) i.e. glide reflections that do not cross a mirror line.
+- If you found none of the above, it is just O