TMP

src/Advanced/Wallpaper/main.typ

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+#import "@local/handout:0.1.0": *
+
+#show: handout.with(
+  title: [Wallpaper Symmetry],
+  by: "Mark",
+)
+
+#include "parts/00 intro.typ"
+#pagebreak()
+
+#include "parts/01 reflect.typ"
+#pagebreak()
+
+#include "parts/02 rotate.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 intro.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 = 0.5
+  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,
+  )
+}
+
+#table(
+  stroke: none,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: (3.5cm, 3.5cm),
+  row-gutter: 2mm,
+  [
+    #cetz.canvas({
+      import cetz.draw: *
+
+      demo(ored)
+      translate(x: 0, y: -1.5)
+      demo(oblue)
+    })
+    #v(1fr)
+    Translation
+  ],
+  [
+    #cetz.canvas({
+      import cetz.draw: *
+
+      circle((-2, 0), radius: 0.1, stroke: none, fill: black)
+      arc(
+        (-2, 0),
+        radius: 1,
+        anchor: "origin",
+        start: 0deg,
+        stop: -30deg,
+        mode: "PIE",
+      )
+
+      demo(ored)
+      rotate(z: -30deg, origin: (-2, 0))
+      demo(oblue)
+    })
+    #v(1fr)
+    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)
+    })
+    #v(1fr)
+    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: 1.5, y: 0)
+
+      demo(oblue)
+
+      set-transform(none)
+      line((-1, 0), (5, 0))
+    })
+    #v(1fr)
+    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 a countable number of reflection, rotation, or glide symmetries. \
+
+#problem()
+Is the empty plane a valid wallpaper?
+
+#solution([
+  No, since it has uncountably many symmetries.
+])

src/Advanced/Wallpaper/parts/01 reflect.typ

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+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+= Mirror Symmetry
+
+#definition()
+A _mirror symmetry_ is a reflection about a line. \
+If $n$ mirror symmetries intersect at a point, we say that point is an _$n$-fold mirror node_.
+
+#v(3mm)
+
+Two mirror nodes are identical if we can map one to the other with a translation and a rotation \
+while preserving the pattern on the wallpaper.
+
+#problem(label: "pat333")
+Find all three three distinct mirror nodes in the following pattern. \
+What is the order of each intersection? \
+#hint([
+  You may notice rotational symmetry in this pattern. \
+  Don't worry about that for now.
+])
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 50mm,
+  image("../res/wolfram/p3m1.svg", height: 100%),
+)
+
+#solution([This is `*333`])
+
+#v(1fr)
+
+
+#definition()
+_Orbifold notation_ gives us a way to write down the symmetries of a wallpaper. \
+We will introduce orbifold notation one symmetry at a time.
+
+#definition()
+In orbifold notation, mirror nodes are denoted by a `*` followed by at least one integer. \
+Every integer $n$ following a `*` denotes a mirror node of order $n$.
+
+#v(3mm)
+
+The order of these integers doesn't matter. `*234` and `*423` are the same signature. \
+However, we usually denote $n$-fold symmetries in descending order (that is, like `*432`).
+
+#problem()
+What is the signature of the wallpaper in @pat333?
+#solution([It is `*333`])
+
+
+// MARK: page
+#v(1fr)
+#pagebreak()
+
+#problem()
+Find the signature of the following pattern.
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 50mm,
+  image("../res/*632-a.png", height: 100%),
+)
+
+
+#solution([
+  It is `*632`:
+  #table(
+    stroke: none,
+    align: center,
+    columns: 1fr,
+    rows: 40mm,
+    image("../res/*632-b.png", height: 100%),
+  )
+])
+
+#v(1fr)
+
+#problem()
+Draw a wallpaper pattern with signature `*2222`
+
+#solution([
+  Sample solutions are below.
+
+  #table(
+    stroke: none,
+    align: center,
+    columns: (1fr, 1fr),
+    rows: 50mm,
+    image("../res/wolfram/pmm.svg", height: 100%),
+    image("../res/escher/pmm.svg", height: 100%),
+  )
+])
+
+#v(1fr)

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
+
+
+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

src/Advanced/Wallpaper/res/escher/pmm.svg

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