Wallpaper groups (#23)

Reviewed-on: #23

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

@@ -0,0 +1,151 @@
+#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. \
+Intuitively, an isometry moves objects on the plane without deforming them.
+
+There are four classes of Euclidean isometries:
+- translations
+- reflections
+- rotations
+- glide reflections
+#note([We can prove 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: -1.0, y: -1.0)
+      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: 2, 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 non-parallel directions (and therefore fills the plane) \
+  #note[
+    "Translational symmetry" means that we can slide the entire wallpaper in some direction, \
+    eventually mapping the pattern to itself.]
+- has a countable number of reflection, rotation, or glide symmetries. \
+
+#v(1fr)
+#pagebreak()
+
+#problem()
+Is a plain square grid a valid wallpaper?
+
+#solution([
+  Yes!
+  - It has translational symmetry in the horizontal and vertical directions
+  - It has a countable number of symmetries---namely, six distinct mirror lines (horizontal, vertical, and diagonal) duplicated once per square.
+  - A square grid is #sym.convolve`442`
+])
+
+#v(1fr)
+
+
+#problem()
+Is the empty plane a valid wallpaper?
+
+#solution([
+  No, since it has uncountably many symmetries.
+])
+
+#v(1fr)