TMP

src/Advanced/Wallpaper/main.typ

@@ -1,5 +1,11 @@
 #import "@local/handout:0.1.0": *
 
+// Resources:
+//
+// https://eschermath.org/wiki/Wallpaper_Patterns.html
+// https://mathworld.wolfram.com/WallpaperGroups.html
+// https://en.wikipedia.org/wiki/Wallpaper_group
+
 #show: handout.with(
   title: [Wallpaper Symmetry],
   by: "Mark",
@@ -10,3 +16,6 @@
 
 #include "parts/01 reflect.typ"
 #pagebreak()
+
+#include "parts/02 rotate.typ"
+#pagebreak()

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

@@ -0,0 +1,115 @@
+#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