Wallpaper groups (#23)

Reviewed-on: #23

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",
@@ -13,3 +19,8 @@
 
 #include "parts/02 rotate.typ"
 #pagebreak()
+
+#include "parts/03 problems.typ"
+#pagebreak()
+
+#include "parts/04 theorem.typ"

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

@@ -5,14 +5,14 @@
 
 #definition()
 A _Euclidean isometry_ is a transformation of the plane that preserves distances. \
-Intuituvely, an isometry moves objects on the plane without deforming them.
+Intuitively, an isometry moves objects on the plane without deforming them.
 
-There are four classes of _Euclidean isometries_:
+There are four classes of Euclidean isometries:
-- Translation
+- translations
-- Reflection
+- reflections
-- Rotation
+- rotations
-- Glide reflection
+- glide reflections
-#note([We can prove that there are no others, but this is beyond the scope of this handout.]) \
+#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) = {
@@ -41,7 +41,7 @@ A simple example of each isometry is shown below:
       import cetz.draw: *
 
       demo(ored)
-      translate(x: 0, y: -1.5)
+      translate(x: -1.0, y: -1.0)
       demo(oblue)
     })
     #v(1fr)
@@ -105,7 +105,7 @@ A simple example of each isometry is shown below:
         (0, 0, 0, 0),
         (0, 0, 0, 0),
       ))
-      translate(x: 1.5, y: 0)
+      translate(x: 2, y: 0)
 
       demo(oblue)
 
@@ -119,13 +119,33 @@ A simple example of each isometry is shown below:
 
 #definition()
 A _wallpaper_ is a two-dimensional pattern that...
-- has translational symmetry in at least two directions
+- has translational symmetry in at least two non-parallel directions (and therefore fills the plane) \
-  #note([(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)

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

@@ -4,51 +4,62 @@
 = Mirror Symmetry
 
 #definition()
-A _mirror symmetry_ is a reflection about a line. \
+A _reflection_ is a transformation of the plane obtained by reflecting all points about a line. \
-If $n$ mirror symmetries intersect at a point, we say that point is an _$n$-fold mirror node_.
+If this reflection maps the wallpaper to itself, we have a _mirror symmetry_. \
 
-#v(3mm)
+If $n$ such mirror lines intersect at a point, they form a _mirror node of order $n$_. \
+#note[Mirror nodes with order 1 do not exist (i.e, $n >= 2$). A line does not intersect itself!]
 
-Two mirror nodes are identical if we can map one to the other with a translation and a rotation \
+#v(2mm)
-while preserving the pattern on the wallpaper.
+
+Two mirror nodes on a wallpaper are identical if we can map one to the other with a translation and a rotation while preserving the pattern on that wallpaper.
 
 #problem(label: "pat333")
-Find all three three distinct mirror nodes in the following pattern. \
+Find all three distinct mirror nodes in the following pattern. \
-What is the order of each intersection? \
+What is the order of each node? \
 #hint([
   You may notice rotational symmetry in this pattern. \
-  Don't worry about that for now.
+  Don't worry about that yet, we'll discuss it later.
 ])
 
 #table(
   stroke: none,
   align: center,
   columns: 1fr,
-  rows: 50mm,
+  rows: 45mm,
   image("../res/wolfram/p3m1.svg", height: 100%),
 )
 
-#solution([This is `*333`])
+#solution([
+  The mirror nodes are:
+  - the center of the amber cross
+  - the center of each right-handed group of three adjacent hexagons
+  - the center of each left-handed group of three adjacent hexagons
+])
 
 #v(1fr)
 
 
 #definition()
-_Orbifold notation_ gives us a way to write down the symmetries of a wallpaper. \
+_Orbifold notation_ gives us a way to describe the symmetries of a wallpaper. \
+It defines a _signature_ that fully describes all the symmetries of a given pattern. \
 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. \
+In orbifold notation, mirror nodes are denoted by a #sym.convolve followed by a list of integer. \
-Every integer $n$ following a `*` denotes a mirror node of order $n$.
+Every integer $n$ following a #sym.convolve denotes a mirror node of order $n$.
 
-#v(3mm)
+#v(2mm)
+
+The order of these integers doesn't matter. #sym.convolve`234` and #sym.convolve`423` are the same signature. \
+However, we usually denote $n$-fold symmetries in descending order (that is, like #sym.convolve`432`). \
+If we have many nodes of the same order, integers may be repeated.
 
-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?
+What is the signature of the wallpaper in @pat333? \
-#solution([It is `*333`])
+#hint[Again, ignore rotational symmetry for now.]
+#solution([It is #sym.convolve`333`])
 
 
 // MARK: page
@@ -62,13 +73,13 @@ Find the signature of the following pattern.
   stroke: none,
   align: center,
   columns: 1fr,
-  rows: 50mm,
+  rows: 60mm,
   image("../res/*632-a.png", height: 100%),
 )
 
 
 #solution([
-  It is `*632`:
+  It is #sym.convolve`632`:
   #table(
     stroke: none,
     align: center,
@@ -81,7 +92,7 @@ Find the signature of the following pattern.
 #v(1fr)
 
 #problem()
-Draw a wallpaper pattern with signature `*2222`
+Draw a wallpaper pattern with signature #sym.convolve`2222`
 
 #solution([
   Sample solutions are below.
@@ -97,3 +108,25 @@ Draw a wallpaper pattern with signature `*2222`
 ])
 
 #v(1fr)
+
+#pagebreak()
+
+
+#remark()
+In an exceptional case, we have two parallel mirror lines. \
+Consider the following pattern:
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 60mm,
+  image("../res/**.png", height: 100%),
+)
+
+The signature of this pattern is #sym.convolve#sym.convolve
+
+#problem()
+Draw another wallpaper pattern with signature #sym.convolve#sym.convolve.
+
+#v(1fr)

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

@@ -4,16 +4,18 @@
 = 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.
+#definition()
-
+A wallpaper may also have $n$-fold rotational symmetry about a point.
-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.
+#v(2mm)
-
+This means there are no more than $n$ rotations around that point that map the wallpaper to itself.
-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
+#v(2mm)
-for by the mirror symmetries.
+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.
-
-By convention we write the rotational symmetries before
-the `*`.
 
+#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,
@@ -25,91 +27,155 @@ the `*`.
 
 
 #problem()
-Mark the three rotation points in Figure 1.
+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 in Figure 2.
+Find the signature of the pattern on the right.
 
-#solution([`3 *3`])
+#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()
 
 
-Some exceptional cases: It is possible to have two different parallel mirror lines. In
+// MARK: glide
-this situation the signature is ∗ ∗
 
+= 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/**.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%),
 )
 
+#solution([
+  This is #sym.diamond.stroked.small`4`#sym.convolve`2`
+])
+
+#v(1fr)
+
+
 
 #pagebreak()
 
-There is another exceptional case with two miracles, where there are two glide reflection
+#problem()
-symmetries along distinct lines. There are other glide reflections, but they can be obtained
+Find two glide reflections in the following pattern.\
-by composing the two marked in the diagram.
+#note[(and thus show that its signature is #sym.times#sym.times.)]
 
 
 #table(
   stroke: none,
   align: center,
-  columns: (1fr, 1fr),
+  columns: 1fr,
-  rows: 60mm,
+  rows: 70mm,
   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`.
+#solution([
-Lastly, if none of the above symmetries appear in the pattern, then there is only regular
+  #table(
-translational symmetry, which we denote by O.
+    stroke: none,
+    align: center,
+    columns: 1fr,
+    rows: 40mm,
+    image("../res/xx-a.png", height: 100%),
+  )
+])
 
-In summary, to find the signature of a pattern:
+#v(1fr)
-- 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/parts/03 problems.typ

@@ -0,0 +1,67 @@
+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+#let pat(img, sol) = {
+  problem()
+
+  table(
+    stroke: none,
+    align: center,
+    columns: (1fr, 1fr),
+    rows: 80mm,
+    image(img, height: 100%), image(img, height: 100%),
+  )
+
+  solution(sol)
+  v(1fr)
+}
+
+= A few problems
+
+Find the signatures of the following patterns. Mark all mirror nodes, rotation centers, and glide reflections. \
+Each pattern is provided twice for convenience.
+
+
+#pat("../res/wolfram/cm.svg", [#sym.times#sym.convolve])
+#pat("../res/wolfram/cmm.svg", [#sym.diamond.stroked`2`#sym.convolve`22`])
+#pagebreak()
+
+
+
+
+#pat("../res/wolfram/p3.svg", [#sym.diamond.stroked`333`])
+#pat("../res/wolfram/p3m1.svg", [#sym.convolve`333`])
+#pagebreak()
+
+#pat("../res/wolfram/p4.svg", [#sym.diamond.stroked`442`])
+#pat("../res/wolfram/p4m.svg", [#sym.convolve`442`])
+#pagebreak()
+
+#pat("../res/wolfram/p6.svg", [#sym.diamond.stroked`632`])
+#pat("../res/wolfram/p6m.svg", [#sym.convolve`632`])
+#pagebreak()
+
+#pat("../res/wolfram/p4g.svg", [#sym.diamond.stroked`4`#sym.convolve`2`])
+#pat("../res/wolfram/p31m.svg", [#sym.diamond.stroked`3`#sym.convolve`3`])
+#pagebreak()
+
+#problem()
+Draw a wallpaper with the signature #sym.convolve`442` \
+#note[Make sure there are no other symmetries!]
+#v(1fr)
+#pagebreak()
+
+
+#pat("../res/wolfram/pgg.svg", [#sym.diamond.stroked`22`#sym.times])
+#pat("../res/wolfram/pmg.svg", [#sym.diamond.stroked`22`#sym.convolve])
+#pagebreak()
+
+#pat("../res/wolfram/pg.svg", [#sym.times#sym.times])
+#pat("../res/wolfram/pm.svg", [#sym.convolve#sym.convolve])
+#pagebreak()
+
+#pat("../res/wolfram/p2.svg", [#sym.diamond.stroked`2222`])
+#pat("../res/wolfram/pmm.svg", [#sym.convolve`2222`])
+#pagebreak()
+
+#pat("../res/wolfram/p1.svg", [#sym.circle.small])

src/Advanced/Wallpaper/parts/04 theorem.typ

@@ -0,0 +1,100 @@
+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+= The Signature-Cost Theorem
+
+#definition()
+First, we'll associate a _cost_ to each type of symmetry in orbifold notation:
+
+#v(4mm)
+#align(
+  center,
+  table(
+    stroke: (1pt, 1pt),
+    align: center,
+    columns: (auto, auto, auto, auto),
+    [*Symbol*], [*Cost*], [*Symbol*], [*Cost*],
+    [#sym.circle.small], [2], [#sym.times or #sym.convolve], [1],
+    [#sym.diamond.stroked.small`2`], [1/2], [#sym.convolve`2`], [1/4],
+    [#sym.diamond.stroked.small`3`], [2/3], [#sym.convolve`3`], [1/3],
+    [#sym.dots], [#sym.dots], [#sym.dots], [#sym.dots],
+    [#sym.diamond.stroked.small`n`],
+    [$(n-1) / n$],
+    [#sym.convolve`n`],
+    [$(n-1) / (2n)$],
+  ),
+)
+
+
+We then calculate the total "cost" of a signature by adding up the costs of each component.
+
+For example, a pattern with signature #sym.convolve`333` has cost 2:
+
+#v(2mm)
+
+$
+  2 / 3 + 2 / 3 + 2 / 3 = 2
+$
+
+#problem()
+Calculate the costs of the following signatures:
+- #sym.diamond.stroked.small`3`#sym.convolve`3`
+- #sym.convolve#sym.convolve
+- #sym.diamond.stroked.small`4`#sym.convolve`2`:
+
+#solution([
+  - #sym.diamond.stroked.small`3`#sym.convolve`3`: $2/3 + 1 + 1/3 = 2$
+  - #sym.convolve#sym.convolve: $1 + 1 = 2$
+  - #sym.diamond.stroked.small`4`#sym.convolve`2`: $3/4 + 1 + 1/4 = 2$
+])
+
+#v(1fr)
+
+#theorem(name: "Signature Cost Theorem")
+The signatures of planar wallpaper patterns are exactly those with total cost 2. \
+#note([We will not prove this theorem today, accept it without proof.])
+
+#problem()
+Consider the 4 symmetries (translation, reflection, rotation, and glide reflection). \
+Which preserve orientation? Which reverse orientation?
+
+#solution([
+  - Reflections and glide reflections reverse orientation (directions of spirals).
+  - Translation and rotation preserve orientation.
+])
+
+#v(1fr)
+#pagebreak()
+
+#problem()
+Use the signature-cost theorem to find all the signatures consisting of only #sym.circle.small or rotational symmetries.
+
+#solution([
+  #sym.diamond.stroked.small`632`, #sym.diamond.stroked.small`442`, #sym.diamond.stroked.small`333`, #sym.diamond.stroked.small`2222`, #sym.circle.small
+])
+
+#v(1fr)
+
+
+#problem()
+Find all the signatures consisting of only mirror symmetries.
+
+#solution([
+  #sym.convolve`632`, #sym.convolve`442`, #sym.convolve`333`, #sym.convolve`2222`, #sym.convolve#sym.convolve
+])
+
+#v(1fr)
+
+
+#problem()
+Find all the remaining signatures. \
+Each must be a mix of of mirror symmetries, rotational symmetries, or glide reflections. \
+#hint([They are all shown in the problems section.])
+
+#solution([
+  #sym.diamond.stroked.small`3`#sym.convolve`3`, #sym.diamond.stroked.small`4`#sym.convolve`2`,
+  #sym.diamond.stroked.small`22`#sym.times, #sym.diamond.stroked.small`22`#sym.convolve,
+  #sym.times#sym.times, #sym.times#sym.convolve
+])
+
+#v(1fr)

tools/typos.toml

@@ -1,6 +1,7 @@
 [default]
 extend-words."LSAT" = "LSAT"
 extend-words."ket" = "ket"
+extend-words."typ" = "typ"
 
 extend-ignore-re = [
 	# spell:disable-line