Wallpaper groups (#23)

Reviewed-on: #23

src/Advanced/Lattices/parts/0 intro.tex

@@ -1,5 +1,6 @@
 \definition{}
-The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates.
+The \textit{integer lattice} $\mathbb{Z}^n$ is the set of points with integer coordinates in $n$ dimensions. \par
+For example, $\mathbb{Z}^3$ is the set of points $(a, b, c)$ where $a$, $b$, and $c$ are integers.
 
 \problem{}
 Draw $\mathbb{Z}^2$.
@@ -8,12 +9,13 @@ Draw $\mathbb{Z}^2$.
 
 
 \definition{}
-We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
+We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$
+if every lattice point can be written as
 $$
 	a_1v_1 + a_2v_2 + ... + a_kv_k
 $$
 for integer coefficients $a_i$. \par
-It is fairly easy to show that $k$ must be at least $n$.
+\textbf{Bonus:} show that $k$ must be at least $n$.
 
 \problem{}
 Which of the following generate $\mathbb{Z}^2$?
@@ -30,8 +32,7 @@ Which of the following generate $\mathbb{Z}^2$?
 \vfill
 
 \problem{}
-Find a set of two vectors that generates $\mathbb{Z}^2$. \\
-Don't say $\{ (0, 1), (1, 0) \}$, that's too easy.
+Find a set of two vectors other than $\{ (0, 1), (1, 0) \}$ that generates $\mathbb{Z}^2$. \\
 
 \vfill
 
@@ -44,7 +45,8 @@ Find a set of vectors that generates $\mathbb{Z}^n$.
 \pagebreak
 
 \definition{}
-A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use.
+Say we have a generating set of a lattice. \par
+The \textit{fundamental region} of this set is the $n$-dimensional parallelogram spanned by its members. \par
 
 \problem{}
 Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.

src/Advanced/Lattices/parts/1 minkowski.tex

@@ -1,6 +1,6 @@
 \section{Minkowski's Theorem}
 
-\theorem{Blichfeldt's Theorem}
+\theorem{Blichfeldt's Theorem}<blich>
 Let $X$ be a finite connected region. If the volume of $X$ is greater than $1$, $X$ must contain two distinct points that differ by an element of $\mathbb{Z}^n$. In other words, there exist distinct $x, y \in X$ so that $x - y \in \mathbb{Z}^n$.
 
 \vspace{2mm}
@@ -9,14 +9,22 @@ Intuitively, this means that you can translate $X$ to cover two lattice points a
 
 
 \problem{}
-Draw a region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector.
+Draw a connected region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector.
 \hint{Area is two-dimensional volume.}
 
 \vfill
 
 
 \problem{}
-The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. Explain the picture and complete the proof.
+Draw a \textit{disconnected} region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points, \par
+and show that no two points in that region differ by an integer vector.
+\note{In other words, show that \ref{blich} indeed requires a connected region.}
+
+\vfill
+
+\problem{}
+The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. \par
+Explain the picture and complete the proof.
 
 \begin{center}
 	\includegraphics[angle=90,width=0.5\linewidth]{proof.png}
@@ -48,10 +56,8 @@ Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points m
 A region $X$ is \textit{convex} if the line segment connecting any two points in $X$ lies entirely in $X$.
 
 \problem{}
-\begin{itemize}
-	\item Draw a convex region in the plane.
-	\item Draw a region that is not convex.
-\end{itemize}
+Draw a convex region in two dimensions. \par
+Then, draw a two-dimensional region that is \textit{not} convex.
 
 \vfill
 \pagebreak
@@ -59,23 +65,28 @@ A region $X$ is \textit{convex} if the line segment connecting any two points in
 
 
 \definition{}
-We say a region $X$ is \textit{symmetric} if for all points $x \in X$, $-x$ is also in $X$.
+We say a region $X$ is \textit{symmetric with respect to the origin} if for all points $x \in X$, $-x$ is also in $X$. \par
+In the following problems, \say{\textit{symmetric}} means \say{symmetric with respect to the origin.}
 
 \problem{}
-\begin{itemize}
-	\item Draw a symmetric region.
-	\item Draw an asymmetric region.
-\end{itemize}
+Draw a symmetric region. \par
+Then, draw an asymmetric region.
+
+\vfill
+
+\problem{}
+Show that a convex symmetric set always contains the origin.
 
 \vfill
 
 \theorem{Minkowski's Theorem}<mink>
-Every convex set in $\mathbb{R}^n$ that is symmetric with respect to the origin and which has a volume greater than $2^n$ contains an integral point that isn't zero.
+Every convex set in $\mathbb{R}^n$ that is symmetric and has a volume \par
+greater than $2^n$ contains an integral point that isn't zero.
 
 
 \problem{}
 Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. \par
-What is the simplest region that has the properties listed above?
+What is a simple class of regions that has the properties listed above?
 
 \vfill
 

src/Advanced/Wallpaper/main.typ

@@ -0,0 +1,26 @@
+#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",
+)
+
+#include "parts/00 intro.typ"
+#pagebreak()
+
+#include "parts/01 reflect.typ"
+#pagebreak()
+
+#include "parts/02 rotate.typ"
+#pagebreak()
+
+#include "parts/03 problems.typ"
+#pagebreak()
+
+#include "parts/04 theorem.typ"

src/Advanced/Wallpaper/meta.toml

@@ -0,0 +1,7 @@
+[metadata]
+title = "Wallpaper Symmetries"
+
+
+[publish]
+handout = true
+solutions = true

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)

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

@@ -0,0 +1,132 @@
+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+= Mirror Symmetry
+
+#definition()
+A _reflection_ is a transformation of the plane obtained by reflecting all points about a line. \
+If this reflection maps the wallpaper to itself, we have a _mirror symmetry_. \
+
+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!]
+
+#v(2mm)
+
+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 distinct mirror nodes in the following pattern. \
+What is the order of each node? \
+#hint([
+  You may notice rotational symmetry in this pattern. \
+  Don't worry about that yet, we'll discuss it later.
+])
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 45mm,
+  image("../res/wolfram/p3m1.svg", height: 100%),
+)
+
+#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 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 #sym.convolve followed by a list of integer. \
+Every integer $n$ following a #sym.convolve denotes a mirror node of order $n$.
+
+#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.
+
+
+#problem()
+What is the signature of the wallpaper in @pat333? \
+#hint[Again, ignore rotational symmetry for now.]
+#solution([It is #sym.convolve`333`])
+
+
+// MARK: page
+#v(1fr)
+#pagebreak()
+
+#problem()
+Find the signature of the following pattern.
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 60mm,
+  image("../res/*632-a.png", height: 100%),
+)
+
+
+#solution([
+  It is #sym.convolve`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 #sym.convolve`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)
+
+#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

@@ -0,0 +1,181 @@
+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.3.1"
+
+= Rotational Symmetry
+
+
+#definition()
+A wallpaper may also have $n$-fold rotational symmetry about a point.
+#v(2mm)
+This means there are no more than $n$ rotations around that point that map the wallpaper to itself.
+#v(2mm)
+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.
+
+#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,
+  align: center,
+  columns: (1fr, 1fr),
+  rows: 50mm,
+  image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
+)
+
+
+#problem()
+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 on the right.
+
+#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()
+
+
+// MARK: glide
+
+= 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/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
+)
+
+#solution([
+  This is #sym.diamond.stroked.small`4`#sym.convolve`2`
+])
+
+#v(1fr)
+
+
+
+#pagebreak()
+
+#problem()
+Find two glide reflections in the following pattern.\
+#note[(and thus show that its signature is #sym.times#sym.times.)]
+
+
+#table(
+  stroke: none,
+  align: center,
+  columns: 1fr,
+  rows: 70mm,
+  image("../res/xx-b.png", height: 100%),
+)
+
+#solution([
+  #table(
+    stroke: none,
+    align: center,
+    columns: 1fr,
+    rows: 40mm,
+    image("../res/xx-a.png", height: 100%),
+  )
+])
+
+#v(1fr)

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)

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

@@ -0,0 +1,86 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20001102//EN" "http://www.w3.org/TR/2000/CR-SVG-20001102/DTD/svg-20001102.dtd">
+<svg xmlns:svg="http://www.w3.org/2000/svg" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10.16cm" height="5.08cm" viewBox="0 0 384 192">
+<title>pmm</title>
+<desc>Exported by Tess 1.70.</desc>
+<clipPath id="mainclip"><rect x="0" y="0" width="384" height="192"/></clipPath>
+<g style="fill-rule:evenodd; stroke-linejoin:round; stroke-linecap:round; clip-path:url(#mainclip)">
+<defs>
+<g id="Tess0p">
+<path d="
+M7.1306642022,-41.0013191628 
+L7.1306642022,-7.1306642022 
+L89.1333025278,-7.1306642022 
+L7.1306642022,-42.7839852134 
+"/></g>
+<g id="Tess0" style="fill:none; stroke:none"><use xlink:href="#Tess0p"/></g>
+<g id="Tess1" style="fill:none; stroke:rgb(0,0,0); stroke-width:1.7826660506"><use xlink:href="#Tess0p"/></g>
+</defs>
+<g transform="translate(192,96) rotate(0) scale(0.5609575611,0.5609575611)">
+<g transform="translate(-405.2631578947,-100)"><use xlink:href="#Tess1"/></g>
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+<g transform="translate(-202.6315789474,200)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,-100)"><use xlink:href="#Tess1"/></g>
+<g><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,100)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,200)"><use xlink:href="#Tess1"/></g>
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+<g transform="translate(-202.6315789474,0) rotate(180)"><use xlink:href="#Tess1"/></g>
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+<g transform="translate(0,-200) rotate(180)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,-100) rotate(180)"><use xlink:href="#Tess1"/></g>
+<g transform="rotate(180)"><use xlink:href="#Tess1"/></g>
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+<g transform="translate(405.2631578947,0) rotate(180)"><use xlink:href="#Tess1"/></g>
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+<g transform="translate(-202.6315789474,-100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(-202.6315789474,0) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
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+<g transform="rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,200) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
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+<g transform="translate(0,-100) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(0,100) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(202.6315789474,-200) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(202.6315789474,-100) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(202.6315789474,0) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+<g transform="translate(202.6315789474,100) scale(1,-1)"><use xlink:href="#Tess1"/></g>
+</g>
+</g>
+</svg>

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