edits

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

@@ -5,9 +5,9 @@
 
 #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:
 - translations
 - reflections
 - rotations
@@ -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
-  #note([(and therefore fills the plane)])
+- 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

@@ -4,16 +4,18 @@
 = 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_.
+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_. \
 
-#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 \
-while preserving the pattern on the wallpaper.
+#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 three distinct mirror nodes in the following pattern. \
+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. \
@@ -24,7 +26,7 @@ What is the order of each node? \
   stroke: none,
   align: center,
   columns: 1fr,
-  rows: 50mm,
+  rows: 45mm,
   image("../res/wolfram/p3m1.svg", height: 100%),
 )
 
@@ -39,21 +41,24 @@ What is the order of each node? \
 
 
 #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 #sym.convolve 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 #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 are repeated.
+If we have many nodes of the same order, integers may be repeated.
+
 
 #problem()
-What is the signature of the wallpaper in @pat333?
+What is the signature of the wallpaper in @pat333? \
+#hint[Again, ignore rotational symmetry for now.]
 #solution([It is #sym.convolve`333`])
 
 
@@ -68,7 +73,7 @@ Find the signature of the following pattern.
   stroke: none,
   align: center,
   columns: 1fr,
-  rows: 50mm,
+  rows: 60mm,
   image("../res/*632-a.png", height: 100%),
 )
 

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

@@ -7,7 +7,7 @@
 #definition()
 A wallpaper may also have $n$-fold rotational symmetry about a point.
 #v(2mm)
-This means there are `n` rotations around that point that map the wallpaper to itself.
+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.
 
@@ -74,7 +74,8 @@ For example, consider the following pattern:
 )
 
 #problem()
-Convince yourself that this pattern has only one mirror symmetry.
+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. \
@@ -100,10 +101,21 @@ Use the following picture to find the glide reflection in the above pattern.
 #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:
 
@@ -111,7 +123,7 @@ Find the signature of the following pattern:
   stroke: none,
   align: center,
   columns: 1fr,
-  rows: 60mm,
+  rows: 50mm,
   image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
 )
 
@@ -167,13 +179,3 @@ Find two glide reflections in the following pattern.\
 ])
 
 #v(1fr)
-
-#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;
-- 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.

src/Advanced/Wallpaper/parts/03 problems.typ

@@ -46,7 +46,8 @@ Each pattern is provided twice for convenience.
 #pagebreak()
 
 #problem()
-Draw a wallpaper with the signature #sym.convolve`442`
+Draw a wallpaper with the signature #sym.convolve`442` \
+#note[Make sure there are no other symmetries!]
 #v(1fr)
 #pagebreak()