From 7b76fd2dc8a6e3862a1112d6dd12e75d592c864f Mon Sep 17 00:00:00 2001 From: Mark Date: Tue, 6 May 2025 21:56:35 -0700 Subject: [PATCH] More edits --- src/Advanced/Wallpaper/parts/00 intro.typ | 10 +- src/Advanced/Wallpaper/parts/01 reflect.typ | 48 ++++- src/Advanced/Wallpaper/parts/02 rotate.typ | 196 +++++++++++++------- 3 files changed, 173 insertions(+), 81 deletions(-) diff --git a/src/Advanced/Wallpaper/parts/00 intro.typ b/src/Advanced/Wallpaper/parts/00 intro.typ index b3bfc7c..f1f85f2 100644 --- a/src/Advanced/Wallpaper/parts/00 intro.typ +++ b/src/Advanced/Wallpaper/parts/00 intro.typ @@ -8,11 +8,11 @@ A _Euclidean isometry_ is a transformation of the plane that preserves distances Intuituvely, an isometry moves objects on the plane without deforming them. There are four classes of _Euclidean isometries_: -- Translation -- Reflection -- Rotation -- Glide reflection -#note([We can prove that there are no others, but this is beyond the scope of this handout.]) \ +- 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) = { diff --git a/src/Advanced/Wallpaper/parts/01 reflect.typ b/src/Advanced/Wallpaper/parts/01 reflect.typ index c35f8bf..93a6472 100644 --- a/src/Advanced/Wallpaper/parts/01 reflect.typ +++ b/src/Advanced/Wallpaper/parts/01 reflect.typ @@ -14,10 +14,10 @@ while preserving the pattern on the wallpaper. #problem(label: "pat333") Find all three 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( @@ -28,7 +28,12 @@ What is the order of each intersection? \ 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) @@ -38,17 +43,18 @@ _Orbifold notation_ gives us a way to write down the symmetries of a wallpaper. 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. \ -Every integer $n$ following a `*` denotes a mirror node of order $n$. +In orbifold notation, mirror nodes are denoted by a #sym.convolve followed by at least one integer. \ +Every integer $n$ following a #sym.convolve denotes a mirror node of order $n$. #v(3mm) -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`). +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. #problem() What is the signature of the wallpaper in @pat333? -#solution([It is `*333`]) +#solution([It is #sym.convolve`333`]) // MARK: page @@ -68,7 +74,7 @@ Find the signature of the following pattern. #solution([ - It is `*632`: + It is #sym.convolve`632`: #table( stroke: none, align: center, @@ -81,7 +87,7 @@ 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 +103,25 @@ 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) diff --git a/src/Advanced/Wallpaper/parts/02 rotate.typ b/src/Advanced/Wallpaper/parts/02 rotate.typ index 837c17d..7964216 100644 --- a/src/Advanced/Wallpaper/parts/02 rotate.typ +++ b/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. - -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 `*`. +#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. +#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, @@ -25,91 +27,153 @@ 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 -this situation the signature is ∗ ∗ +// 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 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() + + +#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%), +) + +#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() -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. +#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, 1fr), - rows: 60mm, + columns: 1fr, + 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`. -Lastly, if none of the above symmetries appear in the pattern, then there is only regular -translational symmetry, which we denote by O. +#solution([ + #table( + stroke: none, + align: center, + columns: 1fr, + rows: 40mm, + image("../res/xx-a.png", height: 100%), + ) +]) +#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 (∗) 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 +- 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.