From 9c6bcecc3056b900ead919ad3acde2389f3b47fe Mon Sep 17 00:00:00 2001 From: Mark Date: Thu, 8 May 2025 18:17:05 -0700 Subject: [PATCH] edits --- src/Advanced/Wallpaper/parts/00 intro.typ | 32 ++++++++++++++++---- src/Advanced/Wallpaper/parts/01 reflect.typ | 31 +++++++++++-------- src/Advanced/Wallpaper/parts/02 rotate.typ | 30 +++++++++--------- src/Advanced/Wallpaper/parts/03 problems.typ | 3 +- 4 files changed, 62 insertions(+), 34 deletions(-) diff --git a/src/Advanced/Wallpaper/parts/00 intro.typ b/src/Advanced/Wallpaper/parts/00 intro.typ index f1f85f2..88a1391 100644 --- a/src/Advanced/Wallpaper/parts/00 intro.typ +++ b/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) diff --git a/src/Advanced/Wallpaper/parts/01 reflect.typ b/src/Advanced/Wallpaper/parts/01 reflect.typ index 93a6472..ed765a7 100644 --- a/src/Advanced/Wallpaper/parts/01 reflect.typ +++ b/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%), ) diff --git a/src/Advanced/Wallpaper/parts/02 rotate.typ b/src/Advanced/Wallpaper/parts/02 rotate.typ index 7964216..3c03ed2 100644 --- a/src/Advanced/Wallpaper/parts/02 rotate.typ +++ b/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. diff --git a/src/Advanced/Wallpaper/parts/03 problems.typ b/src/Advanced/Wallpaper/parts/03 problems.typ index 19b47cc..dd45305 100644 --- a/src/Advanced/Wallpaper/parts/03 problems.typ +++ b/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()