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251d9bb9e5
| Author | SHA1 | Date | |
|---|---|---|---|
| 251d9bb9e5 | |||
| 6e3c665e99 | |||
| 28aaf98594 |
@ -1,11 +1,5 @@
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#import "@local/handout:0.1.0": *
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// Resources:
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//
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// https://eschermath.org/wiki/Wallpaper_Patterns.html
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// https://mathworld.wolfram.com/WallpaperGroups.html
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// https://en.wikipedia.org/wiki/Wallpaper_group
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#show: handout.with(
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title: [Wallpaper Symmetry],
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by: "Mark",
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@ -19,8 +13,3 @@
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#include "parts/02 rotate.typ"
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#pagebreak()
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#include "parts/03 problems.typ"
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#pagebreak()
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#include "parts/04 theorem.typ"
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@ -5,14 +5,14 @@
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#definition()
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A _Euclidean isometry_ is a transformation of the plane that preserves distances. \
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Intuitively, an isometry moves objects on the plane without deforming them.
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Intuituvely, an isometry moves objects on the plane without deforming them.
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There are four classes of Euclidean isometries:
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- translations
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- reflections
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- rotations
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- glide reflections
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#note([We can prove there are no others, but this is beyond the scope of this handout.]) \
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There are four classes of _Euclidean isometries_:
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- Translation
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- Reflection
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- Rotation
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- Glide reflection
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#note([We can prove that there are no others, but this is beyond the scope of this handout.]) \
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A simple example of each isometry is shown below:
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#let demo(c) = {
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@ -41,7 +41,7 @@ A simple example of each isometry is shown below:
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import cetz.draw: *
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demo(ored)
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translate(x: -1.0, y: -1.0)
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translate(x: 0, y: -1.5)
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demo(oblue)
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})
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#v(1fr)
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@ -105,7 +105,7 @@ A simple example of each isometry is shown below:
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(0, 0, 0, 0),
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(0, 0, 0, 0),
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))
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translate(x: 2, y: 0)
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translate(x: 1.5, y: 0)
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demo(oblue)
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@ -119,33 +119,13 @@ A simple example of each isometry is shown below:
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#definition()
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A _wallpaper_ is a two-dimensional pattern that...
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- has translational symmetry in at least two non-parallel directions (and therefore fills the plane) \
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#note[
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"Translational symmetry" means that we can slide the entire wallpaper in some direction, \
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eventually mapping the pattern to itself.]
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- has translational symmetry in at least two directions
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#note([(and therefore fills the plane)])
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- has a countable number of reflection, rotation, or glide symmetries. \
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#v(1fr)
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#pagebreak()
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#problem()
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Is a plain square grid a valid wallpaper?
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#solution([
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Yes!
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- It has translational symmetry in the horizontal and vertical directions
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- It has a countable number of symmetries---namely, six distinct mirror lines (horizontal, vertical, and diagonal) duplicated once per square.
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- A square grid is #sym.convolve`442`
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])
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#v(1fr)
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#problem()
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Is the empty plane a valid wallpaper?
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#solution([
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No, since it has uncountably many symmetries.
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])
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#v(1fr)
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@ -4,62 +4,51 @@
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= Mirror Symmetry
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#definition()
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A _reflection_ is a transformation of the plane obtained by reflecting all points about a line. \
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If this reflection maps the wallpaper to itself, we have a _mirror symmetry_. \
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A _mirror symmetry_ is a reflection about a line. \
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If $n$ mirror symmetries intersect at a point, we say that point is an _$n$-fold mirror node_.
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If $n$ such mirror lines intersect at a point, they form a _mirror node of order $n$_. \
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#note[Mirror nodes with order 1 do not exist (i.e, $n >= 2$). A line does not intersect itself!]
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#v(3mm)
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#v(2mm)
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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.
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Two mirror nodes are identical if we can map one to the other with a translation and a rotation \
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while preserving the pattern on the wallpaper.
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#problem(label: "pat333")
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Find all three distinct mirror nodes in the following pattern. \
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What is the order of each node? \
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Find all three three distinct mirror nodes in the following pattern. \
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What is the order of each intersection? \
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#hint([
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You may notice rotational symmetry in this pattern. \
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Don't worry about that yet, we'll discuss it later.
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Don't worry about that for now.
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])
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#table(
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 45mm,
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rows: 50mm,
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image("../res/wolfram/p3m1.svg", height: 100%),
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)
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#solution([
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The mirror nodes are:
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- the center of the amber cross
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- the center of each right-handed group of three adjacent hexagons
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- the center of each left-handed group of three adjacent hexagons
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])
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#solution([This is `*333`])
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#v(1fr)
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#definition()
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_Orbifold notation_ gives us a way to describe the symmetries of a wallpaper. \
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It defines a _signature_ that fully describes all the symmetries of a given pattern. \
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_Orbifold notation_ gives us a way to write down the symmetries of a wallpaper. \
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We will introduce orbifold notation one symmetry at a time.
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#definition()
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In orbifold notation, mirror nodes are denoted by a #sym.convolve followed by a list of integer. \
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Every integer $n$ following a #sym.convolve denotes a mirror node of order $n$.
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In orbifold notation, mirror nodes are denoted by a `*` followed by at least one integer. \
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Every integer $n$ following a `*` denotes a mirror node of order $n$.
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#v(2mm)
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The order of these integers doesn't matter. #sym.convolve`234` and #sym.convolve`423` are the same signature. \
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However, we usually denote $n$-fold symmetries in descending order (that is, like #sym.convolve`432`). \
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If we have many nodes of the same order, integers may be repeated.
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#v(3mm)
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The order of these integers doesn't matter. `*234` and `*423` are the same signature. \
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However, we usually denote $n$-fold symmetries in descending order (that is, like `*432`).
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#problem()
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What is the signature of the wallpaper in @pat333? \
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#hint[Again, ignore rotational symmetry for now.]
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#solution([It is #sym.convolve`333`])
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What is the signature of the wallpaper in @pat333?
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#solution([It is `*333`])
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// MARK: page
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@ -73,13 +62,13 @@ Find the signature of the following pattern.
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 60mm,
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rows: 50mm,
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image("../res/*632-a.png", height: 100%),
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)
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#solution([
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It is #sym.convolve`632`:
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It is `*632`:
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#table(
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stroke: none,
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align: center,
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@ -92,7 +81,7 @@ Find the signature of the following pattern.
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#v(1fr)
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#problem()
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Draw a wallpaper pattern with signature #sym.convolve`2222`
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Draw a wallpaper pattern with signature `*2222`
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#solution([
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Sample solutions are below.
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@ -108,25 +97,3 @@ Draw a wallpaper pattern with signature #sym.convolve`2222`
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])
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#v(1fr)
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#pagebreak()
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#remark()
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In an exceptional case, we have two parallel mirror lines. \
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Consider the following pattern:
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#table(
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 60mm,
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image("../res/**.png", height: 100%),
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)
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The signature of this pattern is #sym.convolve#sym.convolve
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#problem()
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Draw another wallpaper pattern with signature #sym.convolve#sym.convolve.
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#v(1fr)
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@ -4,18 +4,16 @@
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= Rotational Symmetry
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#definition()
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A wallpaper may also have $n$-fold rotational symmetry about a point.
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#v(2mm)
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This means there are no more than $n$ rotations around that point that map the wallpaper to itself.
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#v(2mm)
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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.
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Another symmetry is n-fold rotational symmetry about a point, whose signature is written n. Multiple bold numbers means multiple points of rotational symmetry.
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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.
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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
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for by the mirror symmetries.
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By convention we write the rotational symmetries before
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the `*`.
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#definition()
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In orbifold notation, rotation is specified similarly to reflection, but uses the prefix #sym.diamond.stroked.small. \
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For example:
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- #sym.diamond.stroked.small`333` denotes a pattern with three distinct centers of rotation of order 3.
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- #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.
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#table(
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stroke: none,
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@ -27,155 +25,91 @@ For example:
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#problem()
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Find the three rotation centers in the left wallpaper. \
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What are their orders?
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#solution([This is #sym.diamond.stroked.small`333`])
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#v(1fr)
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Mark the three rotation points in Figure 1.
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#problem()
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Find the signature of the pattern on the right.
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Find the signature of the pattern in Figure 2.
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#solution([This is #sym.diamond.stroked.small`3`#sym.convolve`3`])
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#v(1fr)
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#remark()
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You may have noticed that we could have an ambiguous classification, since two reflections are equivalent to a translation and a rotation.
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We thus make the following distinction: _rotational symmetry that can be explained by reflection is not rotational symmetry._
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#v(2mm)
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In other words, when classifying a pattern...
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- we first find all mirror symmetries,
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- then all rotational symmetries that are not accounted for by reflection.
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#solution([`3 *3`])
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#pagebreak()
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// MARK: glide
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Some exceptional cases: It is possible to have two different parallel mirror lines. In
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this situation the signature is ∗ ∗
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= Glide Reflections
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#definition()
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Another type of symmetry is the _glide reflection_, denoted #sym.times.
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A glide reflection is the result of a translation along a line followed by reflection about that line.
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For example, consider the following pattern:
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#table(
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 60mm,
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image("../res/**.png", height: 100%),
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)
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#problem()
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Draw another wallpaper pattern with signature `**`
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#pagebreak()
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There are two other types of symmetries. The first called a miracle whose signature is
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written ×. It is the result of a glide reflection, which is translation along a line followed
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by reflection about that line.
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This occurs when there is orientation-reversing symmetry not accounted for by a mirror.
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For example, if we modify Figure 3 slightly we get a signature of ∗×
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#table(
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stroke: none,
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align: center,
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columns: (1fr, 1fr),
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rows: 60mm,
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image("../res/*x-b.png", height: 100%),
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image("../res/*x-a.png", height: 100%),
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)
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#problem()
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Convince yourself that all mirror lines in this pattern are _not_ distinct. /
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In other words, this pattern has only one mirror symmetry.
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Signature ∗×. There is a glide reflection (shown by the by the dotted line)
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taking the clockwise spiral to the counter-clockwise spiral, reversing orientation
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#solution([
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There may seem to be two, but they are identical. \
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We can translate one onto the other.
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])
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#v(1fr)
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#problem()
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Use the following picture to find the glide reflection in the above pattern.
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#table(
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 70mm,
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image("../res/*x-b.png", height: 100%),
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)
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#v(1fr)
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#remark()
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The signature of this wallpaper is #sym.convolve#sym.times.
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#pagebreak()
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#definition()
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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.
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#remark()
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In summary, to find the signature of a pattern:
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- find the mirror lines (#sym.convolve) and the distinct intersections;
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- then find the rotation centers (#sym.diamond.stroked.small) not explained by reflection;
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- then find all glide reflections (#sym.times) that do not cross a mirror line.
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- If we have none of the above, our pattern must be #sym.circle.small.
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#problem()
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Find the signature of the following pattern:
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Find the signatures of the following patterns:
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#table(
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 50mm,
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image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
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)
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#solution([
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This is #sym.convolve#sym.times.
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])
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#v(1fr)
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#problem()
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Find the signature of the following pattern:
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#table(
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stroke: none,
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align: center,
|
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columns: 1fr,
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columns: (1fr, 1fr),
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rows: 60mm,
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image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
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image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
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)
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#solution([
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This is #sym.diamond.stroked.small`4`#sym.convolve`2`
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])
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#v(1fr)
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#pagebreak()
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#problem()
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Find two glide reflections in the following pattern.\
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#note[(and thus show that its signature is #sym.times#sym.times.)]
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There is another exceptional case with two miracles, where there are two glide reflection
|
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symmetries along distinct lines. There are other glide reflections, but they can be obtained
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by composing the two marked in the diagram.
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|
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|
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#table(
|
||||
stroke: none,
|
||||
align: center,
|
||||
columns: 1fr,
|
||||
rows: 70mm,
|
||||
columns: (1fr, 1fr),
|
||||
rows: 60mm,
|
||||
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)
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||||
Figure 7: There are two distinct mirrorless crossings, so the signature is `xx`.
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Lastly, if none of the above symmetries appear in the pattern, then there is only regular
|
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translational symmetry, which we denote by O.
|
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|
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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
|
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|
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@ -1,67 +0,0 @@
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#import "@local/handout:0.1.0": *
|
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#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])
|
||||
@ -1,100 +0,0 @@
|
||||
#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)
|
||||
@ -1,7 +1,6 @@
|
||||
[default]
|
||||
extend-words."LSAT" = "LSAT"
|
||||
extend-words."ket" = "ket"
|
||||
extend-words."typ" = "typ"
|
||||
|
||||
extend-ignore-re = [
|
||||
# spell:disable-line
|
||||
|
||||
Reference in New Issue
Block a user