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@ -8,11 +8,11 @@ A _Euclidean isometry_ is a transformation of the plane that preserves distances
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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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- 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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- 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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A simple example of each isometry is shown below:
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#let demo(c) = {
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@ -14,10 +14,10 @@ while preserving the pattern on the wallpaper.
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#problem(label: "pat333")
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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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What is the order of each node? \
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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 for now.
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Don't worry about that yet, we'll discuss it later.
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])
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#table(
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@ -28,7 +28,12 @@ What is the order of each intersection? \
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image("../res/wolfram/p3m1.svg", height: 100%),
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)
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#solution([This is `*333`])
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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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#v(1fr)
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@ -38,17 +43,18 @@ _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 `*` 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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In orbifold notation, mirror nodes are denoted by a #sym.convolve followed by at least one integer. \
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Every integer $n$ following a #sym.convolve denotes a mirror node of order $n$.
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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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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 are repeated.
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#problem()
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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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#solution([It is #sym.convolve`333`])
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// MARK: page
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@ -68,7 +74,7 @@ Find the signature of the following pattern.
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#solution([
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It is `*632`:
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It is #sym.convolve`632`:
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#table(
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stroke: none,
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align: center,
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@ -81,7 +87,7 @@ 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 `*2222`
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Draw a wallpaper pattern with signature #sym.convolve`2222`
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#solution([
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Sample solutions are below.
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@ -97,3 +103,25 @@ Draw a wallpaper pattern with signature `*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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@ -4,16 +4,18 @@
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= Rotational Symmetry
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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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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 `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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#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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@ -25,91 +27,153 @@ the `*`.
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#problem()
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Mark the three rotation points in Figure 1.
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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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#problem()
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Find the signature of the pattern in Figure 2.
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Find the signature of the pattern on the right.
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#solution([`3 *3`])
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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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#pagebreak()
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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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// MARK: glide
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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/*x-a.png", height: 100%),
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)
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#problem()
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Convince yourself that this pattern has only one mirror symmetry.
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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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#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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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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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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#pagebreak()
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#problem()
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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, 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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)
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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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rows: 60mm,
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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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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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#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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#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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columns: 1fr,
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rows: 70mm,
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image("../res/xx-b.png", height: 100%),
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image("../res/xx-a.png", height: 100%),
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)
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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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#solution([
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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: 40mm,
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image("../res/xx-a.png", height: 100%),
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)
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])
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#v(1fr)
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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 (∗) and the distinct intersections
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- Find the rotational points of symmetry not account for by reflections.
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- Look for any miracles (×) i.e. glide reflections that do not cross a mirror line.
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- If you found none of the above, it is just O
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- find the mirror lines (#sym.convolve) and the distinct intersections;
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- 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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