182 lines
4.1 KiB
Typst
182 lines
4.1 KiB
Typst
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.3.1"
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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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#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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align: center,
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columns: (1fr, 1fr),
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rows: 50mm,
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image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
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)
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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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#problem()
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Find the signature of the pattern on the right.
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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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// 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 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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#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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#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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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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#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,
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rows: 70mm,
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image("../res/xx-b.png", height: 100%),
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)
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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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