101 lines
2.8 KiB
Typst
101 lines
2.8 KiB
Typst
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.3.1"
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= The Signature-Cost Theorem
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#definition()
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First, we'll associate a _cost_ to each type of symmetry in orbifold notation:
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#v(4mm)
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#align(
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center,
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table(
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stroke: (1pt, 1pt),
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align: center,
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columns: (auto, auto, auto, auto),
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[*Symbol*], [*Cost*], [*Symbol*], [*Cost*],
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[#sym.circle.small], [2], [#sym.times or #sym.convolve], [1],
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[#sym.diamond.stroked.small`2`], [1/2], [#sym.convolve`2`], [1/4],
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[#sym.diamond.stroked.small`3`], [2/3], [#sym.convolve`3`], [1/3],
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[#sym.dots], [#sym.dots], [#sym.dots], [#sym.dots],
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[#sym.diamond.stroked.small`n`],
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[$(n-1) / n$],
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[#sym.convolve`n`],
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[$(n-1) / (2n)$],
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),
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)
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We then calculate the total "cost" of a signature by adding up the costs of each component.
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For example, a pattern with signature #sym.convolve`333` has cost 2:
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#v(2mm)
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$
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2 / 3 + 2 / 3 + 2 / 3 = 2
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$
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#problem()
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Calculate the costs of the following signatures:
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- #sym.diamond.stroked.small`3`#sym.convolve`3`
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- #sym.convolve#sym.convolve
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- #sym.diamond.stroked.small`4`#sym.convolve`2`:
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#solution([
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- #sym.diamond.stroked.small`3`#sym.convolve`3`: $2/3 + 1 + 1/3 = 2$
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- #sym.convolve#sym.convolve: $1 + 1 = 2$
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- #sym.diamond.stroked.small`4`#sym.convolve`2`: $3/4 + 1 + 1/4 = 2$
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])
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#v(1fr)
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#theorem(name: "Signature Cost Theorem")
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The signatures of planar wallpaper patterns are exactly those with total cost 2. \
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#note([We will not prove this theorem today, accept it without proof.])
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#problem()
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Consider the 4 symmetries (translation, reflection, rotation, and glide reflection). \
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Which preserve orientation? Which reverse orientation?
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#solution([
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- Reflections and glide reflections reverse orientation (directions of spirals).
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- Translation and rotation preserve orientation.
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])
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#v(1fr)
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#pagebreak()
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#problem()
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Use the signature-cost theorem to find all the signatures consisting of only #sym.circle.small or rotational symmetries.
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#solution([
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#sym.diamond.stroked.small`632`, #sym.diamond.stroked.small`442`, #sym.diamond.stroked.small`333`, #sym.diamond.stroked.small`2222`, #sym.circle.small
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])
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#v(1fr)
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#problem()
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Find all the signatures consisting of only mirror symmetries.
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#solution([
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#sym.convolve`632`, #sym.convolve`442`, #sym.convolve`333`, #sym.convolve`2222`, #sym.convolve#sym.convolve
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])
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#v(1fr)
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#problem()
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Find all the remaining signatures. \
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Each must be a mix of of mirror symmetries, rotational symmetries, or glide reflections. \
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#hint([They are all shown in the problems section.])
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#solution([
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#sym.diamond.stroked.small`3`#sym.convolve`3`, #sym.diamond.stroked.small`4`#sym.convolve`2`,
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#sym.diamond.stroked.small`22`#sym.times, #sym.diamond.stroked.small`22`#sym.convolve,
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#sym.times#sym.times, #sym.times#sym.convolve
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])
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#v(1fr)
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