133 lines
3.1 KiB
Typst
133 lines
3.1 KiB
Typst
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.4.2"
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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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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(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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#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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#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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])
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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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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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#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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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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#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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#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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// MARK: page
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#v(1fr)
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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/*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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#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/*632-b.png", height: 100%),
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)
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])
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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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#solution([
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Sample solutions are below.
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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/wolfram/pmm.svg", height: 100%),
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image("../res/escher/pmm.svg", height: 100%),
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)
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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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