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154 lines
2.9 KiB
Typst
154 lines
2.9 KiB
Typst
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.4.2"
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= Wallpaper Symmetries
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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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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(
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[We can prove there are no others, but this is beyond the scope of this handout.],
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) \
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A simple example of each isometry is shown below:
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#let demo(c) = {
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let s = 0.5
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cetz.draw.line(
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(0, 0),
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(3 * s, 0),
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(3 * s, 1 * s),
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(1 * s, 1 * s),
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(1 * s, 2 * s),
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(0, 2 * s),
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close: true,
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fill: c,
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stroke: black + 0mm * s,
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)
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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, 1fr),
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rows: (3.5cm, 3.5cm),
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row-gutter: 2mm,
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[
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#cetz.canvas({
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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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demo(oblue)
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})
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#v(1fr)
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Translation
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],
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[
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#cetz.canvas({
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import cetz.draw: *
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circle((-2, 0), radius: 0.1, stroke: none, fill: black)
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arc(
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(-2, 0),
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radius: 1,
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anchor: "origin",
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start: 0deg,
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stop: -30deg,
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mode: "PIE",
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)
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demo(ored)
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rotate(z: -30deg, origin: (-2, 0))
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demo(oblue)
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})
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#v(1fr)
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Rotation
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],
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[
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#cetz.canvas({
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import cetz.draw: *
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line((-2, 0), (4, 0))
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translate(x: 0, y: 0.25)
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demo(ored)
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set-transform(none)
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set-transform((
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(1, 0, 0, 0),
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(0, 1, 0, 0),
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(0, 0, 1, 0),
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(0, 0, 0, 1),
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))
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translate(x: 0, y: 0.25)
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demo(oblue)
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})
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#v(1fr)
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Reflection
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],
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[
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#cetz.canvas({
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import cetz.draw: *
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demo(ored)
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set-transform((
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(1, 0, 0, 0),
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(0, 1, 0, 0),
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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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demo(oblue)
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set-transform(none)
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line((-1, 0), (5, 0))
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})
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#v(1fr)
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Glide reflection
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],
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)
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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 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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