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251d9bb9e5
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6e3c665e99 | |||
28aaf98594 |
15
src/Advanced/Wallpaper/main.typ
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#import "@local/handout:0.1.0": *
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#show: handout.with(
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title: [Wallpaper Symmetry],
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by: "Mark",
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)
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#include "parts/00 intro.typ"
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#pagebreak()
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#include "parts/01 reflect.typ"
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#pagebreak()
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#include "parts/02 rotate.typ"
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#pagebreak()
|
7
src/Advanced/Wallpaper/meta.toml
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[metadata]
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title = "Wallpaper Symmetries"
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[publish]
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handout = true
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solutions = true
|
131
src/Advanced/Wallpaper/parts/00 intro.typ
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#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.3.1"
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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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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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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: 0, y: -1.5)
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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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|
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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: 1.5, 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 directions
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#note([(and therefore fills the plane)])
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- has a countable number of reflection, rotation, or glide symmetries. \
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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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])
|
99
src/Advanced/Wallpaper/parts/01 reflect.typ
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@ -0,0 +1,99 @@
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#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.3.1"
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= Mirror Symmetry
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#definition()
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A _mirror symmetry_ is a reflection about a line. \
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If $n$ mirror symmetries intersect at a point, we say that point is an _$n$-fold mirror node_.
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#v(3mm)
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Two mirror nodes are identical if we can map one to the other with a translation and a rotation \
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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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#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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])
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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/wolfram/p3m1.svg", height: 100%),
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)
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#solution([This is `*333`])
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#v(1fr)
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#definition()
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_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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#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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#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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// 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: 50mm,
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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 `*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 `*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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|
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#v(1fr)
|
115
src/Advanced/Wallpaper/parts/02 rotate.typ
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#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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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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|
||||
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#table(
|
||||
stroke: none,
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||||
align: center,
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||||
columns: (1fr, 1fr),
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rows: 50mm,
|
||||
image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
|
||||
)
|
||||
|
||||
|
||||
#problem()
|
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Mark the three rotation points in Figure 1.
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#problem()
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Find the signature of the pattern in Figure 2.
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#solution([`3 *3`])
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#pagebreak()
|
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|
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|
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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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|
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#table(
|
||||
stroke: none,
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||||
align: center,
|
||||
columns: 1fr,
|
||||
rows: 60mm,
|
||||
image("../res/**.png", height: 100%),
|
||||
)
|
||||
|
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#problem()
|
||||
Draw another wallpaper pattern with signature `**`
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||||
|
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|
||||
#pagebreak()
|
||||
|
||||
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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||||
|
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#table(
|
||||
stroke: none,
|
||||
align: center,
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||||
columns: (1fr, 1fr),
|
||||
rows: 60mm,
|
||||
image("../res/*x-b.png", height: 100%),
|
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image("../res/*x-a.png", height: 100%),
|
||||
)
|
||||
|
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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()
|
||||
|
||||
|
||||
#problem()
|
||||
Find the signatures of the following patterns:
|
||||
|
||||
#table(
|
||||
stroke: none,
|
||||
align: center,
|
||||
columns: (1fr, 1fr),
|
||||
rows: 60mm,
|
||||
image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
|
||||
image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
|
||||
)
|
||||
|
||||
|
||||
#pagebreak()
|
||||
|
||||
There is another exceptional case with two miracles, where there are two glide reflection
|
||||
symmetries along distinct lines. There are other glide reflections, but they can be obtained
|
||||
by composing the two marked in the diagram.
|
||||
|
||||
|
||||
#table(
|
||||
stroke: none,
|
||||
align: center,
|
||||
columns: (1fr, 1fr),
|
||||
rows: 60mm,
|
||||
image("../res/xx-b.png", height: 100%),
|
||||
image("../res/xx-a.png", height: 100%),
|
||||
)
|
||||
|
||||
Figure 7: There are two distinct mirrorless crossings, so the signature is `xx`.
|
||||
Lastly, if none of the above symmetries appear in the pattern, then there is only regular
|
||||
translational symmetry, which we denote by O.
|
||||
|
||||
In summary, to find the signature of a pattern:
|
||||
- Find the mirror lines (∗) and the distinct intersections
|
||||
- Find the rotational points of symmetry not account for by reflections.
|
||||
- Look for any miracles (×) i.e. glide reflections that do not cross a mirror line.
|
||||
- If you found none of the above, it is just O
|
BIN
src/Advanced/Wallpaper/res/**.png
Normal file
After Width: | Height: | Size: 535 KiB |
BIN
src/Advanced/Wallpaper/res/*632-a.png
Normal file
After Width: | Height: | Size: 484 KiB |
BIN
src/Advanced/Wallpaper/res/*632-b.png
Normal file
After Width: | Height: | Size: 280 KiB |
BIN
src/Advanced/Wallpaper/res/*x-a.png
Normal file
After Width: | Height: | Size: 666 KiB |
BIN
src/Advanced/Wallpaper/res/*x-b.png
Normal file
After Width: | Height: | Size: 160 KiB |
BIN
src/Advanced/Wallpaper/res/3*3.png
Normal file
After Width: | Height: | Size: 533 KiB |
BIN
src/Advanced/Wallpaper/res/333.png
Normal file
After Width: | Height: | Size: 568 KiB |
86
src/Advanced/Wallpaper/res/escher/pmm.svg
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@ -0,0 +1,86 @@
|
||||
<?xml version="1.0" standalone="no"?>
|
||||
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20001102//EN" "http://www.w3.org/TR/2000/CR-SVG-20001102/DTD/svg-20001102.dtd">
|
||||
<svg xmlns:svg="http://www.w3.org/2000/svg" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10.16cm" height="5.08cm" viewBox="0 0 384 192">
|
||||
<title>pmm</title>
|
||||
<desc>Exported by Tess 1.70.</desc>
|
||||
<clipPath id="mainclip"><rect x="0" y="0" width="384" height="192"/></clipPath>
|
||||
<g style="fill-rule:evenodd; stroke-linejoin:round; stroke-linecap:round; clip-path:url(#mainclip)">
|
||||
<defs>
|
||||
<g id="Tess0p">
|
||||
<path d="
|
||||
M7.1306642022,-41.0013191628
|
||||
L7.1306642022,-7.1306642022
|
||||
L89.1333025278,-7.1306642022
|
||||
L7.1306642022,-42.7839852134
|
||||
"/></g>
|
||||
<g id="Tess0" style="fill:none; stroke:none"><use xlink:href="#Tess0p"/></g>
|
||||
<g id="Tess1" style="fill:none; stroke:rgb(0,0,0); stroke-width:1.7826660506"><use xlink:href="#Tess0p"/></g>
|
||||
</defs>
|
||||
<g transform="translate(192,96) rotate(0) scale(0.5609575611,0.5609575611)">
|
||||
<g transform="translate(-405.2631578947,-100)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(-405.2631578947,0)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-405.2631578947,100)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(-405.2631578947,200)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-202.6315789474,-100)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-202.6315789474,0)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(-202.6315789474,100)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(-202.6315789474,200)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,-100)"><use xlink:href="#Tess1"/></g>
|
||||
<g><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,100)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,200)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(202.6315789474,-100)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(202.6315789474,0)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(202.6315789474,100)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(202.6315789474,200)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(-202.6315789474,-200) rotate(180)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(-202.6315789474,0) rotate(180)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-202.6315789474,100) rotate(180)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(0,-200) rotate(180)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,-100) rotate(180)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="rotate(180)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,100) rotate(180)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(202.6315789474,-200) rotate(180)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(202.6315789474,-100) rotate(180)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(405.2631578947,-200) rotate(180)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(405.2631578947,-100) rotate(180)"><use xlink:href="#Tess1"/></g>
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<g transform="translate(405.2631578947,100) rotate(180)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-202.6315789474,-100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(-202.6315789474,0) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-202.6315789474,100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(-202.6315789474,200) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,-100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(0,200) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
|
||||
<g transform="translate(202.6315789474,-100) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
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||||
<g transform="translate(202.6315789474,0) rotate(180) scale(1,-1)"><use xlink:href="#Tess1"/></g>
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src/Advanced/Wallpaper/res/wolfram/p3.svg
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src/Advanced/Wallpaper/res/wolfram/p31m.svg
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src/Advanced/Wallpaper/res/wolfram/pm.svg
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src/Advanced/Wallpaper/res/wolfram/pmg.svg
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src/Advanced/Wallpaper/res/wolfram/pmm.svg
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