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src/Advanced/Wallpaper/main.typ Normal file
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#import "@local/handout:0.1.0": *
#show: handout.with(
title: [Wallpaper Symmetry],
by: "Mark",
)
#include "parts/00 intro.typ"
#pagebreak()
#include "parts/01 reflect.typ"
#pagebreak()

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[metadata]
title = "Wallpaper Symmetries"
[publish]
handout = true
solutions = true

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#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.3.1"
= Wallpaper Symmetries
#definition()
A _Euclidean isometry_ is a transformation of the plane that preserves distances. \
Intuituvely, an isometry moves objects on the plane without deforming them.
There are four classes of _Euclidean isometries_:
- Translation
- Reflection
- Rotation
- Glide reflection
#note([We can prove that there are no others, but this is beyond the scope of this handout.]) \
A simple example of each isometry is shown below:
#let demo(c) = {
let s = 0.5
cetz.draw.line(
(0, 0),
(3 * s, 0),
(3 * s, 1 * s),
(1 * s, 1 * s),
(1 * s, 2 * s),
(0, 2 * s),
close: true,
fill: c,
stroke: black + 0mm * s,
)
}
#table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: (3.5cm, 3.5cm),
row-gutter: 2mm,
[
#cetz.canvas({
import cetz.draw: *
demo(ored)
translate(x: 0, y: -1.5)
demo(oblue)
})
#v(1fr)
Translation
],
[
#cetz.canvas({
import cetz.draw: *
circle((-2, 0), radius: 0.1, stroke: none, fill: black)
arc(
(-2, 0),
radius: 1,
anchor: "origin",
start: 0deg,
stop: -30deg,
mode: "PIE",
)
demo(ored)
rotate(z: -30deg, origin: (-2, 0))
demo(oblue)
})
#v(1fr)
Rotation
],
[
#cetz.canvas({
import cetz.draw: *
line((-2, 0), (4, 0))
translate(x: 0, y: 0.25)
demo(ored)
set-transform(none)
set-transform((
(1, 0, 0, 0),
(0, 1, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1),
))
translate(x: 0, y: 0.25)
demo(oblue)
})
#v(1fr)
Reflection
],
[
#cetz.canvas({
import cetz.draw: *
demo(ored)
set-transform((
(1, 0, 0, 0),
(0, 1, 0, 0),
(0, 0, 0, 0),
(0, 0, 0, 0),
))
translate(x: 1.5, y: 0)
demo(oblue)
set-transform(none)
line((-1, 0), (5, 0))
})
#v(1fr)
Glide reflection
],
)
#definition()
A _wallpaper_ is a two-dimensional pattern that...
- has translational symmetry in at least two directions
#note([(and therefore fills the plane)])
- has a countable number of reflection, rotation, or glide symmetries. \
#problem()
Is the empty plane a valid wallpaper?
#solution([
No, since it has uncountably many symmetries.
])

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#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.3.1"
= Mirror Symmetry
#definition()
A _mirror symmetry_ is a reflection about a line. \
If $n$ mirror symmetries intersect at a point, we say that point is an _$n$-fold mirror node_.
#v(3mm)
Two mirror nodes are identical if we can map one to the other with a translation and a rotation \
while preserving the pattern on the wallpaper.
#problem(label: "pat333")
Find all three three distinct mirror nodes in the following pattern. \
What is the order of each intersection? \
#hint([
You may notice rotational symmetry in this pattern. \
Don't worry about that for now.
])
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 50mm,
image("../res/wolfram/p3m1.svg", height: 100%),
)
#solution([This is `*333`])
#v(1fr)
#definition()
_Orbifold notation_ gives us a way to write down the symmetries of a wallpaper. \
We will introduce orbifold notation one symmetry at a time.
#definition()
In orbifold notation, mirror nodes are denoted by a `*` followed by at least one integer. \
Every integer $n$ following a `*` denotes a mirror node of order $n$.
#v(3mm)
The order of these integers doesn't matter. `*234` and `*423` are the same signature. \
However, we usually denote $n$-fold symmetries in descending order (that is, like `*432`).
#problem()
What is the signature of the wallpaper in @pat333?
#solution([It is `*333`])
// MARK: page
#v(1fr)
#pagebreak()
#problem()
Find the signature of the following pattern.
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 50mm,
image("../res/*632-a.png", height: 100%),
)
#solution([
It is `*632`:
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 40mm,
image("../res/*632-b.png", height: 100%),
)
])
#v(1fr)
#problem()
Draw a wallpaper pattern with signature `*2222`
#solution([
Sample solutions are below.
#table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: 50mm,
image("../res/wolfram/pmm.svg", height: 100%),
image("../res/escher/pmm.svg", height: 100%),
)
])
#v(1fr)

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