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21
src/Advanced/Wallpaper/main.typ
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#import "@local/handout:0.1.0": *
// Resources:
//
// https://eschermath.org/wiki/Wallpaper_Patterns.html
// https://mathworld.wolfram.com/WallpaperGroups.html
// https://en.wikipedia.org/wiki/Wallpaper_group
#show: handout.with(
title: [Wallpaper Symmetry],
title: [Wallpaper Symmetries],
by: "Mark",
)
#include "parts/00 intro.typ"
#include "parts/00 arithmetic.typ"
#pagebreak()
#include "parts/01 reflect.typ"
#pagebreak()
#include "parts/02 rotate.typ"
#pagebreak()
#include "parts/03 problems.typ"
#pagebreak()
#include "parts/04 theorem.typ"

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src/Advanced/Wallpaper/parts/00 arithmetic.typ Normal file
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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 = 0.75
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,
)
}
#box(
height: 10cm,
align(
center,
table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: (1fr, 1fr),
cetz.canvas({
import cetz.draw: *
demo(ored)
translate(x: 0, y: -2.5)
demo(oblue)
})
+ [Translation],
cetz.canvas({
import cetz.draw: *
circle((-2, 0), radius: 0.2, stroke: none, fill: black)
demo(ored)
rotate(z: -45deg, origin: (-2, 0))
demo(oblue)
})
+ [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)
})
+ [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: 2.5, y: 0)
demo(oblue)
set-transform(none)
line((-1, 0), (5, 0))
})
+ [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 countably many reflection, rotation, or glide symmetries. \
#problem()
Is the empty plane a valid wallpaper?
#solution([
No, since it has an uncountable number of symmetries.
])
#pagebreak()
#definition()
A _mirror symmetry_ is a reflection about a line.
Its signature is `*`.
An integer $n$ following `*` denotes $n$-fold mirror symmetry, the intersection of $n$ mirror lines.
Two intersections of mirror lines are considered the same if we can perform a translation and rotation that sends one to the other, while leaving the pattern the same. There are various possible combinations of mirror symmetries.
This flower pattern has signature `*632`: there are
three distinct point of intersecting mirror lines with 6, 3, and 2 mirror lines respectively.
#table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: 50mm,
image("../res/*632-a.png", height: 100%),
image("../res/*632-b.png", height: 100%),
)
#problem()
Design a wallpaper pattern with signature `*2222`
#pagebreak()
Another symmetry is n-fold rotational symmetry about a point, whose signature is written n. Multiple bold numbers means multiple points of rotational symmetry.
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.
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
for by the mirror symmetries.
By convention we write the rotational symmetries before
the `*`.
#table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: 50mm,
image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
)
#problem()
Mark the three rotation points in Figure 1.
#problem()
Find the signature of the pattern in Figure 2.
#solution([`3 *3`])
#pagebreak()
Some exceptional cases: It is possible to have two different parallel mirror lines. In
this situation the signature is ∗ ∗
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 60mm,
image("../res/**.png", height: 100%),
)
#problem()
Draw another wallpaper pattern with signature `**`
#pagebreak()
There are two other types of symmetries. The first called a miracle whose signature is
written ×. It is the result of a glide reflection, which is translation along a line followed
by reflection about that line.
This occurs when there is orientation-reversing symmetry not accounted for by a mirror.
For example, if we modify Figure 3 slightly we get a signature of ∗×
#table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: 60mm,
image("../res/*x-b.png", height: 100%),
image("../res/*x-a.png", height: 100%),
)
Signature ∗×. There is a glide reflection (shown by the by the dotted line)
taking the clockwise spiral to the counter-clockwise spiral, reversing orientation
#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

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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. \
Intuitively, an isometry moves objects on the plane without deforming them.
There are four classes of Euclidean isometries:
- translations
- reflections
- rotations
- glide reflections
#note([We can prove 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: -1.0, y: -1.0)
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: 2, 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 non-parallel directions (and therefore fills the plane) \
#note[
"Translational symmetry" means that we can slide the entire wallpaper in some direction, \
eventually mapping the pattern to itself.]
- has a countable number of reflection, rotation, or glide symmetries. \
#v(1fr)
#pagebreak()
#problem()
Is a plain square grid a valid wallpaper?
#solution([
Yes!
- It has translational symmetry in the horizontal and vertical directions
- It has a countable number of symmetries---namely, six distinct mirror lines (horizontal, vertical, and diagonal) duplicated once per square.
- A square grid is #sym.convolve`442`
])
#v(1fr)
#problem()
Is the empty plane a valid wallpaper?
#solution([
No, since it has uncountably many symmetries.
])
#v(1fr)

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src/Advanced/Wallpaper/parts/01 reflect.typ
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#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.3.1"
= Mirror Symmetry
#definition()
A _reflection_ is a transformation of the plane obtained by reflecting all points about a line. \
If this reflection maps the wallpaper to itself, we have a _mirror symmetry_. \
If $n$ such mirror lines intersect at a point, they form a _mirror node of order $n$_. \
#note[Mirror nodes with order 1 do not exist (i.e, $n >= 2$). A line does not intersect itself!]
#v(2mm)
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.
#problem(label: "pat333")
Find all three distinct mirror nodes in the following pattern. \
What is the order of each node? \
#hint([
You may notice rotational symmetry in this pattern. \
Don't worry about that yet, we'll discuss it later.
])
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 45mm,
image("../res/wolfram/p3m1.svg", height: 100%),
)
#solution([
The mirror nodes are:
- the center of the amber cross
- the center of each right-handed group of three adjacent hexagons
- the center of each left-handed group of three adjacent hexagons
])
#v(1fr)
#definition()
_Orbifold notation_ gives us a way to describe the symmetries of a wallpaper. \
It defines a _signature_ that fully describes all the symmetries of a given pattern. \
We will introduce orbifold notation one symmetry at a time.
#definition()
In orbifold notation, mirror nodes are denoted by a #sym.convolve followed by a list of integer. \
Every integer $n$ following a #sym.convolve denotes a mirror node of order $n$.
#v(2mm)
The order of these integers doesn't matter. #sym.convolve`234` and #sym.convolve`423` are the same signature. \
However, we usually denote $n$-fold symmetries in descending order (that is, like #sym.convolve`432`). \
If we have many nodes of the same order, integers may be repeated.
#problem()
What is the signature of the wallpaper in @pat333? \
#hint[Again, ignore rotational symmetry for now.]
#solution([It is #sym.convolve`333`])
// MARK: page
#v(1fr)
#pagebreak()
#problem()
Find the signature of the following pattern.
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 60mm,
image("../res/*632-a.png", height: 100%),
)
#solution([
It is #sym.convolve`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 #sym.convolve`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)
#pagebreak()
#remark()
In an exceptional case, we have two parallel mirror lines. \
Consider the following pattern:
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 60mm,
image("../res/**.png", height: 100%),
)
The signature of this pattern is #sym.convolve#sym.convolve
#problem()
Draw another wallpaper pattern with signature #sym.convolve#sym.convolve.
#v(1fr)

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src/Advanced/Wallpaper/parts/02 rotate.typ
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#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.3.1"
= Rotational Symmetry
#definition()
A wallpaper may also have $n$-fold rotational symmetry about a point.
#v(2mm)
This means there are no more than $n$ rotations around that point that map the wallpaper to itself.
#v(2mm)
As before, two points of rotational symmetry are identical if we can perform a translation and rotation that maps one to the other without changing the wallpaper.
#definition()
In orbifold notation, rotation is specified similarly to reflection, but uses the prefix #sym.diamond.stroked.small. \
For example:
- #sym.diamond.stroked.small`333` denotes a pattern with three distinct centers of rotation of order 3.
- #sym.diamond.stroked.small`4`#sym.convolve`2` denotes a pattern with one rotation center of order 4 and one mirror node of order 2.
#table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: 50mm,
image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
)
#problem()
Find the three rotation centers in the left wallpaper. \
What are their orders?
#solution([This is #sym.diamond.stroked.small`333`])
#v(1fr)
#problem()
Find the signature of the pattern on the right.
#solution([This is #sym.diamond.stroked.small`3`#sym.convolve`3`])
#v(1fr)
#remark()
You may have noticed that we could have an ambiguous classification, since two reflections are equivalent to a translation and a rotation.
We thus make the following distinction: _rotational symmetry that can be explained by reflection is not rotational symmetry._
#v(2mm)
In other words, when classifying a pattern...
- we first find all mirror symmetries,
- then all rotational symmetries that are not accounted for by reflection.
#pagebreak()
// MARK: glide
= Glide Reflections
#definition()
Another type of symmetry is the _glide reflection_, denoted #sym.times.
A glide reflection is the result of a translation along a line followed by reflection about that line.
For example, consider the following pattern:
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 60mm,
image("../res/*x-a.png", height: 100%),
)
#problem()
Convince yourself that all mirror lines in this pattern are _not_ distinct. /
In other words, this pattern has only one mirror symmetry.
#solution([
There may seem to be two, but they are identical. \
We can translate one onto the other.
])
#v(1fr)
#problem()
Use the following picture to find the glide reflection in the above pattern.
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 70mm,
image("../res/*x-b.png", height: 100%),
)
#v(1fr)
#remark()
The signature of this wallpaper is #sym.convolve#sym.times.
#pagebreak()
#definition()
If none of the above symmetries appear in a pattern, then we only have simple translational symmetry. We denote this with the signature #sym.circle.small.
#remark()
In summary, to find the signature of a pattern:
- find the mirror lines (#sym.convolve) and the distinct intersections;
- then find the rotation centers (#sym.diamond.stroked.small) not explained by reflection;
- then find all glide reflections (#sym.times) that do not cross a mirror line.
- If we have none of the above, our pattern must be #sym.circle.small.
#problem()
Find the signature of the following pattern:
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 50mm,
image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
)
#solution([
This is #sym.convolve#sym.times.
])
#v(1fr)
#problem()
Find the signature of the following pattern:
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 60mm,
image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
)
#solution([
This is #sym.diamond.stroked.small`4`#sym.convolve`2`
])
#v(1fr)
#pagebreak()
#problem()
Find two glide reflections in the following pattern.\
#note[(and thus show that its signature is #sym.times#sym.times.)]
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 70mm,
image("../res/xx-b.png", height: 100%),
)
#solution([
#table(
stroke: none,
align: center,
columns: 1fr,
rows: 40mm,
image("../res/xx-a.png", height: 100%),
)
])
#v(1fr)

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src/Advanced/Wallpaper/parts/03 problems.typ
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#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.3.1"
#let pat(img, sol) = {
problem()
table(
stroke: none,
align: center,
columns: (1fr, 1fr),
rows: 80mm,
image(img, height: 100%), image(img, height: 100%),
)
solution(sol)
v(1fr)
}
= A few problems
Find the signatures of the following patterns. Mark all mirror nodes, rotation centers, and glide reflections. \
Each pattern is provided twice for convenience.
#pat("../res/wolfram/cm.svg", [#sym.times#sym.convolve])
#pat("../res/wolfram/cmm.svg", [#sym.diamond.stroked`2`#sym.convolve`22`])
#pagebreak()
#pat("../res/wolfram/p3.svg", [#sym.diamond.stroked`333`])
#pat("../res/wolfram/p3m1.svg", [#sym.convolve`333`])
#pagebreak()
#pat("../res/wolfram/p4.svg", [#sym.diamond.stroked`442`])
#pat("../res/wolfram/p4m.svg", [#sym.convolve`442`])
#pagebreak()
#pat("../res/wolfram/p6.svg", [#sym.diamond.stroked`632`])
#pat("../res/wolfram/p6m.svg", [#sym.convolve`632`])
#pagebreak()
#pat("../res/wolfram/p4g.svg", [#sym.diamond.stroked`4`#sym.convolve`2`])
#pat("../res/wolfram/p31m.svg", [#sym.diamond.stroked`3`#sym.convolve`3`])
#pagebreak()
#problem()
Draw a wallpaper with the signature #sym.convolve`442` \
#note[Make sure there are no other symmetries!]
#v(1fr)
#pagebreak()
#pat("../res/wolfram/pgg.svg", [#sym.diamond.stroked`22`#sym.times])
#pat("../res/wolfram/pmg.svg", [#sym.diamond.stroked`22`#sym.convolve])
#pagebreak()
#pat("../res/wolfram/pg.svg", [#sym.times#sym.times])
#pat("../res/wolfram/pm.svg", [#sym.convolve#sym.convolve])
#pagebreak()
#pat("../res/wolfram/p2.svg", [#sym.diamond.stroked`2222`])
#pat("../res/wolfram/pmm.svg", [#sym.convolve`2222`])
#pagebreak()
#pat("../res/wolfram/p1.svg", [#sym.circle.small])

100
src/Advanced/Wallpaper/parts/04 theorem.typ
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#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.3.1"
= The Signature-Cost Theorem
#definition()
First, we'll associate a _cost_ to each type of symmetry in orbifold notation:
#v(4mm)
#align(
center,
table(
stroke: (1pt, 1pt),
align: center,
columns: (auto, auto, auto, auto),
[*Symbol*], [*Cost*], [*Symbol*], [*Cost*],
[#sym.circle.small], [2], [#sym.times or #sym.convolve], [1],
[#sym.diamond.stroked.small`2`], [1/2], [#sym.convolve`2`], [1/4],
[#sym.diamond.stroked.small`3`], [2/3], [#sym.convolve`3`], [1/3],
[#sym.dots], [#sym.dots], [#sym.dots], [#sym.dots],
[#sym.diamond.stroked.small`n`],
[$(n-1) / n$],
[#sym.convolve`n`],
[$(n-1) / (2n)$],
),
)
We then calculate the total "cost" of a signature by adding up the costs of each component.
For example, a pattern with signature #sym.convolve`333` has cost 2:
#v(2mm)
$
2 / 3 + 2 / 3 + 2 / 3 = 2
$
#problem()
Calculate the costs of the following signatures:
- #sym.diamond.stroked.small`3`#sym.convolve`3`
- #sym.convolve#sym.convolve
- #sym.diamond.stroked.small`4`#sym.convolve`2`:
#solution([
- #sym.diamond.stroked.small`3`#sym.convolve`3`: $2/3 + 1 + 1/3 = 2$
- #sym.convolve#sym.convolve: $1 + 1 = 2$
- #sym.diamond.stroked.small`4`#sym.convolve`2`: $3/4 + 1 + 1/4 = 2$
])
#v(1fr)
#theorem(name: "Signature Cost Theorem")
The signatures of planar wallpaper patterns are exactly those with total cost 2. \
#note([We will not prove this theorem today, accept it without proof.])
#problem()
Consider the 4 symmetries (translation, reflection, rotation, and glide reflection). \
Which preserve orientation? Which reverse orientation?
#solution([
- Reflections and glide reflections reverse orientation (directions of spirals).
- Translation and rotation preserve orientation.
])
#v(1fr)
#pagebreak()
#problem()
Use the signature-cost theorem to find all the signatures consisting of only #sym.circle.small or rotational symmetries.
#solution([
#sym.diamond.stroked.small`632`, #sym.diamond.stroked.small`442`, #sym.diamond.stroked.small`333`, #sym.diamond.stroked.small`2222`, #sym.circle.small
])
#v(1fr)
#problem()
Find all the signatures consisting of only mirror symmetries.
#solution([
#sym.convolve`632`, #sym.convolve`442`, #sym.convolve`333`, #sym.convolve`2222`, #sym.convolve#sym.convolve
])
#v(1fr)
#problem()
Find all the remaining signatures. \
Each must be a mix of of mirror symmetries, rotational symmetries, or glide reflections. \
#hint([They are all shown in the problems section.])
#solution([
#sym.diamond.stroked.small`3`#sym.convolve`3`, #sym.diamond.stroked.small`4`#sym.convolve`2`,
#sym.diamond.stroked.small`22`#sym.times, #sym.diamond.stroked.small`22`#sym.convolve,
#sym.times#sym.times, #sym.times#sym.convolve
])
#v(1fr)

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@ -1,86 +0,0 @@
<?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>
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1
tools/typos.toml
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@ -1,7 +1,6 @@
[default]
extend-words."LSAT" = "LSAT"
extend-words."ket" = "ket"
extend-words."typ" = "typ"
extend-ignore-re = [
# spell:disable-line