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2025-02-26 08:46:48 -08:00
441c458d02 Added "Gods, Demons, and Mortals" 2025-02-26 08:05:49 -08:00
967d842025 Added short warning to typst lib 2025-02-26 08:05:48 -08:00
101 changed files with 188 additions and 150854 deletions

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@ -1,7 +1,7 @@
[tinymist]: https://marketplace.visualstudio.com/items?itemName=myriad-dreamin.tinymist
[latex-workshop]: https://marketplace.visualstudio.com/items?itemName=James-Yu.latex-workshop
[CC BY-NC-SA 4.0]: https://creativecommons.org/licenses/by-nc-sa/4.0
[betalupi.com/handouts]: https://betalupi.com/handouts
[betalupi.com/handouts]: https://static.betalupi.com/ormc
[ORMC]: https://circles.math.ucla.edu/circles/
[Overleaf]: https://overleaf.com
[Typst.app]: https://typst.app
@ -26,7 +26,7 @@ By submitting or editing a handout in this repository, you agree to release it u
## 🛠️ Contributing
If you want to use one of these handouts for a class, see [betalupi.com/handouts]. \
If you want to use one of these handouts for a class, see [`betalupi.com/handouts`](https://betalupi.com/handouts). \
You only need to read this section if you want to edit these handouts.
### Setup
@ -56,7 +56,7 @@ This repository is organized as follows:
- [`./tools`](./tools) contains build scripts, [`./.github`](./.github) configures automation. \
You can ignore everything in these directories.
All handouts in this repository are based on `handout.cls` or `handout@0.1.0`.
All handouts in this repository are based on `ormc_handout.cls` or `handout@0.1.0`.
- If you're using Typst (preferred), read [`docs-typst.md`](./docs-typst.md)
- If you're still using LaTeX, read [`docs-latex.md`](./docs-latex.md).
@ -95,8 +95,8 @@ _(I do not recommend this. The default toolchain makes it easier to share improv
### For LaTeX:
1. Get the handout's directory (i.e, download the whole repo as a zip and extract the folder you want.)
2. Download [`./resources/handout.cls`](./resources/handout.cls)
3. Put this `handout.cls` in the same directory as the handout.
2. Download [`./resources/ormc_handout.cls`](./resources/ormc_handout.cls)
3. Put this `ormc_handout.cls` in the same directory as the handout.
4. Fix the include path at the top of `main.tex`:
You'll need to replace
@ -104,7 +104,7 @@ You'll need to replace
```latex
\documentclass[
...
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
```
with
@ -112,11 +112,11 @@ with
```latex
\documentclass[
...
]{handout}
]{ormc_handout}
```
5. Make a new overleaf project with the resulting directory.
6. **Do not use pdflatex**, it misbehaves with `handout`. Tell Overleaf to use XeLaTeX.
6. **Do not use pdflatex**, it misbehaves with `ormc_handout`. Tell Overleaf to use XeLaTeX.
### For Typst:
@ -124,5 +124,5 @@ Out-of-band typst compilation isn't supported. Clone the repository and use vsco
This is because typst can't import packages from a relative path.
If you _really_ want it, standalone typst compilation _is_ possible. \
Follow the LaTeX instructions, but fix `handout@0.1.0` instead of `handout`. \
Follow the LaTeX instructions, but fix `handout@0.1.0` instead of `ormc_handout`. \
You'll figure it out.

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@ -1,25 +1,23 @@
# LaTeX documentation
All LaTeX handouts are based on [`handout.cls`](./lib/tex/handout.cls). \
All LaTeX handouts are based on [`ormc_handout.cls`](./lib/tex/ormc_handout.cls). \
This class is based on `article.cls`, and should work with most LaTeX packages.
The best way to start a new document is to make a copy of an existing one.
- [Advanced/Cryptography](./src/Advanced/Cryptography) is a good example of a simple handout.
- [Advanced/DFAs](./src/Advanced/DFAs) is a good example of a handout with graphs.
- [Advanced/Geometric Optimization](./src/Advanced/Geometric%20Optimization) is a good example of a handout with geometry.
## Notes
## Notes
- Compile your handouts with XeLaTeX. \
`pdflatex` is known to misbehave with `handout.cls`. \
`pdflatex` is known to misbehave with `ormc_handout.cls`. \
This will happen by default if you use vscode. \
If you use Overleaf, you'll have to configure it manually (see document settings).
## Document Options
Document options are passed to `\documentclass`, as follows:
```latex
\documentclass[
% Show solutions is `solutions` is provided,
@ -39,35 +37,32 @@ Document options are passed to `\documentclass`, as follows:
% This should only be used for single-page handouts
% (e.g, warm-ups)
nopagenumber
]{handout}
]{ormc_handout}
```
Use `geometry` to change margins and page dimensions. US letter is the default.
## Utilities
## Utilities
- `\say{text}`: Puts text in quotes, handling details like period spacing. Courtesy of `dirtytalk`.
- `\note[Type]{text}`: Makes a note.
- `\hint{text}`: Shorthand for `\note[Hint]{text}`
## Sections
The usual LaTeX title-customization techniques _WILL NOT WORK_ with this class. \
The usual LaTeX title-customization techniques *WILL NOT WORK* with this class. \
Don't even try to load `titlesec`.
`handout.cls` supports two levels of sections:
`ormc_handout.cls` supports two levels of sections:
- `\section`, for large parts of the handout
- `\definition`, `\theorem`, `\proposition`, `\example`, `\remark`, `\problem`, and `\problempart`
All these macros have the following syntax: `\problem{title}<label>`
- `title` is the problem's title, and may be empty.
- `label` is the problem's label. This is optional. \
If a label is provided, this section may be referenced with `\ref{label}`.
Examples:
- `\problem{}`
- `\problem{Bonus}`
- `\problem{}<gcd>`, which may be referenced with `\ref{gcd}`

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@ -4,27 +4,23 @@ See [typst.app/docs](https://typst.app/docs) for typst's documentation. \
All typst handouts are based on [`handout@0.1.0`](./lib/typst/local/handout/0.1.0).
The best way to start a new document is to make a copy of an existing one.
- [Advanced/Tropical Polynomials](./src/Advanced/Tropical%20Polynomials) is a good place to start.
- [Warm-Ups/Painting](./src/Warm-Ups/Painting) is a good example of tikz-like pictures.
## Notes
## Notes
- Typst's equivalent of tikz is cetz ([homepage](https://cetz-package.github.io), [docs](https://cetz-package.github.io/docs/api))
- Typst handouts are always compiled with solutions. \
Handouts without solutions are automatically compiled and published at [betalupi.com/handouts](https://betalupi.com/handouts). \
Handouts without solutions are automatically compiled and published at [betalupi.com/handouts](https://static.betalupi.com/ormc). \
If you'd like to compile a student handout manually, run the following command in a handout directory:
```bash
typst compile main.typ --package-path ../../../lib/typst --input show_solutions=false
```
Where `package_path` is a relative path to [./lib/typst](./lib/typst).
## Document Options
All typst handouts start with the following:
```typst
#show: handout.with(
// Should match `meta.toml`
@ -42,12 +38,10 @@ All typst handouts start with the following:
```
## Notable commands
- `#v(1fr)`: Like LaTeX's `\vfill`. Creates whitespace that grows automatically. \
`fr` means "fraction". `#v(2fr)` will fill twice as much space as `#v(1fr)` on the same page.
## Utilities
- `#note([content], type: "Note type")`: Makes a note. `type` is optional.
- `#hint([content])`: Shorthand for `#note([content], type: "Hint")`
- `#solution([content])`: A pretty box for solutions. Hidden in student handouts.
@ -56,13 +50,12 @@ All typst handouts start with the following:
- `#if_no_solutions([content])`: Shows content only if we **aren't** showing solutions.
## Sections
High-level sections are denoted with `=`. \
Subsections start with `==`, subsubsections with `===`, and so on. \
**`handout@0.1.0` is only designed to use `=`, subsections might be ugly.**
`handout@0.1.0` also provides the following commands:
`handout@0.1.0` also provides the following commands:
- `problem`
- `definition`
- `theorem`
@ -70,13 +63,11 @@ Subsections start with `==`, subsubsections with `===`, and so on. \
- `remark`
These all have the same syntax: `#problem("title", label: "label")`
- `title` is the problem's title, and may be omitted.
- `label` is the problem's label. This is optional. \
If a label is provided, this problem can be referenced with `@label`
**Examples:**
- `#problem()`
- `#problem("Bonus")`
- `#problem(label: "gcd")`, which may be referenced with `@gcd`

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@ -1,5 +1,5 @@
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{../../../lib/tex/macros}[2025/03/02 Handout macros]
\ProvidesPackage{../../../lib/tex/macros}[2023/10/16 ORMC Macros]
\RequirePackage{hyperref}
\RequirePackage{pgf}

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@ -1,5 +1,5 @@
\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{../../../lib/tex/handout}[2025/03/02 2.0.2 Mark's handout class]
\ProvidesClass{../../../lib/tex/ormc_handout}[2023/05/29 2.0.2 ORMC Handout]
@ -13,7 +13,7 @@
\@twocolumnfalse
\@twosidefalse
\@mparswitchfalse
% Handout-specific
% ORMC-specific
\newif{\if@solutions} % If false, solutions and instructor notes are hidden.
\newif{\if@singlenumbering} % If true, the same counter is used for all objects.
\newif{\if@nopagenumber} % If true, don't number pages.
@ -25,7 +25,7 @@
\DeclareOption{10pt}{\renewcommand\@ptsize{0}}
\DeclareOption{11pt}{\renewcommand\@ptsize{1}}
\DeclareOption{12pt}{\renewcommand\@ptsize{2}}
% Handout-specific options
% ORMC-specific options
\DeclareOption{solutions}{\@solutionstrue}
\DeclareOption{nosolutions}{\@solutionsfalse}
\DeclareOption{multinumbering}{\@singlenumberingfalse}
@ -37,7 +37,7 @@
\DeclareOption{showwarning}{\@nowarningfalse}
\DeclareOption{hidewarning}{\@nowarningtrue}
\DeclareOption{unfinished}{\@unfinishedtrue}
\DeclareOption*{\ClassWarning{handout}{\CurrentOption ignored}}
\DeclareOption*{\ClassWarning{ormc_handout}{\CurrentOption ignored}}
\@unfinishedfalse
\ExecuteOptions{
@ -629,12 +629,12 @@
% Keep track of the current background color.
% Useful for transparent tikz drawings.
\def\bgcolor{white}
\def\ORMCbgcolor{white}
% Make a box environment.
% These can safely be nested.
% Args: title, back color, frame color.
\newenvironment{hobox}[3]{
\newenvironment{ORMCbox}[3]{
% \linehack draws a line across a tcolorbox.
% tcolorbox only supports two sections, but
% this hack allows us to have more.
@ -653,7 +653,7 @@
% Keep track of the current background color.
% Useful for transparent tikz drawings.
\def\bgcolor{#2}
\def\ORMCbgcolor{#2}
\begin{tcolorbox}[
enhanced,
@ -675,21 +675,21 @@
}
\newenvironment{examplesolution}{
\begin{hobox}{Example Solution}{black!10!white}{black!65!white}
\begin{ORMCbox}{Example Solution}{black!10!white}{black!65!white}
} {
\end{hobox}
\end{ORMCbox}
}
\if@solutions
\newenvironment{solution}{
\begin{hobox}{Solution}{ored!10!white}{ored}
\begin{ORMCbox}{Solution}{ored!10!white}{ored}
} {
\end{hobox}
\end{ORMCbox}
}
\newenvironment{instructornote}{
\begin{hobox}{Note for Instructors}{ocyan!10!white}{ocyan}
\begin{ORMCbox}{Note for Instructors}{ocyan!10!white}{ocyan}
} {
\end{hobox}
\end{ORMCbox}
}
\else
\excludecomment{solution}

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@ -49,28 +49,19 @@
// Text style
set text(font: "New Computer Modern")
set par(
leading: 0.5em,
spacing: 0.5em,
leading: 0.55em,
first-line-indent: 0mm,
hanging-indent: 0mm,
justify: true,
spacing: 0.5em,
)
//
// List style
show list: set block(spacing: 0.5em, below: 1em)
set list(
indent: 4mm,
body-indent: 1.5mm,
// Manually set spacing,
// `tight` has no effect.
spacing: 2mm,
)
set enum(
indent: 4mm,
body-indent: 1.5mm,
spacing: 2mm,
tight: false,
indent: 5mm,
spacing: 3mm,
)
//
@ -103,9 +94,7 @@
// Make handout title
{
import "header.typ": make_header, solution_warning, short_solution_warning
import "solution.typ": solutions_state, reset_solutions
reset_solutions()
import "solution.typ": show_solutions
let url = link(
"https://betalupi.com/handouts",
@ -120,8 +109,8 @@
top_right: url,
)
context {
if solutions_state.get() {
if show_solutions {
if short_warning {
short_solution_warning()
} else {
@ -129,7 +118,6 @@
}
}
}
}
// Include rest of document
doc

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@ -1,63 +1,34 @@
#import "misc.typ": ored, oblue
/// If false, hide instructor info.
/// If true, show it.
///
/// Solutions are shown by default. This behavior
/// is less surprising than hiding content by default.
#let solutions_state = state("solutions_state", true)
/// Force solutions to be hidden after this point.
///
/// This function produces content that must be
/// included in the document. If it is not included,
/// this function will have no effect.
#let hide_solutions() = {
return solutions_state.update(x => false)
}
/// Force solutions to be shown after this point.
///
/// This function produces content that must be
/// included in the document. If it is not included,
/// this function will have no effect.
#let show_solutions() = {
return solutions_state.update(x => true)
}
/// Reset the solution flag to its default value.
/// This value is determined by compile flags:
/// Compile with the following command to hide solutions:
/// `typst compile main.typ --input show_solutions=false`
///
/// Solutions are shown by default.
///
/// This function produces content that must be
/// included in the document. If it is not included,
/// this function will have no effect.
#let reset_solutions() = {
/// Solutions are shown by default. This behavior
/// is less surprising than hiding content by default.
#let show_solutions = {
if "show_solutions" in sys.inputs {
if (
// Show solutions unless they're explicitly disabled
not (
sys.inputs.show_solutions == "false" or sys.inputs.show_solutions == "no"
) {
return solutions_state.update(x => false)
}
)
} else {
// Show solutions by default
true
}
}
#let if_solutions(content) = context {
if solutions_state.get() { content }
#let if_solutions(content) = {
if show_solutions { content }
}
#let if_no_solutions(content) = context {
if not solutions_state.get() { content }
#let if_no_solutions(content) = {
if not show_solutions { content }
}
#let if_solutions_else(if_yes, if_no) = context {
if solutions_state.get() { if_yes } else { if_no }
#let if_solutions_else(if_yes, if_no) = {
if show_solutions { if_yes } else { if_no }
}
#let solution(content) = {

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\input{tikzset.tex}

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@ -24,7 +24,7 @@
label/.style = {
rectangle,
% For automatic red background in solutions
fill = \bgcolor,
fill = \ORMCbgcolor,
draw = none,
rounded corners = 0mm
},
@ -32,7 +32,7 @@
% Nodes
edg/.style = {
midway,
fill = \bgcolor,
fill = \ORMCbgcolor,
text = gray
},
int/.style = {},

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@ -5,7 +5,7 @@
shortwarning,
singlenumbering,
unfinished
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{multicol}

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@ -4,7 +4,7 @@
solutions,
singlenumbering,
shortwarning
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{multicol}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\input{tikzset.tex}

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@ -24,7 +24,7 @@
label/.style = {
rectangle,
% For automatic red background in solutions
fill = \bgcolor,
fill = \ORMCbgcolor,
draw = none,
rounded corners = 0mm
},

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\input{tikzset.tex}

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@ -24,7 +24,7 @@
label/.style = {
rectangle,
% For automatic red background in solutions
fill = \bgcolor,
fill = \ORMCbgcolor,
draw = none,
rounded corners = 0mm
},

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
@ -16,11 +16,6 @@
\title{Error-Correcting Codes}
\subtitle{Prepared by Mark on \today}
% TODO:
% ISBN section is tedious, could use some work.
% Check problem 5
\begin{document}
\maketitle

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@ -131,7 +131,7 @@ Take a valid ISBN-13 and swap two adjacent digits. When will the result be a val
\vfill
\problem{}<isbnnocorrect>
\problem{}<isbn-nocorrect>
\texttt{978-008-2066-466} was a valid ISBN until I changed a single digit. \par
Can you find the digit I changed? Can you recover the original ISBN?

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@ -1,6 +1,6 @@
\section{Error Correction}
As we saw in \ref{isbnnocorrect}, the ISBN check-digit scheme does not allow us to correct errors. \par
As we saw in \ref{isbn-nocorrect}, the ISBN check-digit scheme does not allow us to correct errors. \par
QR codes feature a system that does. \par
\vspace{1mm}

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@ -3,7 +3,7 @@
\documentclass[
nosolutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

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@ -4,7 +4,7 @@
solutions,
singlenumbering,
nopagenumber
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\uptitlel{Advanced 2}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

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@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\uptitlel{Advanced 2}

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@ -4,7 +4,7 @@
solutions,
singlenumbering,
shortwarning
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{tikz}

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@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\input{tikxset}

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@ -22,7 +22,7 @@
label/.style = {
circle,
% For automatic red background in solutions
fill = \bgcolor,
fill = \ORMCbgcolor,
draw = none
},
%

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@ -3,7 +3,7 @@
\documentclass[
nosolutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

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@ -1,7 +1,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{units}
\input{src/tikzset}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../../lib/tex/handout}
]{../../../../lib/tex/ormc_handout}
\usepackage{../../../../lib/tex/macros}
\usepackage{units}
\input{tikzset}

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@ -462,7 +462,7 @@ Thus,
\end{equation*}
\begin{hobox}{Review: Multiplying Vectors by Matrices}{black!10!white}{black!65!white}
\begin{ORMCbox}{Review: Multiplying Vectors by Matrices}{black!10!white}{black!65!white}
\begin{equation*}
Av =
\begin{bmatrix}
@ -480,7 +480,7 @@ Thus,
\end{equation*}
Note that each element of $Av$ is the dot product of a row in $A$ and a column in $v$.
\end{hobox}
\end{ORMCbox}
\problem{}
Compute the following product:

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@ -154,7 +154,7 @@ The \textit{Hadamard Gate} is given by the following matrix: \par
\end{equation*}
\note{Note that we divide by $\sqrt{2}$, since $H$ must be orthonormal.}
\begin{hobox}{Review: Matrix Multiplication}{black!10!white}{black!65!white}
\begin{ORMCbox}{Review: Matrix Multiplication}{black!10!white}{black!65!white}
Matrix multiplication works as follows:
\begin{equation*}
@ -196,7 +196,7 @@ The \textit{Hadamard Gate} is given by the following matrix: \par
This is exactly the first column of the matrix product. \par
Also, note that each element of $Ac_0$ is the dot product of a row in $A$ and a column in $c_0$.
\end{hobox}
\end{ORMCbox}
\problem{}

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@ -13,7 +13,7 @@
line width = 0.35mm
},
qubit/.style = {
fill = \bgcolor,
fill = \ORMCbgcolor,
line width = 0.35mm
},
wire/.style = {

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{ifthen}

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@ -1,6 +1,5 @@
\definition{}
The \textit{integer lattice} $\mathbb{Z}^n$ is the set of points with integer coordinates in $n$ dimensions. \par
For example, $\mathbb{Z}^3$ is the set of points $(a, b, c)$ where $a$, $b$, and $c$ are integers.
The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates.
\problem{}
Draw $\mathbb{Z}^2$.
@ -9,13 +8,12 @@ Draw $\mathbb{Z}^2$.
\definition{}
We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$
if every lattice point can be written as
We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
$$
a_1v_1 + a_2v_2 + ... + a_kv_k
$$
for integer coefficients $a_i$. \par
\textbf{Bonus:} show that $k$ must be at least $n$.
It is fairly easy to show that $k$ must be at least $n$.
\problem{}
Which of the following generate $\mathbb{Z}^2$?
@ -32,7 +30,8 @@ Which of the following generate $\mathbb{Z}^2$?
\vfill
\problem{}
Find a set of two vectors other than $\{ (0, 1), (1, 0) \}$ that generates $\mathbb{Z}^2$. \\
Find a set of two vectors that generates $\mathbb{Z}^2$. \\
Don't say $\{ (0, 1), (1, 0) \}$, that's too easy.
\vfill
@ -45,8 +44,7 @@ Find a set of vectors that generates $\mathbb{Z}^n$.
\pagebreak
\definition{}
Say we have a generating set of a lattice. \par
The \textit{fundamental region} of this set is the $n$-dimensional parallelogram spanned by its members. \par
A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use.
\problem{}
Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.

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@ -1,6 +1,6 @@
\section{Minkowski's Theorem}
\theorem{Blichfeldt's Theorem}<blich>
\theorem{Blichfeldt's Theorem}
Let $X$ be a finite connected region. If the volume of $X$ is greater than $1$, $X$ must contain two distinct points that differ by an element of $\mathbb{Z}^n$. In other words, there exist distinct $x, y \in X$ so that $x - y \in \mathbb{Z}^n$.
\vspace{2mm}
@ -9,22 +9,14 @@ Intuitively, this means that you can translate $X$ to cover two lattice points a
\problem{}
Draw a connected region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector.
Draw a region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector.
\hint{Area is two-dimensional volume.}
\vfill
\problem{}
Draw a \textit{disconnected} region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points, \par
and show that no two points in that region differ by an integer vector.
\note{In other words, show that \ref{blich} indeed requires a connected region.}
\vfill
\problem{}
The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. \par
Explain the picture and complete the proof.
The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. Explain the picture and complete the proof.
\begin{center}
\includegraphics[angle=90,width=0.5\linewidth]{proof.png}
@ -56,8 +48,10 @@ Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points m
A region $X$ is \textit{convex} if the line segment connecting any two points in $X$ lies entirely in $X$.
\problem{}
Draw a convex region in two dimensions. \par
Then, draw a two-dimensional region that is \textit{not} convex.
\begin{itemize}
\item Draw a convex region in the plane.
\item Draw a region that is not convex.
\end{itemize}
\vfill
\pagebreak
@ -65,28 +59,23 @@ Then, draw a two-dimensional region that is \textit{not} convex.
\definition{}
We say a region $X$ is \textit{symmetric with respect to the origin} if for all points $x \in X$, $-x$ is also in $X$. \par
In the following problems, \say{\textit{symmetric}} means \say{symmetric with respect to the origin.}
We say a region $X$ is \textit{symmetric} if for all points $x \in X$, $-x$ is also in $X$.
\problem{}
Draw a symmetric region. \par
Then, draw an asymmetric region.
\vfill
\problem{}
Show that a convex symmetric set always contains the origin.
\begin{itemize}
\item Draw a symmetric region.
\item Draw an asymmetric region.
\end{itemize}
\vfill
\theorem{Minkowski's Theorem}<mink>
Every convex set in $\mathbb{R}^n$ that is symmetric and has a volume \par
greater than $2^n$ contains an integral point that isn't zero.
Every convex set in $\mathbb{R}^n$ that is symmetric with respect to the origin and which has a volume greater than $2^n$ contains an integral point that isn't zero.
\problem{}
Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. \par
What is a simple class of regions that has the properties listed above?
What is the simplest region that has the properties listed above?
\vfill

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

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@ -3,7 +3,7 @@
\documentclass[
nosolutions,
singlenumbering,
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{units}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
%shortwarning
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\graphicspath{ {./images/} }

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@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{amsmath}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\input{tikxset.tex}

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@ -24,7 +24,7 @@
label/.style = {
rectangle,
% For automatic red background in solutions
fill = \bgcolor,
fill = \ORMCbgcolor,
draw = none,
rounded corners = 0mm
},

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@ -4,7 +4,7 @@
solutions,
singlenumbering,
shortwarning
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\input{diagram}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
shortwarning
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{chessfss}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{units}

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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usetikzlibrary{calc}

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@ -1,26 +0,0 @@
#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],
by: "Mark",
)
#include "parts/00 intro.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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@ -1,7 +0,0 @@
[metadata]
title = "Wallpaper Symmetries"
[publish]
handout = true
solutions = true

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@ -1,151 +0,0 @@
#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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@ -1,132 +0,0 @@
#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)

View File

@ -1,181 +0,0 @@
#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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@ -1,67 +0,0 @@
#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])

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@ -1,100 +0,0 @@
#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>
<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>
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@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -24,7 +24,7 @@
label/.style = {
rectangle,
% For automatic red background in solutions
fill = \bgcolor,
fill = \ORMCbgcolor,
draw = none,
rounded corners = 0mm
},

View File

@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -1,6 +1,6 @@
% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[solutions]{../../../lib/tex/handout}
\documentclass[solutions]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -3,7 +3,7 @@
\documentclass[
solutions,
singlenumbering
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{units}

View File

@ -190,7 +190,7 @@ For example, see the proof of the statement in \ref{binomsum} on the next page.
\pagebreak
\begin{hobox}{Alternative Proof}{ogrape!10!white}{ogrape}
\begin{ORMCbox}{Alternative Proof}{ogrape!10!white}{ogrape}
Consider the following problem: \par
How many ways are there to write a number $x$ as an ordered sum of positive integers? \par
\note{
@ -234,7 +234,7 @@ For example, see the proof of the statement in \ref{binomsum} on the next page.
We've found that the number of ways to split $x$ can be written as either
$\sum_{n = 1}^{x-1} \binom{x-1}{n}$ or $2^{x-1}$,
and therefore $\sum_{n = 1}^{x-1} \binom{x-1}{n} = 2^{x-1}$.
\end{hobox}
\end{ORMCbox}
\pagebreak

View File

@ -2,7 +2,7 @@
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -1,6 +1,6 @@
% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[solutions]{../../../lib/tex/handout}
\documentclass[solutions]{../../../lib/tex/ormc_handout}
\usepackage{adjustbox}
\usepackage{../../../lib/tex/macros}

View File

@ -1,6 +1,6 @@
% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[solutions]{../../../lib/tex/handout}
\documentclass[solutions]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -3,7 +3,7 @@
shortwarning,
singlenumbering,
nopagenumber
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\geometry{top = 20mm}

View File

@ -3,7 +3,7 @@
hidewarning,
singlenumbering,
nopagenumber
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -3,7 +3,7 @@
hidewarning,
singlenumbering,
nopagenumber
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

View File

@ -3,7 +3,7 @@
hidewarning,
singlenumbering,
nopagenumber
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\def\utri#1#2{
@ -102,7 +102,7 @@
\vfill
\begin{hobox}{Notes}{ogrape!10!white}{ogrape}
\begin{ORMCbox}{Notes}{ogrape!10!white}{ogrape}
\begin{itemize}
\item
Each player needs to make multiple groups before the first player's second turn. \par
@ -111,7 +111,7 @@
\item Groups are easiest to secure near the corners or along the edges. Play there first. \par
\end{itemize}
\end{hobox}
\end{ORMCbox}

View File

@ -3,7 +3,7 @@
\documentclass[
solutions,
shortwarning
]{../../../lib/tex/handout}
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}

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