\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{decorations.markings}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Knot Theory},
    pdfpagemode=FullScreen,
    }
% \usepackage{ulem}

% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
\newcommand{\normal}{\unlhd} % normal subgroup of
\newcommand{\F}{\mathbb{F}} % random field
\newcommand{\Isom}{\mathrm{Isom}} % isometries group

\DeclareMathOperator{\img}{im} % image
\DeclareMathOperator{\sgn}{sgn} % sign of permutation
\DeclareMathOperator{\ord}{ord} % order of group element
\DeclareMathOperator{\lcm}{lcm} % least common multiple
% \DeclareMathOperator{\gcd}{gcd} % greatest common divisor

% \newcommand{\bpmat}{\begin{pmatrix}}
% \newcommand{\epmat}{\end{pmatrix}}


% knot theory
\newcommand{\lk}{\mathrm{lk}} % linking number


% SOME ENVIRONMENTS

\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
    }

\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
\newcommand{\ee}{\end{enumerate}}




\title{MAT 150A HW06}
\author{[add your name here]}
\date{Due Tuesday, 11/21/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\begin{reminder} 
Your homework submission \textbf{\large must be typed} (TeX'ed) up in full sentences, with proper mathematical formatting. 

\begin{comment}
The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{http://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{http://detexify.kirelabs.org/classify.html}
\end{itemize}
\end{comment}
\end{reminder}


\paragraph{Covered in this HW}
Chapter 6: isometries of the plane, symmetries of plane figures, lattices; translations, point groups, crystallographic restriction


\paragraph{Grading} 
Some of the (parts of) problems will be graded in detail out of several points, and necessary feedback will be given. The rest will be graded out of 2 points. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that a conjugate of a glide reflection in $\Isom(\R^2)$ is also a glide reflection, and that the glide vectors have the same length.

\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
We used $\R^2$ to describe the points on the plane. We could equivalently use $\C$, the complex plane. Since we use the same notion of distance for points in the complex plane, as metric spaces, $\R^2$ is the same as $\C$. The generators of $\Isom(\R^2) = \Isom(\C)$ on page 160, Equation (6.3.1), in the textbook, or slide 10 in \href{http://www.melissa-zhang.com/Teaching/FA2023/150A/Lecture17.pdf}{Lecture 17}.

Write formulas for the generators of $\Isom(\C)$ in terms of the complex variable $z = x + iy$.



\solution





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

Let $G$ denote the group of symmetries of the following finite wallpaper pattern constructed from equilateral triangles of side length 1:

\begin{center}
\includegraphics[width=3in]{150A-HWs/images/triangles-wallpaper.png} \\
{\footnotesize Source: {http://mathworld.wolfram.com/TriangularGrid.html}}
\end{center}



\bea
\item Determine the point group $\overline{G}$ of $G$, and find the index in $G$ of the subgroup of translations $L$.
\item Find translation vectors $a,b \in \R^2$ realizing $L$ as the lattice $\Z a + \Z b$. 
\ee

\solution




\end{document}
