%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentstyle[12pt]{amsart}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   ENVIRONMENTS
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%% \theoremstyle{plain} %% This is the default
\newtheorem{th}{Theorem}[section]
\newtheorem{cor}[th]{Corollary}
\newtheorem{lm}[th]{Lemma}
\newtheorem{prp}[th]{Proposition}
\newtheorem{clm}[th]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[th]{Definition}
\newtheorem{exmp}[th]{Example}
\newtheorem{prb}[th]{Problem}
%
\theoremstyle{remark}
\newtheorem{rem}[th]{Remark}
%
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Math macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Logical symbols
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\satf}{\models}
\newcommand{\id}{{\text{\sl id}}}       % The identity map
\newcommand{\miff}{\Longleftrightarrow} % math `if and only if'
\newcommand{\mimplies}{\Longrightarrow} % math `implies'
\newcommand{\mimplied}{\Longleftarrow}  % math `implied by'
\newcommand{\miffdef}{\stackrel{\text{\normalshape
   def}}{\miff}}                        % defining iff
\newcommand{\eqdef}{\stackrel{\text{\normalshape
   def}}{=}}                            % defining equation
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
%
%\newcommand{\neareq}{\stackrel{\ae}{=}}
%\newcommand{\aeqv}{\,\stackrel{\text{\normalshape ae}}{\equiv}\,}
%\newcommand{\aeq}{\,\stackrel{\text{\normalshape ae}}{=}\,}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%  Classical math
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\tmod}[1]{\text{\normalshape mod~$#1$}}

\newcommand{\BZ}{\Bbb Z} % integers
\newcommand{\BQ}{\Bbb Q} % rationals
\newcommand{\BR}{\Bbb R} % reals
\newcommand{\BC}{\Bbb C} % complex numbers
\newcommand{\re}{\text{\normalshape Re}\,}    % real part
\newcommand{\im}{\text{\normalshape Im}\,}    % imaginary part
%
\newcommand{\meet}{\wedge}
\newcommand{\join}{\vee}
\newcommand{\union}{\cup}
\newcommand{\inter}{\cap}  % intersection
\newcommand{\iso}{\cong}   % isomorphic
\newcommand{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\newcommand{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\bold{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\Ker}{\text{\normalshape Ker}\,}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%  Operators on algebras
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%  \m is used to produce models (algebras) as follows:
%%%  typing `\m a'  produces in bold face, A.
%%%  \vr does the same for varieties
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\m}[1]{\mbox{${\bold{\uppercase{#1}}}$}}
\newcommand{\sm}[1]{\mbox{{\scriptsize\bf{\uppercase{#1}}}}}
\newcommand{\vr}[1]{{\cal{\uppercase {#1}}}}
%
\newcommand{\Var}{{\bold{V}}}
\newcommand{\Ho}{{\bold{H}}}
\newcommand{\Su}{{\bold{S}}}
\newcommand{\Pd}{{\bold{P}}}
\newcommand{\Iso}{{\bold{I}}}
\newcommand{\Ps}{{\bold{P}}_{\text{\normalshape s}}}
\newcommand{\fin}[1]{{\bold{#1}}_{\text{\normalshape fin}}}
%
\newcommand{\Pol}{{\text{\normalshape Pol}\,}}
\newcommand{\Clo}{{\text{\normalshape Clo}\,}}
\newcommand{\Con}{{\text{\normalshape Con}\,}}
\newcommand{\Conb}{{\bold{Con}\,}}       % Con(A) as a lattice
%\newcommand{\Cg}[1]{\text{\normalshape Cg}^{\m{#1}}}
%\newcommand{\Sg}[1]{\text{\normalshape Sg}^{\m{#1}}}
\newcommand{\Cg}[1]{\text{\normalshape Cg}^{\sm{#1}}}
\newcommand{\Sg}[1]{\text{\normalshape Sg}^{\sm{#1}}}
\newcommand{\cg}{\text{\normalshape Cg}\,}
\newcommand{\sg}{\text{\normalshape Sg}\,}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  macros for tame congruence theory.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\M}[3]{\text{\normalshape
   M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\normalshape
   Min}_{\m{#1}}}
%
\newcommand{\typ}{{\text{\normalshape typ}\,}}
\newcommand{\ann}{{\text{\normalshape ann}\,}}
\newcommand{\Sep}{{\text{\normalshape Sep}\,}}
\newcommand{\ssb}{{\text{\normalshape ssb}\,}}
\newcommand{\sst}{{\text{\normalshape sst}\,}}
%
\newcommand{\utyp}{{\bold{1}}\,} % unary
\newcommand{\atyp}{{\bold{2}}\,} % affine
\newcommand{\btyp}{{\bold{3}}\,} % boolean
\newcommand{\ltyp}{{\bold{4}}\,} % lattice
\newcommand{\styp}{{\bold{5}}\,} % semilattice
\newcommand{\ityp}{{\bold{i}}\,}
\newcommand{\jtyp}{{\bold{j}}\,}
%
\newcommand{\solv}{\stackrel{s}{\sim}}
\newcommand{\ssolv}{\stackrel{ss}{\sim}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Abbreviations
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\s}[1]{\underset{\mbox{\scriptsize\sl #1}}{\textstyle\bold \sim}}
%\newcommand{\s}[1]{\underset{#1}{\textstyle\bold \sim}}

\setlength{\oddsidemargin}{0.0in}
\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{0.0in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\textheight}{9in}

\begin{document}

\noindent
{\Large\bf The Classical Construction Problems}

\bigskip

The following problems concern straightedge and compass
constructions in the plane.

\bigskip

\noindent
{\large\bf Trisecting the angle:}
Is there a construction method which accepts as
input an angle $\alpha$ and produces as output
the angle $\alpha/3$?
(There are two ways to interpret this: one may ask for
a uniform construction method that works for all $\alpha$,
or one may allow different constructions for different
$\alpha$.)

\bigskip

\noindent
{\large\bf Squaring the circle:}
Given a fixed circle, 
is it possible to construct a square of equal area?
(More precisely: Given a segment of length $r$, 
is it possible to construct a segment 
of length $s$ where $s$ is chosen so that
a square of side length 
$s$ has area equal to the area of a circle of radius $r$?)

\bigskip

\noindent
{\large\bf Duplicating the cube:}
Given cube of volume $V$, is it possible to
construct a cube of volume $2V$?
(More precisely:
Given a segment of length $c$, is it possible to
construct a segment of length $d$ where $d$ is chosen
so that a cube of side length $d$ has twice the volume 
of a cube of side length $c$?)

\bigskip

\noindent
{\large\bf Constructing a regular $n$-gon:}
For which values of $n$ is it possible to construct
a regular $n$-gon? (An $n$-gon is {\it regular}
if all pairs of adjacent sides meet at the same
angle, which must be $\pi(1-\frac{2}{n})$.)

\bigskip


\bigskip

\noindent
{\Large\bf What Exactly Is A `Construction'?}

\bigskip

The intuitive description of a construction is as follows.
We start with whatever set of lines, circles, points, or segments
we are given, and consider these to be `previously
constructed'. We are now permitted a finite number
of steps in which we may

\begin{enumerate}
\item[(i)] draw a line through any
two previously constructed points to produce a new
`constructed' line;
\item[(ii)] draw a circle centered at any previously constructed
point, whose radius has length equal to the length of 
any previously constructed segment, to produce a new
`constructed' circle;
\item[(iii)] draw a point at the intersection
of any two previously constructed lines,
circles, or a line and a circle, to produce a new
`constructed' point;
\item[(iv)] identify a segment on a previously constructed
line which lies between two previously constructed
points as a new `constructed' segment.
\end{enumerate}

\bigskip

To be precise, we will say that a {\em construction} 
is a finite list, for example
\[\lb P_1, P_2, c_1, c_2, P_3, P_4, \ell_1, \ell_2, P_5\rb,\]
of lines, circles, points and segments, where each element
on the list is constructed from previous elements
by the rules described above.
For the problem of trisecting an angle, we may identify
the angle $\alpha$ with a pair of lines which cross
at angle $\alpha$. So, the trisection problem is:
Given two lines which cross 
at angle $\alpha$, can we construct two lines which
cross at angle $\alpha/3$?

\end{document}
