% Math 370 hw3 Spring 2014
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\title{History of Mathematics \\
Homework 3
}
\author{Ethan Bolker}

\begin{document}

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This homework is due Thursday, February 20.

\begin{enumerate}

\item Let $P$ and $Q$ be two points in the unit disk. Find a Euclidean
  construction for the (hyperbolic/pseudo) line joining them. That is,
  find a straightedge-and-compass construction for the circle through
  those points that meets the unit circle at a right angle. (This is
  not easy. After trying it yourself (or with a classmate) feel free
  to try to find a solution on the web. If you do, you should write it
  out in your own words (and pictures) and tell me where and how you
  found it.) 

\item

Calculate the (hyperbolic/pseudo) distance from the center of the unit
disk to a point at (Euclidean) distance $r<1$ from the
center. 

Recall that distances at a point $P$ at distance $r < 1$ from the
center of the unit disk are stretched by a factor $1/(1-r^2)$.

\emph{Hint.} Your argument should lead to an integral that you can
evaluate using what you learned in Calculus. If you don't remember
what you need you should be able to look it up. 

\item
Use your answer to the previous problem to show that lines through the
center are infinitely long - in other words, that the distance from
the center approaches $\infty$ as $r \rightarrow 1$.

\item At
  \url{http://math.stackexchange.com/questions/675522/whats-the-intuition-behind-pythagoras-theorem}
  you will find a question and several answers about
the Pythagorean Theorem. I read through them rapidly, but found no
mention of the fact that the theorem depends on the parallel
postulate.

I think we should provide another answer to point that out. (I have
enough reputation to do that.) For this homework, write something that
I might post. 

Here's a hint. What's a square in spherical or hyperbolic geometry?
Can you compute the area of a square by squaring the length of its
side?

\item Build a paper model of the hyperbolic plane. There are
  instructions at
  \url{http://euler.slu.edu/escher/index.php/Hyperbolic_Paper_Exploration}.

\begin{itemize}
\item  Answer the two questions there. 
\item Let $ABC$ be one of the equilateral triangles you taped together
  to make your hyperbolic plane. Let $D$ be the foot of the
  perpendicular from $C$ to $AB$. Try to compute the angle of
  parallelism $\Pi(CD)$. (I do not know the answer. I don't even know
  whether the question makes sense. You should be able to find some
  upper and lower bounds for that angle by experiment.)
\end{itemize}

Bring your model to class if you can. That might be hard to do, since
you can't flatten it out!

\end{enumerate}

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Here is the \LaTeX{} source for this document. You can cut it from the
pdf and use it to start your answers. I used the \verb!\jobname! macro
for the source file name, so you can call your file by any name you like.

\verbatiminput{\jobname}

\end{document}
