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\title{Homework 1 --- Scissors congruence and equidecomposability in Euclidean space}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

Homework is assigned on Tuesdays; it is due at the start of class two weeks
after it is assigned. Problems marked ``Hard'' are extra credit; in other
words, doing all the ordinary problems correctly will earn full credit. But
by doing hard problems, it is possible to make mistakes on, or fail to
complete, all the ordinary problems, and still earn full credit.

\begin{prob}[Tensor products]
Identify the more common names of the following tensor products of $\Z$--modules
(remember that $\Z$--modules are just Abelian groups)
\begin{enumerate}
\item{$\Z\otimes_\Z \Z/n\Z$ for $n$ a positive integer}
\item{$\Z/n\Z \otimes_\Z \Z/m\Z$ for $n,m$ positive integers}
\item{$\Q\otimes_\Z \Q$}
\item{$\Z^n\otimes_\Z\Q$}
\end{enumerate}
\end{prob}

\begin{prob}
Find a {\em bounded} subset of the plane which is congruent to a proper subset of
itself.
\end{prob}

\begin{prob}
Calculate the Dehn invariants of the regular platonic solids of side length $1$.
What is the relation between the Dehn invariant of the octahedron and the tetrahedron?
Show that an octahedron and two tetrahedra of the same edge length can be glued up
to make a parallelepiped which tiles Euclidean space.
\end{prob}

\begin{prob}
Let $R$ and $S$ be two squares in the plane of equal side length. Find a decomposition
of $R$ into pieces which can be moved by {\em translation alone} to give a decomposition
of $S$. Try to find a decomposition with as few pieces as possible.
\end{prob}

\begin{prob}
The {\em rhombic triacontahedron} is obtained in the following way from a regular
dodecahedron: erect a little five--sided pyramid on each of the twelve faces of the
dodecahedron. For a certain choice of pyramid, each of the five sides of the pyramid
on one face of the dodecahedron joins up with a side of the pyramid on a neighboring
face of the dodecahedron to make a rhombus. Explicitly, if $F_1,F_2$ are two faces which
share an edge $a,b$, and if $c_1,c_2$ are the apex of the pyramid erected on $F_1,F_2$
respectively, then if the heights of the pyramids are chosen to be a certain (equal)
number, the points $c_1,c_2,a,b$ are in the same plane and span a rhombus.
Thus this solid has thirty congruent rhombic faces. Draw a good picture of this solid.
Show that the diagonals of the rhombus faces have a ratio of $\frac {1 + \sqrt{5}} 2$ ---
that is, the golden ratio. Show that the dihedral angles of the faces are {\em rational}
multiples of $\pi$, and therefore that the Dehn invariant of this solid is zero.
\end{prob}

\begin{rmk}
The rhombic triacontahedron can be decomposed into parallelepipeds (which already shows
it has Dehn invariant zero). These parallelepipeds tile three dimensional space in a
non--periodic fashion, which is a three--dimensional version of Penrose tiles.
\end{rmk}

\begin{prob}
Let $D$ be the unit disk. Show that there is no partition of $D$ into two disjoint
sets $D = X \coprod Y$ such that $X$ and $Y$ are congruent. Hint: where does the
center go?
\end{prob}

\begin{prob}
Using the fact that the group of orientation--preserving
isometries of the plane is isomorphic to a semi--direct product of $\R^2$
and $S^1$, show that this group does not contain a subgroup isomorphic to $F_2$,
the free group on two generators. Hint: show that $\Isom^+(\E^2)$ has a
homomorphism to an abelian group with abelian kernel, and that $F_2$ does not.
\end{prob}

\begin{prob}[Zylev's theorem; Hard]
Show that if two polyhedra are stably scissors congruent, then they are scissors
congruent. That is, if  $P \coprod A \sim Q \coprod A$ then $P \sim Q$. (Note: the
notation $X \coprod Y$ means the disjoint union of $X$ and $Y$.)
\end{prob}

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