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\begin{document}


\title{Two Unrelated Topics of Interest for the Purpose of Demonstration}
\author{Michael Trott}
\email{info@wolfram.com}
\affiliation{Wolfram Research, Inc., 100 Trade Center Drive, Champaign
Illinois 61820-7237}\label{I1}
\date{\today}
\begin{abstract}
This paper offers a demonstration of the appearance and general
document features of the Physical Review style sheet as adapted
from the American Physical Society specifications for journal submissions.
Although the topics in this paper were written specifically for
{\itshape Publicon}, it is hoped they will nonetheless be regarded
with interest by the physics community. {\copyright} 2004 Wolfram
Research, Inc.
\end{abstract}
\pacs{0000-1111-222}
\maketitle

\section{HIGH-PRECISION VALUE FOR THE QUARTIC ANHARMONIC OSCILLATOR
GROUND STATE}

\subsection{Introduction}

As is well known, only a very limited number of one--dimensional
potentials allow for an exact solution of the Schr\"odinger equation.
This means that for many model potentials one has to resort to numerical
solution methods. For judging their accuracy, reliability, and speed
it is important to have high--precision values of certain nonexactly
solvable potentials. The most investigated of such potentials is
the quartic anharmonic oscillator \cite{{Tymczak, Graffi, Sergeev,
Suzuki, Chebot, Shanley, Hatsuda, Weniger, Ho96, Cizek, Bay, Voros99,
Khan, Antons, Skala, Meiszn, Delaba, Fernan, Vinett}} described
by
\begin{equation}
-\psi _{k}^{\prime \prime }( z) +z^{4}\psi _{k}( z) =\varepsilon
_{k}\psi _{k}( z) %
\label{XRef-Equation-22712250}
\end{equation}

The eigenfunctions to the eigenvalues $\varepsilon _{k}$ decay exponentially
for $z\rightarrow \pm \infty $. 

\subsection{The Hill Determinant Method}

A classical method to solve Sturm--Liouville problems of type \ref{XRef-Equation-22712250}
is to calculate the eigenvalues of a truncated version of the corresponding
Hill determinant. Using the harmonic oscillator basis $\phi _{n}(
z) $ we write $\psi _{0}( z) =\sum \limits_{k=0}^{\infty }\alpha
_{k}\phi _{k}( z) $ where
\begin{gather}
-\phi _{k}^{\prime \prime }( z) +z^{2}\phi _{k}( z) =\epsilon _{k}\phi
_{k}( z) 
\\\phi _{n}( z) =\frac{1}{\sqrt{\sqrt{\pi }2^{n}n!}}e^{\frac{-z^{2}}{2}}H_{n}(
z) 
\end{gather}

Forming the matrix elements $h_{m,n}=\int _{-\infty }^{\infty }\phi
_{m}( z) (-\phi _{n}^{{\prime\prime}}( z) +z^{4}\phi _{n}( z) )dz$.
For $n\geq m$ we obtain
\begin{equation}
h_{m,n}=\begin{cases}
0 & n-m>4 \\
\begin{array}{l}
 2^{\frac{m-n}{2}-4} \sqrt{\frac{m!}{n!}} \left( 32 n \delta _{m,n-2}
\left( n-1\right) ^{2}+16 \left( n-3\right)  \left( n-2\right) 
n \delta _{m,n-4} \left( n-1\right) +\right.  \\
 \left. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4 \left( 2 n \left(
3 n+5\right) +5\right)  \delta _{m,n}+8 \left( n+1\right)  \delta
_{m,n+2}+\delta _{m,n+4}\right) 
\end{array} & \textup{else} \\
\end{cases}
\end{equation}

A rough estimation shows that one obtains about 0.2 digits per harmonic
oscillator state. So by taking into account the first 500 eigenstates
and carrying out the calculation with about five thousand digits
one obtains about 120 reliable digits for $\varepsilon _{0}$. (This
calculation takes about 20 minutes on a 2000 vintage workstation
using \textit{Mathematica} 4 \cite{Wolfram99}.) 
\begin{multline*}
\varepsilon _{0}=1.0603620904841828996470460166926635455152087285289779332162452416959435\\
63044344421126896299134671703510546244358582525580982763829\ldots
\end{multline*}

The Hill determinant approach allows in addition to the calculation
of the eigenvalues, the calculation of the eigenvectors. The following
graphic visualizes the matrix of eigenvectors of $(h_{m,n})_{1\leq
n,m\leq 100}$. The graphic shows that the lowest eigenfunctions
are quite similar to the harmonic oscillator eigenfunctions. Higher
states are complicated mixtures of harmonic oscillator states. The
overall ``checkerboard''--like structure results from the fact that
the contribution of the antisymmetric (symmetric) harmonic oscillator
states to the symmetric (antisymmetric) anharmonic oscillator states
is identical zero. The very high states are dominated by truncation
effects and do not correctly mimic the anharmonic oscillator states.
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr1.eps}

\end{center}
\caption{The matrix of eigenvectors of $(h_{m,n})_{1\leq n,m\leq
100}$.}

\end{figure}

\subsection{The New Algorithm}

To get a very high--precision approximation of
\begin{equation}
-\psi ^{{\prime\prime}}( x) +z^{4}\psi ( x) =\lambda  \psi ( x)
\end{equation}

we start with the series expansion 
\begin{equation}
\psi ( x) =y_{n}( x) =\sum \limits_{k=0}^{n}a_{k}( \lambda ) x^{k}
\end{equation}

For the ground state we choose (ignoring normalization) $\psi (
0) =1, \psi ^{\prime }( 0) =0$. For ``suitable chosen'' $x^{*}$
we then find high--precision approximations for the zeros of $y_{n}(
x^{*}) $ and $y_{n}^{\prime }( x^{*}) $. These zeros then bound
$\lambda _{0}$ from below and above. 

Using the differential equation one obtains the following recursion
relation for the $a_{k}( \lambda ) $:
\begin{equation}
a_{k}( \lambda ) =\frac{a_{m-6}( \lambda ) -\lambda  a_{m-2}( \lambda
) }{m^{2}-m}
\end{equation}

For large $n$ ($n\rightarrow \infty $) we want the function $y_{n}(
x) $ to vanish as $x\rightarrow \infty $. For a $\lambda$ smaller
than the smallest possible $\lambda$, the function $y_{n}( x^{*})
$ will not have a zero, but the function $y_{n}^{\prime }( x^{*})
$ will have a zero for a certain $x^{*}$. For a $\lambda$ larger
than the smallest possible $\lambda$, the function $y_{n}( x^{*})
$ will have a zero, but the function $y_{n}^{\prime }( x^{*}) $
will not have a zero for a certain $x^{*}$. This allows to find
a bounding interval for $\lambda _{0}$. The next two graphics show
$y_{80}( x) $ and $y_{80}^{\prime }( x) $ for 10 equidistant values
for $\lambda$ from the interval $[1.05,1.08]$ to visualize this
bounding process. (For more details, see \cite{Trott00A}.)
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr2.eps}

\end{center}
\caption{$y_{80}( x) $ and $y_{80}^{\prime }( x) $ for 10 equidistant
values for $\lambda$ from the interval $[1.05,1.08]$.}

\end{figure}

It is straightforward to implement the calculation of the bounding
interval for $\lambda _{0}$ in {\itshape Mathematica} in a three--line
program (see \cite{Trott00}). Using \texttt{FindRoot} we calculate
high--precision values for the zeros of $y_{n}( \xi ) $ and $y_{n}^{\prime
}( \xi ) $.


{\ttfamily
$\begin{array}{l}
\text{$\lambda$Bounds}\text{[}\text{n\_}\text{,}\text{ }\text{$\xi$\_}\text{,}\text{
}\text{opts\_\_\_}\text{]}\text{ }\text{:=}\text{ }\\
\text{\ \ \ }\,\text{Function}\text{[}\text{f}\text{,}\text{ }\text{$\lambda$}\text{
}\text{/.}\text{ }\text{FindRoot}\text{[}\text{f}\text{[}\text{n}\text{,}\text{
}\text{$\lambda$}\text{,}\text{ }\text{$\xi$}\text{]}\text{ }\text{==}\text{
}\text{0}\text{,}\text{ }\\
\text{\ \ \ \ \ \ \ }\,\text{\{}\text{$\lambda$}\text{,}\text{ }\text{106}\,\text{/}\,\text{100}\text{,}\text{
}\text{107}\,\text{/}\,\text{100}\text{\}}\text{,}\text{ }\text{opts}\text{]}\text{]}\text{
}\text{/@}\text{ }\text{\{}\text{y}\text{,}\text{ }\text{yPrime}\text{\}}\text{;}\text{
}\end{array}$}



The calculation of $y_{n}( \xi ) $ and $y_{n}^{\prime }( \xi ) $
is also straightforward based on a recursive calculations of the
$a_{k}( \lambda ) $. 


{\ttfamily
$\begin{array}{l}
\text{y}\text{[}\text{n\_}\text{,}\text{ }\text{$\lambda$\_Real}\text{,}\text{
}\text{$\xi$\_}\text{]}\text{ }\text{:=}\text{ }\text{Module}\text{[}\text{\{}\text{a6}\text{,}\text{
}\text{a4}\text{,}\text{ }\text{a2}\text{,}\text{ }\text{ak}\text{,}\text{
}\text{$\sigma$}\text{\}}\text{,}\text{ }\\
\text{\ \ \ }\,\text{\{}\text{a6}\text{,}\text{ }\text{a4}\text{,}\text{
}\text{a2}\text{\}}\text{ }\text{=}\text{ }\text{\{}\text{1}\text{,}\text{
}\text{-}\,\text{$\lambda$}\,\text{/}\,\text{2}\text{,}\text{ }\text{$\lambda$}\,\text{\textasciicircum
}\,\text{2}\,\text{/}\,\text{24}\text{\}}\text{;}\text{ }\\
\text{\ \ \ \ }\,\text{$\sigma$}\text{ }\text{=}\text{ }\text{a6}\text{
}\text{+}\text{ }\text{a4}\,\text{*}\,\text{$\xi$}\,\text{\textasciicircum
}\,\text{2}\text{ }\text{+}\text{ }\text{a2}\,\text{*}\,\text{$\xi$}\,\text{\textasciicircum
}\,\text{4}\text{;}\text{ }\\
\text{\ \ \ \ }\,\text{Do}\text{[}\text{ak}\text{ }\text{=}\text{
}\text{a6}\text{ }\text{-}\text{ }\text{$\lambda$}\,\text{*}\,\text{(}\text{a2}\,\text{/}\,\text{(}\text{k}\,\text{*}\,\text{(}\text{k}\text{
}\text{-}\text{ }\text{1}\text{)}\text{)}\text{)}\text{;}\text{
}\\
\text{\ \ \ \ \ \ }\,\text{\{}\text{a6}\text{,}\text{ }\text{a4}\text{,}\text{
}\text{a2}\text{\}}\text{ }\text{=}\text{ }\text{\{}\text{a4}\text{,}\text{
}\text{a2}\text{,}\text{ }\text{ak}\text{\}}\text{;}\text{ }\\
\text{\ \ \ \ \ \ }\,\text{$\sigma$}\text{ }\text{=}\text{ }\text{$\sigma$}\text{
}\text{+}\text{ }\text{ak}\,\text{*}\,\text{$\xi$}\,\text{\textasciicircum
}\,\text{k}\text{,}\text{ }\text{\{}\text{k}\text{,}\text{ }\text{6}\text{,}\text{
}\text{n}\text{,}\text{ }\text{2}\text{\}}\text{]}\text{;}\text{
}\text{$\sigma$}\text{]}\end{array}$}




{\ttfamily
$\begin{array}{l}
\text{y}\text{[}\text{n\_}\text{,}\text{ }\text{$\lambda$\_Real}\text{,}\text{
}\text{$\xi$\_}\text{]}\text{ }\text{:=}\text{ }\text{Module}\text{[}\text{\{}\text{a6}\text{,}\text{
}\text{a4}\text{,}\text{ }\text{a2}\text{,}\text{ }\text{ak}\text{,}\text{
}\text{$\sigma$}\text{\}}\text{,}\text{ }\\
\text{\ \ \ }\,\text{\{}\text{a6}\text{,}\text{ }\text{a4}\text{,}\text{
}\text{a2}\text{\}}\text{ }\text{=}\text{ }\text{\{}\text{1}\text{,}\text{
}\text{-}\,\text{$\lambda$}\,\text{/}\,\text{2}\text{,}\text{ }\text{$\lambda$}\,\text{\textasciicircum
}\,\text{2}\,\text{/}\,\text{24}\text{\}}\text{;}\text{ }\\
\text{\ \ \ \ }\,\text{$\sigma$}\text{ }\text{=}\text{ }\text{a6}\text{
}\text{+}\text{ }\text{2}\,\text{*}\,\text{$\xi$}\,\text{*}\,\text{a4}\text{
}\text{+}\text{ }\text{4}\,\text{*}\,\text{$\xi$}\,\text{\textasciicircum
}\,\text{3}\,\text{*}\,\text{a2}\text{;}\text{ }\\
\text{\ \ \ \ }\,\text{Do}\text{[}\text{ak}\text{ }\text{=}\text{
}\text{a6}\text{ }\text{-}\text{ }\text{$\lambda$}\,\text{*}\,\text{(}\text{a2}\,\text{/}\,\text{(}\text{k}\,\text{*}\,\text{(}\text{k}\text{
}\text{-}\text{ }\text{1}\text{)}\text{)}\text{)}\text{;}\text{
}\\
\text{\ \ \ \ \ \ }\,\text{\{}\text{a6}\text{,}\text{ }\text{a4}\text{,}\text{
}\text{a2}\text{\}}\text{ }\text{=}\text{ }\text{\{}\text{a4}\text{,}\text{
}\text{a2}\text{,}\text{ }\text{ak}\text{\}}\text{;}\text{ }\\
\text{\ \ \ \ \ \ }\,\text{$\sigma$}\text{ }\text{=}\text{ }\text{$\sigma$}\text{
}\text{+}\text{ }\text{k}\,\text{*}\,\text{ak}\,\text{*}\,\text{$\xi$}\,\text{\textasciicircum
}\,\text{(}\text{k}\text{ }\text{-}\text{ }\text{1}\text{)}\text{,}\text{
}\text{\{}\text{k}\text{,}\text{ }\text{6}\text{,}\text{ }\text{n}\text{,}\text{
}\text{2}\text{\}}\text{]}\text{;}\text{ }\text{$\sigma$}\text{]}\end{array}$}



Calculating now \texttt{$\lambda$Bounds[16000, 16, }\textit{{\itshape
startingValues}}\texttt{, WorkingPrecision$ \rightarrow\ \ $6000,
AccuracyGoal$ \rightarrow\ \ $600,} \texttt{MaxIterations$ \rightarrow\ \ $100]}
(where \textit{startingValues} has been obtained from a call to
\texttt{$\lambda$Bounds} recursively, one gets in a few minutes
a 1184 digit approximation to the ground state energy of the quartic
anharmonic oscillator. 
\begin{multline*}
\varepsilon _{0}=1.0603620904841828996470460166926635455152087285289779332162452416959435\\
630443444211268962991346717035105462443585825255808798082102931470131768363738\\
249357892262460047081754469601416374884172822569059357577908880617887902636015\\
493956902751961489009429348735844094426948979012139714642909519233545338283470\\
335057576151120257039888523720240221841103086573731091398915453658410311167940\\
583354860009227440069631126702388622971429699610592155832266713769355086736100\\
008318300275179262335739139061361807764985969618149941279280927284070795610604\\
407229468099491362757292738727913689027984247222617169444889547513704380684054\\
391877877295323424587437254317832319060381068741604403437453014684727813918612\\
940470431034013510716071103530089298232754276615189869505650471602527560895262\\
621910256882009644102878156400527052929324050763826502825911247736253847185471\\
440257228543848529745045857097884024906699957047684458770917620291243752732549\\
071164334402302947306923981908956853745359884460160023132919330593958693049166\\
442816339461633242870042614612377430099522342042085977356901535654168503089418\\
513487957341065854797194675964667966134676885864379526545195605682867159583388\\
84743467012042420714919290048732\ldots 
\end{multline*}



Statistical analysis of the number does not show any regularity.



\subsection{Summary}

A power series based approach to the high--precision calculation
of the ground state of the anharmonic oscillator was presented.
{\itshape Mathematica} code to carry out the calculation, as well
as results were given. The method can straightforwardly be used
to calculate ten thousands of digits of the quartic anharmonic,
as well as other anharmonic oscillators. Work concerning the application
of the method to higher states is in progress. 

\ All calculations and visualizations have been carried out in {\itshape
Mathematica} 4.

\subsection{Local Density of States for the Harmonic Oscillator}

The next graphic shows the local density of states ${\mathcal D}_{E}(
x) =\langle x|\theta ( E-\hat{H}) |x\rangle $ for the harmonic oscillator
\cite{Trott00A}. 
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr3.eps}

\end{center}
\caption{The local density of states ${\mathcal D}_{E}( x) =\langle
x|\theta ( E-\hat{H}) |x\rangle $ for the harmonic oscillator.}

\end{figure}

\subsection{Acknowledgments}

The author would like to thank Andr\'e Kuzniarek for making a prerelease
version of the {\itshape Publicon} typesetting system. This work
was supported by Wolfram Research, Inc.\cite{Note1}

\section{ARE BRILLOUIN ZONES OF HIGH ORDER FRACTAL?}

\subsection{Introduction}

Brillouin zones are among the most popular objects a solid state
physicists deals with \cite{{Brillo, Ashcro76, Crackw, Landsb}}.
Despite the fundamental importance for the explanation of most properties
of crystalline solids, Brillouin zones as an own subject have rarely
been investigated (the only ones we are aware of are \cite{{Bieber,
Jones, Skriga, Veerman}}). For electronic properties, mostly the
low order Brillouin zones matter, as an ownstanding subject the
high order Brillouin zones are interesting. Mathematically, the
$(n+1)$th Brillouin zone is the set of points that a line to them
crosses exactly $n$ bisector planes. In computational geometry a
$n$th order Brillouin zones is also called $n$th degree Voronoi
region or \textit{n}th nearest point Voronoi diagrams. The most
important fact for high order Brillouin zones is that their shape
approaches that of a thin spherical shell and their volume is a
constant. Here, for the first time we report on some computational
results of higher order Brillouin zones. All calculations and visualizations
were done with \textit{Mathematica} 4 \cite{Wolfram99}. 

Recursive definition of Brillouin zones: Given a lattice $\Lambda
$ in $\mathbb{R}^{d}$with lattice points $h_{i}$ ($i$ being a multiindex)
the first Brillouin zone ${\mathcal B}{\mathcal Z}_{1}$ is the closure
of the set of all points $x$ such that $|x-0|\leq $$|x-h_{i}|$ for
all $h_{i}${\bfseries $ \neq\ \ $}$0$. The $n$th order Brillouin
zones is the closure of the set of all points $x$ such that $|x-0|\leq
$$|x-h_{i}|$ for all $h_{i}$$ \neq\ \ $$0$ and $x\notin {\mathcal
B}{\mathcal Z}_{n-1}$. 

\subsection{2D Hexagonal lattice}

Figure \ref{XRef-Figure-227122430} shows the first twenty Brillouin
zones of a 2D hexagonal lattice. It is interesting to observe that
the first, third, and fourth Brillouin zones have the shape of an
hexagon. For higher orders the shape becomes much more complicated.
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr4.eps}

\end{center}\label{XRef-Figure-227122430}
\caption{The first twenty Brillouin zones of a 2D hexagonal lattice.}

\end{figure}

Figure \ref{XRef-Figure-227122413} shows the 200th Brillouin zone
in one symmetry unit (inside an angle of $30^{\mbox{}^{\circ}}$).
One sees many small and a few quite large Landsberg zones. The distribution
$p( A) $ of the area of the Landsberg zones in the limit $n\rightarrow
\infty $ might be an interesting subject to study. 
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr5.eps}

\end{center}\label{XRef-Figure-227122413}
\caption{The 200th Brillouin zone in one symmetry unit (inside an
angle of $30^{\mbox{}^{\circ}}$).}

\end{figure}

A good numerical fit to number of faces (line segments) $\sharp
_{n}$ of the $n$th Brillouin zone is $\sharp _{n}\propto n^{1.15}$.
Figure \ref{XRef-Figure-227122355} shows the circumference of the
Brillouin zones normalized to the circumference of a circle with
the same radius. Does the ratio approach a finite value in the limit
$n\rightarrow \infty $? 
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr6.eps}

\end{center}\label{XRef-Figure-227122355}
\caption{The circumference of the Brillouin zones normalized to
the circumference of a circle with the same radius.}

\end{figure}

\subsection{3D Cubic Lattices}

In \cite{Trott00B} we gave a complete implementation for the effective
calculation of higher order Brillouin zones of the three cubic lattice
in $\mathbb{R}^{3}$. Figure \ref{XRef-Figure-227122323} shows the
(outside of) 15th Brillouin zone for the simple cubic, Figure \ref{XRef-Figure-22712238}
shows the 18th for the face--centered cubic and Figure \ref{XRef-Figure-227122254}
shows the 10th for the body--centered cubic lattice. The higher
order Brillouin zones show quite complicated behavior. Large faces
alternate with small ones, relatively plane regions alternate with
quite structured ones. The appearance of the $n+1$th Brillouin zone
is typically completely independent of the appearance of the $n$th
Brillouin zone. 
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr7.eps}

\end{center}\label{XRef-Figure-227122323}
\caption{The (outside of) 15th Brillouin zone for the simple cubic.}

\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr8.eps}

\end{center}\label{XRef-Figure-22712238}
\caption{The (outside of) 18th Brillouin zone for the face--centered
cubic.}

\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr9.eps}

\end{center}\label{XRef-Figure-227122254}
\caption{The (outside of) 10th Brillouin zone for the body--centered
cubic lattice.}

\end{figure}

Using this implementation we analysed various properties of higher
order Brillouin zones. Figure \ref{XRef-Figure-22712228} shows the
area of the Brillouin zones normalized to the area of a sphere of
the same volume. The order of the three curves from the bottom is
sc, fcc, bcc. 
\begin{figure}[h]
\begin{center}
\includegraphics{APS_PhysRevSample.tex_gr10.eps}

\end{center}\label{XRef-Figure-22712228}
\caption{The area of the Brillouin zones normalized to the area
of a sphere of the same volume.}

\end{figure}

The last result to be given here is the number of faces of the Brillouin
zones. By a face we mean any connected planar part of the outside
facing side of a Brillouin zone (point contacts separate faces).
Table \ref{XRef-Table-227122227} gives the results for the first
25 Brillouin zones.
\begin{table}
\caption{The results for the first 25 Brillouin zones.} 
\begin{ruledtabular}

\begin{tabular}{llll}
\textit{n} & \textit{sc} & \textit{fcc} & \textit{bcc}\\
\hline
1 & 6 & 14 & 12\\
2 & 12 & 72 & 48\\
3 & 72 & 96 & 30
\end{tabular}
\end{ruledtabular}\label{XRef-Table-227122227}
\end{table}
\begin{table}
\caption{The results for the first 25 Brillouin zones.} 
\begin{ruledtabular}

\begin{tabular}{ll}
\textit{n} & \textit{sc}\\
\hline
1 & 6
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}
\caption{The results for the first 25 Brillouin zones.} 
\begin{ruledtabular}

\begin{tabular}{llll}
\textit{n} & \textit{sc} & \textit{fcc}\\
\hline
1 & 6 & 14\\
2 & 12 & 72
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}
\caption{The results for the first 25 Brillouin zones.} 
\begin{ruledtabular}

\begin{tabular}{lllll}
\textit{n} & \textit{sc} & \textit{fcc} & \textit{bcc} & \textit{bcc}\\
\hline
1 & 6 & 14 & 12 & 12\\
2 & 12 & 72 & 48 & 48\\
3 & 72 & 96 & 30 & 30
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Summary}

Preliminary results about some computational results about higher
order Brillouin zones have been presented. Further work is in progress
and will be published elsewhere.

All calculations and visualizations have been carried out in \textit{Mathematica}
4. 

\subsection{Acknowledgments}

The author would like to take the opportunity to thank Amy Young
for assistance in typesetting this paper.

\appendix
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