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Preprint version of a paper published in Ferroelectrics 305 (2004) 173-178.
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aperiodic order, model sets, shelling
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\begin{document}
\begin{center}
{\bf\large  Shelling of Homogeneous Media}

UWE GRIMM$^{\,\rm a}_{}$ and MICHAEL BAAKE$^{\,\rm b}_{}$\\[1ex]
$^{\rm a}$ Department of Applied Mathematics, The Open University,\\
Walton Hall, Milton Keynes MK7 6AA, UK.\\[0.5ex]
$^{\rm b}$ Institut f\"{u}r Mathematik,
Universit\"{a}t Greifswald,\\ Jahnstr. 15a,
17487 Greifswald, Germany.
\end{center}

\abstract{%
A homogeneous medium is characterised by a point set in Euclidean
space (for the atomic positions, say), together with some
self-averaging property. Crystals and quasicrystals are homogeneous,
but also many structures with disorder still are.  The corresponding
shelling is concerned with the number of points on shells around an
arbitrary, but fixed centre. For non-periodic point sets, where the
shelling depends on the chosen centre, a more adequate quantity is the
{\em averaged\/} shelling, obtained by averaging over points of the
set as centres. For homogeneous media, such an average is still well
defined, at least almost surely (in the probabilistic sense). Here, we
present a two-step approach for planar model sets.}

\section{\normalsize Introduction}

The discovery of quasicrystals and the challenge to describe their
structure led to a revived interest in model
sets\myref{M1,BM,M2,B}. These sets, also called cut-and-project sets,
are pure point diffractive\myref{M2,B} (under some mild assumptions),
and can be regarded as generalisations of lattices to an aperiodic
setting.  Recently, various combinatorial properties of the
corresponding point sets in Euclidean space have been
investigated\myref{BG1,BG2,BG3}. Among those, which are natural
generalisations of the lattice case, is the shelling problem, which is
considered in this article.

The shelling structure of a point set consists of the number of points
on shells around an arbitrary, but fixed centre. For a lattice
$\varGamma$, the answer does not depend on the centre, as long as it
is in $\varGamma$. However, the corresponding statement is no longer
true for non-periodic point sets. In fact, it might even happen that
no two centres give the same result.  In such cases, a more adequate
quantity is the {\em averaged}\/ shelling, obtained by taking the
average over all points of the set as centres.

This radial distribution function is a characteristic geometric
quantity that reflects itself in the corresponding (powder)
diffraction spectrum and related objects of physical interest. The
underlying combinatorial and algebraic structure is well understood
for periodic crystals, but less so for non-periodic arrangements such
as mathematical quasicrystals or model sets. Here, we concentrate on
the case of planar model sets. In this case, the answer consists of a
universal part that encodes properties of the underlying cyclotomic
number field, and a non-universal part that depends on the details of
the model set construction.

\section{\normalsize Planar Lattices}

Before we consider aperiodic model sets, we briefly summarise the
result for two periodic planar cases with irreducible point symmetry,
the square lattice and the triangular lattice.  This is of course well
known\myref{CS}, but we shall follow an approach that generalises to
the model set case. We start with the square lattice.

To this end, consider the vertex set of the square lattice as a subset
of the complex plane, i.e., as the set of Gaussian integers
$\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}$, which are the integers
in the cyclotomic field $\QQ(i)$.  The number $c(r^2)$ of lattice
points on circles of radius $r$ around the origin is then the number
of solutions of the equation $x\overline{x}=r^{2}$ with
$x\in\mathbb{Z}[i]$, where $\overline{x}$ denotes the complex
conjugate of $x$. Note that, if there are solutions at all, the number
$r^2$ must be an integer. Since $r^2=0$ is trivial (with $c(0)=1$), we
restrict to $r^2>0$ from now on.

To compute the number of solutions, we employ prime factorisation. The
cyclotomic field $\QQ(i)$ has class number one, so prime factorisation
is unique, which means that, up to units, we can uniquely factorise
$r^2$ in terms of primes in $\QQ$ or in $\QQ(i)$. In the complex field
extension from $\QQ$ to $\QQ(i)$, three situations can occur. If a
prime $p\in\mathbb{Z}$ is also a prime in $\mathbb{Z}[i]$, it is
called {\em inert}. If $p\in\mathbb{Z}$ is, up to a unit, the square
of a prime in $\mathbb{Z}[i]$, i.e., $p=\varepsilon \pi^2$ with $\pi$
a prime in $\mathbb{Z}[i]$ and $\varepsilon\in\{\pm i,\pm 1\}$, we say
that $p$ {\em ramifies}. Finally, if $p=\varepsilon\pi\overline{\pi}$
where the primes $\pi,\overline{\pi}\in\mathbb{Z}[i]$ are not related
to each other by a unit, the prime $p$ {\em splits}. The situation for
the Gaussian integers is summarised in Table~\ref{tab1}.  An example
for a splitting prime is $5 = (2+i)(2-i)$.

\begin{table}[b]
\vspace{-3mm}
\caption{Splitting of primes in the field extension from 
$\QQ$ to $\QQ(i)$.\label{tab1}}\vspace{-1ex}
\begin{center}
\begin{tabular}{@{}ccc@{}}
\hline
\multicolumn{1}{c}{\makebox[16ex]{Primes $p$ in 
$\mathbb{Z}$\rule[-1.2ex]{0ex}{4ex}}} &
\multicolumn{1}{c}{\makebox[24ex]{Primes $\pi$ in $\mathbb{Z}[i]$}} & \\
\hline\rule{0ex}{2.8ex}%
$p=2$ & 
$2=-i(1+i)^2=\varepsilon\pi^2$ & ramifies \\
$p\equiv 1\,\bmod 4$ & 
$p=\pi\overline{\pi}$ & {\bf splits}\\
$p\equiv 3\bmod 4$ & 
$p=\pi$ & inert\rule[-1.2ex]{0ex}{1.2ex}\\
\hline
\end{tabular}
\end{center}
\vspace*{-1ex}
\end{table}

Consider the prime factorisation of a positive integer $r^2$ within
$\QQ(i)$, up to units. The number of solutions of the equation
$x\overline{x}=r^{2}$ is obtained by counting in how many ways we can
distribute the prime factors of $r^2$ on $x$ and $\overline{x}$, such
that they are complex conjugates of each other. If $r^2$ contains an
{\em inert}\/ prime factor $p$, we can only do this if it occurs with
an even power, say $p^{2t}$, in which case both $x$ and $\overline{x}$
must contain a factor $p^{t}$. If $r^2$ contains a {\em ramified}\/
prime factor $p^{t}\sim\pi^{2t}$, we also have only one choice; both
$x$ and $\overline{x}$ contain the factor $\pi^{t}$ (up to a unit, to
accommodate complex conjugation). Finally, if $r^2$ is divisible by
the power $p^t=\pi^t{\overline{\pi}}^t$ of a {\em splitting}\/ prime
$p$, we have (t+1) possibilities: $x$ can contain a factor
$\pi^s{\overline{\pi}}^{t-s}$ for $s=0,1,\ldots,t$, with
$\overline{x}$ containing the remaining factor
$\pi^{t-s}{\overline{\pi}}^{s}$. Since we have four units, it is
obvious that solutions come in groups of four.

Consequently, for $r^2>0$, the shelling number $c(r^2)$ vanishes
unless $r^2$ is an integer such that all inert primes factors of $r^2$
occur with even powers, whence
\begin{equation}
c(r^2) \; = \; 4 \prod_{\stackrel{\scriptstyle p\mid r^2}
            {\scriptstyle p \; {\rm splits}}} \!\!\big(t(p)+1\big)\, ,
\label{shell}
\end{equation}
where the product is over all splitting primes that divide $r^2$, and
$t(p)$ is the maximum power such that $p^{t(p)}$ divides $r^2$.  The
actual values can then be derived using the splitting structure as
given in Table~\ref{tab1}. Noting that $a(r^2)=c(r^2)/4$ is a
multiplicative arithmetic function, we can encapsulate the result
neatly in terms of a Dirichlet series generating function, which turns
out\myref{BG3} to be the Dedekind zeta function of the cyclotomic
field $\QQ(i)$.

The situation is very similar for the triangular lattice, which we
consider as the set of Eisenstein integers
$\mathbb{Z}[\xi_3]=\{a+b\xi_3\mid a,b\in\mathbb{Z}\}$, where
$\xi_3=\exp(2\pi i/3)$, hence $1+\xi_3^{}+\xi_3^2=0$. Now, there are
six units, $\xi_{3}^k$ and $\xi_{3}^k(1+\xi^{}_{3})$, $k\in\{0,1,2\}$.
The splitting structure is given in Table~\ref{tab2}; an example of a
splitting prime is $13=(3+2i)(3-2i)$.  The shelling numbers are again
given by equation (\ref{shell}), but with the prefactor $4$ replaced
by $6$.

\begin{table}[h]
\caption{Splitting of primes in the field extension from 
$\QQ$ to $\QQ(\xi_{3})$.\label{tab2}}\vspace{-1ex}
\begin{center}
\begin{tabular}{@{}ccc@{}}
\hline
\multicolumn{1}{c}{\makebox[16ex]{Primes $p$ in 
$\mathbb{Z}$\rule[-1.2ex]{0ex}{4ex}}} &
\multicolumn{1}{c}{\makebox[24ex]{Primes $\pi$ in 
$\mathbb{Z}[\xi_{3}]$}} & \\
\hline\rule{0ex}{2.8ex}%
$p=3$ & 
$3=(1+\xi_{3})(1-\xi_{3})^2=\varepsilon\pi^2$ & ramifies \\
$p\equiv 1\,\bmod 3$ & 
$p=\pi\overline{\pi}$ & {\bf splits}\\
$p\equiv 2\bmod 3$ & 
$p=\pi$ & inert\rule[-1.2ex]{0ex}{1.2ex}\\
\hline
\end{tabular}
\end{center}
\vspace*{-1ex}
\end{table}


\begin{table}[b]
\caption{Splitting of primes in the field extension from 
$\QQ(\tau)$ to $\QQ(\xi)$.\label{tab3}}\vspace{-1ex}
\begin{center}
\begin{tabular}{@{}c|ccc@{}}
\hline
\multicolumn{1}{c|}{\makebox[16ex]{Primes $p$ in 
$\mathbb{Z}$\rule[-1.2ex]{0ex}{4ex}}} &
\multicolumn{1}{c}{\makebox[16ex]{Primes $P$ in $\mathbb{Z}[\tau]$}} &
\multicolumn{1}{c}{\makebox[16ex]{Primes $\pi$ in $\mathbb{Z}[\xi]$}} & \\
\hline\rule{0ex}{2.8ex}%
$p=5$ & $5=(\sqrt{5})^2$ & $\sqrt{5}\sim (1-\xi)^2$ & ramifies\\
$p\equiv 1\,\bmod 5$ & $p=P_{1}P_{2}$ & $P_{i}=\pi_{i}\overline{\pi}_{i}$ &
{\bf splits}\\
$p\equiv \pm 2\bmod 5$ & $p=P$ & $P=\pi$ & inert\\
$p\equiv -1\bmod 5$ & $p=P_{1}P_{2}$ & $P_{i}=\pi_{i}$ &
inert\rule[-1.2ex]{0ex}{1.2ex}\\
\hline
\end{tabular}
\end{center}
\vspace*{-1ex}
\end{table}

\section{\normalsize Shelling of Planar Modules}

In fact, the basic argument generalises\myref{BG1,BG2,BG3} to any
planar module $\mathbb{Z}[\xi_{n}]$ with $\xi_{n}$ a primitive $n$\/th
root of unity, provided that unique prime factorisation holds, which
limits it to the cases where the cyclotomic field $\QQ(\xi_{n})$ has
class number one. This leaves $29$ cases of interest, $n\in\{3, 4, 5,
7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25,\break 27, 28, 32,
33, 35, 36, 40, 44, 45, 48, 60, 84$\}, which correspond to planar
modules with $N$-fold rotational symmetry, where $N=n$ for even $n$,
and $N=2n$ for odd $n$, is the number of units. Apart from the
crystallographic cases $n=3$ and $n=4$, these modules correspond to
dense point sets in the plane. Nevertheless, the shelling numbers for
these point sets are again given by equation (\ref{shell}), with the
prefactor $4$ now replaced by the corresponding value of $N$. Of
course, the appropriate splitting structure of the primes needs to be
used. Note that, for the module, there is again no difference between
central and averaged shelling.

As an explicit example, we consider the case $n=5$, where we may
choose $\xi=\xi_{5}=\exp(2\pi i/5)$ as the primitive root. We now have
three fields that enter; besides $\QQ$ and the cyclotomic field
$\QQ(\xi)$, the third is the maximal real subfield of the latter,
$\QQ(\tau)=\QQ(\sqrt{5})$, where
$\tau=\xi+\overline{\xi}=(1+\sqrt{5})/2$ is the golden number. The
possible squared radii of the circles are now in $\mathbb{Z}[\tau]$
(i.e., they are integers in the field $\QQ(\tau)$), so the splitting
that we have to consider in equation (\ref{shell}) is that from
$\QQ(\tau)$ to $\QQ(\xi)$.  The splitting structure is given in
Table~\ref{tab3}.

As mentioned before, this module corresponds to a dense point set in
the plane.  The model sets of interest\myref{M2,B} are suitable Delone
subsets of modules of this type. Therefore, the shelling problem for
the module determines the {\em maximum}\/ number that may occur on
shells of a given radius, the actual number depending on how the
selection of points takes place.


\section{\normalsize Averaged shelling of model sets}

We consider the example of the vertex set of the T\"{u}bingen triangle
tiling. This set is given by selecting all $x\in\mathbb{Z}[\xi]$ for
which $x^{\star}$, the image under the map $\star:\; \xi\mapsto \xi^2$
which is a Galois automorphism of $\QQ(\xi)$, falls into a domain $W$
which is a regular decagon of edge length $\tau/\sqrt{2+\tau}$, i.e.,
$\varLambda=\{x\in\mathbb{Z}[\xi]\mid x^{\star}\in W\}$. This choice
corresponds to a binary triangle tiling with edge lengths $1/\tau=\tau
-1$ and $1$.

To calculate the averaged shelling, we use Weyl's theorem of uniform
dis\-tri\-bution\myref{M3}. 
%
%
\begin{table}[hb]
\caption{Averaged shelling for the T\"{u}bingen triangle 
tiling.\label{tab4}}\vspace{-1ex}
\begin{center}
\begin{tabular}{@{}cccccr@{}c@{}l@{}}
\hline\rule[-1.2ex]{0ex}{4ex}
$r^2$ & representative & orbit & $t^2=\sigma(r^2)$ & type &
\multicolumn{3}{c}{av.\ shelling} \\
\hline\rule{0ex}{2.4ex}%
$2-\tau$  & $\xi+\overline{\xi}$       & $10$ & $1+\tau$  & $1$ & 
$6$&$-$&$2\tau$ \\
$1$       & $1$                   & $10$ & $1$       & $1$ & 
$28$&$-$&$14\tau$ \\
$3-\tau$  & $\xi^2-{\overline{\xi}^2}$   & $10$ & $2+\tau$  & $2$ & 
$-50$&$+$&$32\tau$ \\
$5-2\tau$ & $2+\xi^2$             & $20$ & $3+2\tau$ & $3$ & 
$-32$&$+$&$20\tau$ \\
$4-\tau$  & $1-\xi-{\overline{\xi}^2}$   & $20$ & $3+\tau$  & $3$ & 
$112$&$-$&$68\tau$ \\
$1+\tau$  & $\xi^2+{\overline{\xi}^2}$   & $10$ & $2-\tau$  & $1$ & 
$20$&$-$&$8\tau$ \\
$2+\tau$  & $\xi-\overline{\xi}$       & $10$ & $3-\tau$  & $2$ & 
$-44$&$+$&$30\tau$ \\
$4$       & $2$                   & $10$ & $4$       & $1$ & 
$-18$&$+$&$12\tau$ \\
$3+\tau$  & $1-\xi-\xi^2$         & $20$ & $4-\tau$  & $3$ & 
$32$&$-$&$16\tau$ \\
$5$       & $1+2\xi+2\overline{\xi}$   & $10$ & $5$       & $1$ & 
$4$&$-$&$2\tau$ \\
$3+2\tau$ & $2+\xi$               & $20$ & $5-2\tau$ & $3$ & 
$-264$&$+$&$168\tau$ \\
$2+3\tau$ & $1-\xi^2-\overline{\xi}^2$ & $10$ & $5-3\tau$ & $1$ & 
$102$&$-$&$58\tau$ \\
$6+\tau$  & $1-2\xi-\xi^2$        & $20$ & $7-\tau$  & $3$ & 
$260$&$-$&$160\tau$ \\
$5+2\tau$ & $2-\xi^2$             & $20$ & $7-2\tau$ & $3$ & 
$288$&$-$&$176\tau$ \\
$7+\tau$  & $3+\xi+\xi^2$         & $20$ & $8-\tau$  & $3$ & 
$-168$&$+$&$104\tau$% 
\rule[-0.8ex]{0ex}{0.8ex} \\
\hline
\end{tabular}
\end{center}
\vspace*{-1ex}
\end{table}
%
%
The frequency of a difference $x\in\varLambda-\varLambda$ is given by
the the relative overlap area, or covariogram, of the window $W$ and a
copy shifted by $x^{\star}$. Due to the dihedral $D_{10}$ symmetry of
the window, the result is the same for the entire $D_{10}$ orbit of
$x^{\star}$, thus it is sufficient to consider one representative, and
multiply the covariogram by the length of the orbit.  The result for
the T\"{u}bingen triangle tiling is given in Table~\ref{tab4}, which
contains all possible radii $r\le 3$, completing and extending a
previously published table\myref{BG3} where one possible radius was
missed. Here, $t^2=\sigma(r^2)=\sigma(x\overline{x})$ is the squared
length of $x^{\star}$, and $\sigma$ is algebraic conjugation in the
field $\QQ(\tau)$, which maps $\sqrt{5}\mapsto -\sqrt{5}$, hence
$\sigma(\tau)=-1/\tau=1-\tau$.

The averaged shelling numbers are in $\mathbb{Z}[\tau]$ (and probably
even in $2\mathbb{Z}[\tau]$), which can be understood from topological
properties of the tiling\myref{FHK,G} and its symmetry. The frequency
module of the tiling, which is the integer span of the frequencies of
all finite clusters, is constrained by the existence of topological
invariants\myref{FHK,G}. Here\myref{G}, it is
$\frac{1}{10}\mathbb{Z}[\tau]$. As all averaged shelling numbers are
finite sums with integer coefficients and weights from the frequency
module, the restriction is inherited.

Explicit results for the averaged shelling numbers were also obtained
for the eightfold symmetric Ammann-Beenker model set\myref{BG1} and
for the vertex set of the twelvefold symmetric shield
tiling\myref{BG2}.  These point sets are obtained from the modules
$\mathbb{Z}[\exp(2\pi i/8)]$ and $\mathbb{Z}[\exp(2\pi i/12)]$, with a
regular octagon and a regular dodecagon as windows, respectively.  In
the natural choice of length scales, the Ammann-Beenker tiling
contains squares and $45$ degree rhombi of egde length one. The shield
tiling is made up of equilateral triangles, squares and
`shield'-shaped hexagons, all of the same edge length
$\sqrt{2-\sqrt{3}}$.  

\begin{table}[h]
\caption{Averaged shelling for planar tilings.\label{tab5}}\vspace{-1ex}
\begin{center}
\begin{tabular}{@{}cccccc@{}}
\hline\rule[-1.2ex]{0ex}{4ex}
$r$     & square & triangular  & Ammann-%
                                 Beenker & T\"{u}bingen & shield  \\
\hline\rule{0ex}{2.4ex}%
$0.518$ &         &            &         &              & $4.536$ \\[-0.51ex]
$0.618$ &         &            &         & $2.764$      &         \\[-0.51ex]
$0.732$ &         &            &         &              & $2.000$ \\[-0.51ex]
$0.765$ &         &            & $1.172$ &              &         \\[-0.51ex]
$0.897$ &         &            &         &              & $0.536$ \\[-0.51ex]
$1.000$ & $4.000$ & $6.000$    & $4.000$ & $5.348$      & $8.000$ \\[-0.51ex]
$1.176$ &         &            &         & $1.777$      &         \\[-0.51ex]
$1.239$ &         &            &         &              & $3.072$ \\[-0.51ex]
$1.328$ &         &            &         & $0.361$      &         \\[-0.51ex]
$1.414$ & $4.000$ &            & $2.485$ &              & $6.431$ \\[-0.51ex]
$1.506$ &         &            &         &              & $6.000$ \\[-0.51ex]
$1.543$ &         &            &         & $1.974$      &         \\[-0.51ex]
$1.618$ &         &            &         & $7.056$      &         \\[-0.51ex]
$1.674$ &         &            &         &              & $0.210$ \\[-0.51ex]
$1.732$ &         & $6.000$    & $3.029$ &              & $5.238$ \\[-0.51ex]
$1.848$ &         &            & $4.887$ &              &         \\[-0.51ex]
$1.880$ &         &            &         &              & $1.525$ \\[-0.51ex]
$1.902$ &         &            &         & $4.541$      &         \\[-0.51ex]
$1.932$ &         &            &         &              & $9.895$ \\[-0.51ex]
$2.000$ & $4.000$ & $6.000$    & $0.828$ & $1.416$      & $4.309$%   
\rule[-0.8ex]{0ex}{0.8ex} \\
\hline
\end{tabular}
\end{center}
\vspace*{-1ex}
\end{table}

For the Ammann-Beenker tiling and the shield tiling, the averaged
shelling numbers lie in index-$2$ submodules of $\mathbb{Z}[\sqrt{2}]$
and $\frac{1}{2}\mathbb{Z}[1/\sqrt{3}]$, respectively. Details will be
given elsewhere\myref{BGG}. To give a rough impression of the
differences between these tilings, their averaged shelling numbers for
radii $r\le 2$ are compared in Table~\ref{tab5}. Clearly, an
increasing number of radii with small occupation, which are absent in
the lattice case, appear as the value of $N$ increases. The growth of
the number of possible radii reflects the increasing local complexity
of the tiling.  Of course, this comparison is somewhat arbitrary, as
we might have scaled the tilings differently, for instance such that
the shortest distance is the same in all cases. Nevertheless, it
becomes evident that nonperiodic systems tend to have a larger number
of occupied shells, as expected.

An analogous approach is possible for other combinatorial quantities,
such as averaged coordination numbers\myref{BGG}.

\rule{0.cm}{0.3cm}

\noindent {\bf Acknowledgments} 
The authors gratefully acknowledge financial support by The Royal
Society (UG) and by Deutsche Forschungsgemeinschaft (MB).

\small
\begin{thebibliography}{99}

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    ed.\ Trebin, H.-R.,
    Wiley-VCH, Berlin, pp.~160--171 (2003).
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\bibitem{BGG}
    Baake, M., G\"{a}hler, F., Grimm, U., in preparation.
\end{thebibliography}

\end{document}
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