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\begin{center}
{\Large\textbf{Coloured Plane Groups}}
\footnote{0567-7394/80/060884-05\$01.00\quad
\copyright 1980 International Union of Crystallography}
\end{center}
\begin{center}
\textsc{By J.D. Jarratt} \\
\textit{Pure Mathematics Department, University of Sydney,
Sydney, Australia}\\
\ \\
\textsc{and R.R.E. Schwarzenberger}
\ \\
\textit{Science Education Department, University of Warwick,
Coventry, England}
\\ \ \\
\textit{(Received 9 August 1979; accepted 4 June 1980)}
\end{center}
\section*{Abstract}
The 46 black and white plane groups are well known.
The corresponding colour groups with more than two
colours are extremely numerous. We give a listing of
the 935 groups with $N$ colours for $N$ lying between 2
and 15 inclusive.
\section{Introduction}
Consider an $n$-dimensional space group $G$ whose
elements permute $N$ colours transitively and let $G_1$ be
the subgroup keeping the first colour fixed. Then the
index of $G_1$ in $G$ is $N$ and the colours correspond
naturally to the cosets. The effect of any member of $G$
under group multiplication on the cosets is the same as
its effect on the colours. For these reasons a \textit{coloured
space group with $N$ colours} is defined to be a pair $G\supset
G_1$ consisting of a space group $G$ and a subgroup $G_1$
which is also a space group, of index $N$. The pairs $G\supset
G_1$ and $G'\supset G_1'$ are equivalent if there is an
isomorphism between $G$ and $G'$ which maps $G_1$ onto
$G_1'$. This implies that there is actually an affine
transformation $f$ such that $G' = fGf^{-1}$ and $G_1' =
fG_1f^{-1}$. If $N = 2$ then $G\supset G_1$ is called a \textit{black and
white group}. These definitions go back to Heesch and
Shubnikov; they can be found (in slightly different
form) in the paper of van der Waerden \& Burckhardt
(1961).
There are two well known books by Shubnikov \&
Belov (1964) and by Loeb (1971) which discuss the
case $N = 2$ in detail and which include coloured
pictures iliustrating various coloured place groups.
Both give complete descriptions of the 46 black and
white groups but for $N = 2$ the listings begun in these
books are far from complete. In this paper we describe
a method for obtaining a complete listing for any given
value of $N$ and give explicit results for all $N$ up to 15.
For further remarks on the significance of coloured
groups in the enumeration of space, groups and in the
study of twinning we refer to Schwarzenberger (1980).
Other recent work on coloured space groups by
Senechal (1975) and Harker (1976) has led to the
development of arithmetic algorithms for the determination
of coloured space groups. Senschal (1979)
uses such an algorithm to count coloured plane groups
for various values of $N$. When $N$ is prime she shows
that the number is 14, 15, 13, 16 when $N = 5$, 7, 11, 1
modulo 12 in agreement with our computations; when
$N$ is composite there were some discrepancies between
our preliminary results even for $N = 4$. Meanwhile,
Wieting (1980) has developed two quite different
methods of computation: one using generators and
relations for $N\le 5$, and the other using pairs of plane
ornamental groups for $N\le 60$. Comparison of the
preliminary results both of Senechal and of ourselves
with the results of Wieting made us aware of several
errors which have been corrected in the present version.
We are grateful to Senechal and Wieting for their
generous cooperation but accept sole responsibility for
any errors which remain.
\section{The Hermann decomposition}
Let $G\supset G_1$ be a coloured group and $T\supset T_1$ the
corresponding pair of lattices. These groups yield point
groups $H = G/T$ and $H_1 = G_1/T_1$ with homomorphisms
\[
\begin{matrix}
0 & \rightarrow & T & \rightarrow & G & \overset{\rightarrow}{p}
&H&\rightarrow&1\\
& & \uparrow & & \uparrow & & \uparrow \\
0 & \rightarrow &T_1& \rightarrow &G_1&\rightarrow&H_1&\rightarrow&1\\
\end{matrix}
\]
where vertical arrows denote embeddings of subgroups.
Following Hermann (1929) we call the
coloured group $G\supset G_1$ \textit{lattice equivalent} if $T = T_1$ and
\textit{class equivalent} if $H = H_1$. The main result, which is
due to Hermann and holds for arbitrary dimension $n$,
is:
\textit{Theorem.} Any coloured group $G\supset G_1$ can be
expressed uniquely as the composition of a lattice
equivalent coloured group $G\supset G'$ and a class
equivalent coloured group $G'\supset G_1$.
\textit{Proof.} A subgroup $G'$ of $G$ satisfies the required
conditions if and only if it has lattice $T$ and $p(G') = H_1$.
There is one and only one subgroup with these
properties, namely $G' = p^{-1}(H_1)$. The pair $G\supset G'$ is
then lattice equivalent of index $r$ and the pair $G'\supset G_1$ is
class equivalent of index $s$ where $r\times s = N$.
\textit{Remark.} If two coloured groups are equivalent then
so are their lattice equivalent and class equivalent parts.
The converse is not true (see \S 4).
Table 1 lists the numbers of distinct coloured plane
groups corresponding to various factorizations $N = r \times
s$ for $n = 2$. An indication of the method used to obtain
these results is given in \S\S 3 and 4.
\section{The lattice equivalent and class equivalent cases}
The lattice equivalent coloured groups $G\supset G'$ are finite
in number for given dimension $n$. To obtain the list for
$n = 2$ it is sufficient to consider the possible pairs $H\supset
H_1$ of point groups. For completeness, and for use in
\S 4, we list the 52 lattice equivalent coloured plane
groups explicitly in Table 2. The class equivalent
coloured groups $G'\supset G_1$ depend on the possible pairs
$T\supset T_1$ of lattices. We consider these according to the
Bravais type of $T$; the number which occur is infinite
but is finite for given $s$. The groups $G'\supset G_1$ which
occur depend on the choice of integers $p$, $q$; in the
tabulations which follow, the symbol for $G_1$ is placed
below the symbol for $G'$ to indicate existence of the
corresponding coloured group $G'\supset G_1$.
\begin{center}
Table 1. \textit{Number of coloured plane groups with}\\
\textit{$N = r\times s$ colours for $N = 2,\dots,15$}
\end{center}
\[
\begin{array}{rlccclr}
& \text{Lattice} & & & & \text{Class} & \\
N&\text{equivalent}&&\text{Other}&&\text{equivalent}&\text{Total}\\
2&(2\times1)\ \ 26&&&&(1\times2)\ \ 20&46 \\
3&(3\times1)\ \ \ 5&&&&(1\times3)\ \ 18&23 \\
4&(4\times1)\ \ 12&&(2\times2)\ 46&&(1\times4)\ \ 38&96 \\
5&&&&&(1\times5)\ \ 14&14 \\
6&(6\times1)\ \ \ 6&(3\times2)\ 11&&(2\times3)\ \ 44&(1\times6)\ \ 29&90 \\
7&&&&&(1\times7)\ \ 15&15 \\
8&(8\times1)\ \ \ 2&(4\times2)\ 26&&(2\times4)\ \ 98&(1\times8)\ \ 44&170 \\
9&&&(3\times3)\ 10&&(1\times9)\ \ 30&40 \\
10&&&&(2\times5)\ 45&(1\times10)\ 30&75 \\
11&&&&&(1\times11)\ 13&13 \\
12&(12\times1)\ \ 1&(6\times2)\ \ 9&&(2\times6)\ 98&(1\times12)\ 58&221 \\
& &(4\times3)\ 25&&(3\times4)\ 30&& \\
13&&&&&(1\times13)\ 16&16 \\
14&&&&(2\times7)\ 53&(1\times14)\ 29&82 \\
15&&&&(3\times5)\ 10&(1\times15)\ 24&34 \\
\ \\
\text{Total}&\multicolumn{1}{c}{52}&&\multicolumn{1}{c}{505}&&
\multicolumn{1}{c}{378}&935
\end{array}
\]
\begin{center}
Table 2. \textit{The 52 lattice equivalent groups $G\supset G'$}
\end{center}
{\tiny
\[
\begin{array}{lllllllllll}
r=2 &G &P2 &Pm &Pg &Cm &Pmm&Pmg &Pgg &Cmm &P4 \\
&G'&P1 &P1 &P1 &P1 &P2 &P2 &P2 &P2 &P2 \\
&G &Pmm &Pmg &Pmg &Pgg &P4mm&P4gm&Cmm &P4mm&P4gm\\
&G'&Pm &Pm &Pg &Pg &Pmm &Pgg &Cm &Cmm &Cmm \\
&G &P4mm&P4gm&P31m&P3m1&P6 &P6mm&P6mm&P6mm& \\
&G'&P4 &P4 &P3 &P3 &P3 &P31m&P3m1&P6 & \\
r=3 &G &P3 &P6 &P31m&P3m1&P6mm& & & & \\
&G'&P1 &P2 &Cm &Cm &Cmm & & & & \\
r=4 &G &Pmm &Pmg &Pgg &Cmm &P4 &P4mm&P4gm& & \\
&G'&P1 &P1 &P1 &P1 &P1 &P2 &P2 & & \\
&G &P4mm&P4gm&P4mm&P4gm&P6mm& & & & \\
&G'&Pm &Pg &Cm &Cm &P3 & & & \\
r=6 &G &P31m&P3m1&P6 &P6mm&P6mm&P6mm& & & \\
&G'&P1 &P1 &P1 &P2 &Cm1 &C1m & & & \\
r=8,\ 12&G &P4mm&P4gm&P6mm& & & & & & \\
&G'&P1 &P1 &P1 & & & & & &
\end{array}
\]
}
\renewcommand{\thesubsection}{(\roman{subsection})}
\subsection{\textit{$T = P$ parallelogram}}
Each sublattice $P_1$ is determined by the highest
common factor $d = \text{h.c.f.}(p,q)$ of a factorization $s =
pq$. For $s\le 15$ the possible values of $d$ are 1 (for all $s$),
2 (for $s = 4$, 8, 12) and 3 (for $s = 9$). Each value of a
gives two coloured groups:
\begin{align*}
G' &= P1\ \ \ \ P2 \\
G_1 & =P_11\ \ \ P_12
\end{align*}
\subsection{\textit{$T = P$ rectangle (primitive orthogonal)}}
Each primitive sublattice $P_1$ compatible with reflections
is determined by a factorization $s = pq$ and
generators $(p,0)$, $(0q)$ with respect to orthogonal
coordinates. If $p \ne q$ there are eight coloured groups
whereas if $p = q$ there are five because of the
equivalences marked $\sim$:
{\scriptsize
\[
\hspace{-1mm}\begin{array}{rllllllllllll}
&p,q\ \text{odd}&Pm1&&P1m&Pg1&&P1g&Pgm&&Pgm&Pmm&Pmg \\
G'=&p,q\ \text{even}&Pm1&&P1m&Pm1&&P1m&Pmm&&Pmm&Pmm&Pmm\\
&p-q\ \text{odd}&Pm1&&P1m&Pm1&&P1g&Pmm&&Pmg&Pmm&Pmg\\
G_1=&&P_1m&\sim&P_1m&P_1g&\sim&P_1g&P_1gm&\sim&P_1mg&P_1mm&P_1gg
\end{array}
\]
}
Similarly each centred sublattice $C_1$ compatible with
reflections is determined by a factorization $s = 2pq$ and
generators $(2p,0)$, $(0,2q)$, $(p,q)$. If $p \ne q$ there are three
coloured groups reducing to two if $p = q$:
\begin{align*}
G' &= Pm1\ \ \ \,P1m\ \ Pmm \\
G_1 &= C_1m \sim C_1m\ \ C_1mm
\end{align*}
\subsection{\textit{$T = C$ diamond (centred orthogonal)}}
Each primitive sublattice $P_1$ compatible with relections
is determined by a factorization $s = 2pq$ and
generators $(p,0)$, $(0,q)$. If $p \ne q$ there are eight coloured
groups reducing to five if $p = q$:
\begin{align*}
G' &= Cm1\ \ \ C1m\ \ Cm1\ \ C1m\ \ Cmm\ \ \ Cmm\ Cmm\ Cmm \\
G_1 &= P_1m \sim P_1m\ \ P_1g \sim P_1g\ \ \ P_1mg \sim P_1gm\
P_1mm\ P_1gg.
\end{align*}
Similarly each centred subiattice $C_1$ compatible with
reflections is determined by a factorization $s = pq$
where $p$, $q$ have the same parity and generators $(p,0)$.
$(0,q)$, $(\frac{1}{2}p,\frac{1}{2}q)$. If $p \ne q$ there
are three coloured groups reducing to two if $p = q$:
\begin{align*}
G' &= Cm1\ \ \ \,C1m\ \ Cmm \\
G_1 &= C_1m \sim C_1m\ \ C_1mm
\end{align*}
\subsection{\textit{$T = P$ square}}
The possible sublattices invariant under rotations of
order 4 are
\medskip
\noindent $P_1$ with generators $(p,0)$, $(0,p)$ and $s = p^2$
\medskip
\noindent $P_1$ with generators $(2p,0)$, $(0,2p)$, $(p,p)$ and $s = 2p^2$
\medskip
\noindent $P_1$ with generators $(p,q)$, $(-q,p)$ and $s=p^2+q^2$, $p\ne q$
\medskip
\noindent of which only the first two are invariant also under
reflections. In the range $2 \le s \le 15$ the coloured groups
which arise are:
\[
\begin{array}{rcllll}
G' &=&P4 & P4mm & P4mm & P4gm \\
G_1&=&P_14 & P_14mm & P_14gm & P_14gm \\
s &=&2,4,5,8, & 2,4,8,9& 2,4,8 & 9 \\
& &9,10,13
\end{array}
\]
\subsection{\textit{$T = P$ hexagonal}}
The possible sublattices invariant under rotations of
order 3 or 6 are (with generators now expressed relative
to inclined axes).
Finally, counting up the various ways of expressing
$s = 2$,\dots, 15 in terms of $p$, $q$ we obtain the 378
class equivalent coloured plane groups enumerated in
Table 3.
\medskip
\noindent $P_1$ with generators $(p,0)$, $(0,p)$ and $s = p^2$
\medskip
\noindent $P_1$ with generators $(3p,0)$, $(0,3p)$, $(p,p)$ and $s = 3p^2$
\medskip
\noindent $P_1$ with generators $(p,q)$, $(-q,p+q)$ and
$s=p^2+pq+q^2$, $p\ne q$
\medskip
\noindent of which only the first two are invariant under reflections.
In the range $2 \le s \le 15$ the coloured groups which arise are:
\[
\begin{array}{rclllllll}
G'&=&P3 &P31m &P31m &P3m1 &P3m1 &P6 &P6mm \\
G &=&P_13 &P_131m&P_13m1&P_13m1&P_131m&P_16 &P_16mm \\
s &=&3,4,7,9,&4,9 &3,12 &4,9 &3,12 &3,4,7,9,&3,4,9,12 \\
& &12,13 & & & & &12,13 &
\end{array}
\]
\begin{center}
Table 3. \textit{Number of class equivalent groups $G'\supset G_1$ with
$s$ colours arranged according to Bravais type}
\end{center}
{\tiny
\[
\begin{array}{llcrrrrrrrrrrrrrr}
&G &T_1&s=2& 3& 4& 5& 6& 7& 8& 9&10&11&12&13&14&15\\
\ \\
(i)&P1^* & & 1& 1& 2& 1& 1& 1& 2& 2& 1& 1& 2& 1& 1& 1\\
&P2^* & & 1& 1& 2& 1& 1& 1& 2& 2& 1& 1& 2& 1& 1& 1\\
(ii)&Pm &P_1& 3& 2& 5& 2& 6& 2& 7& 3& 6& 2&10& 2& 6& 4\\
& &C_1& 1& -& 2& -& 2& -& 3& -& 2& -& 4& -& 2& -\\
&Pmm^* &P_1& 2& 1& 5& 1& 4& 1& 6& 2& 4& 1& 8& 1& 4& 2\\
& &C_1& 1& -& 1& -& 1& -& 2& -& 1& -& 2& -& 1& -\\
&Pg & & 1& 2& 1& 2& 2& 2& 1& 3& 2& 2& 2& 2& 2& 4\\
&Pmg & & 2& 2& 2& 2& 4& 2& 2& 3& 4& 2& 4& 2& 4& 4\\
&Pgg & & -& 1& -& 1& -& 1& -& 2& -& 1& -& 1& -& 2\\
(iii)&Cm &P_1& 2& -& 4& -& 4& -& 6& -& 4& -& 8& -& 4& -\\
& &C_1& -& 2& 1& 2& -& 2& 2& 3& -& 2& 2& 2& -& 4\\
&Cmm^* &P_1& 3& -& 4& -& 4& -& 7& -& 4& -& 8& -& 4& -\\
& &C_1& -& 1& 1& 1& -& 1& 1& 2& -& 1& 1& 1& -& 2\\
(iv)&P4 & & 1& -& 1& 1& -& -& 1& 1& 1& -& -& 1& -& -\\
&P4mm & & 2& -& 2& -& -& -& 2& 1& -& -& -& -& -& -\\
&P4gm & & -& -& -& -& -& -& -& 1& -& -& -& -& -& -\\
(v)&P3^* & & -& 1& 1& -& -& 1& -& 1& -& -& 1& 1& -& -\\
&P6 & & -& 1& 1& -& -& 1& -& 1& -& -& 1& 1& -& -\\
&P31m & & -& 1& 1& -& -& -& -& 1& -& -& 1& -& -& -\\
&P3m1^*& & -& 1& 1& -& -& -& -& 1& -& -& 1& -& -& -\\
&P6mm & & -& 1& 1& -& -& -& -& 1& -& -& 1& -& -& -\\
\text{Total} &&& 20&18&38&14&29&15&44&30&30&13&58&16&29&24
\end{array}
\]
\begin{center}
$^*$Consult Table 4 before using this information in conjunction with
Table 2.
\end{center}
}
\section{The mixed case}
Previous sections have dealt with all coloured plane
groups arising from factorizations $N = r \times s$ with $r = 1$
(class equivalent, $s \le 15$) or $s = 1$ (lattice equivalent).
Unless $N$ is prime there are further coloured groups to
be obtained from a careful comparison of Table 2 and
Table 3. Thus, for each of the 52 lattice equivalent
coloured groups $G\supset G'$ listed in Table 2, we determine
from Table 3 the number of possible class equivalent
groups $G'\supset G_1$ so as to obtain the total number of
compositions $G \supset G' \supset G_1$). There is one difficulty: it
may be necessary to distinguish between subgroups $G_1$
of $G'$ which, although equivalent in $G'$, are not
equivalent in $G$ (that is, the coloured groups $G'\supset G_1$
are equivalent although the compositions $G\supset G_1$ are
not). An example may clarify this phenomenon.
Consider the lattice equivalent coloured group $P6\supset P2$
of index 3 from Table 2 and the unique class equivalent
coloured group $P2\supset P_12$ of index 2 from Table 3. The
group P6 defines centres of rotation of order six (6
centres) and, between these, centres of rotation of order
two (2 centres). Which of these occur as 2 centres for
the group $P_12$? Either a mixture of 6 centres and 2
centres of $P6$ or else only 2 centres of P6, giving two
distinct coloured groups $P6\supset P_12$. Note that it is not
possible for the 2 centres of $P_12$ to consist only of 6
centres of $P6$ (although this can happen when $P2\supset P_12$
is of index 4 instead of index 2). In this way we obtain
the entry 2 in the column of Table 4 for $s = 2$. Other
entries of Table 4 are obtained by a similar argument.
By using Tables 2, 3 and 4 together we obtain the
figures in the middle column of Table 1.
For any of the seventeen plane groups G we may,
using Tables 2, 3 and 4, find the total number of
coloured groups $G\supset G_1$ of index $N = 2$,\dots, 15. The
results are listed in Table 5.
\begin{center}
Table 4. \textit{Coloured groups $G'\supset G_1$ giving inequivalent
compositions $G\supset G'\supset G_1$ for $s=2$,\dots, 7}
\end{center}
{\small
\[
\begin{array}{lcrclrrrrrr}
\multicolumn{1}{c}{G}&r&G'&\supset&G_1&s=2&3&4&5&6&7\\
\ \\
Pm,Pg,Pmg&2,4&&&&3&3&6&4&9&5\\
Pmm,Pgg,Cmm,&4,8&&&&2&2&4&3&5&3\\
P4,P4mm,P4gm&&&&&&&&&&\\
Cm&2&P1&\supset&P_11&2&3&5&4&7&5\\
P3,P31m,P3m1,&3,6,12&&&&1&2&3&2&3&2\\
P6,P6mm&&&&&\\
\ \\
Pmm,Pmg &2 & & & &2&2&4&3&5&3\\
Pmg &2 & & & &3&3&6&4&9&5\\
Cmm,P4,P4mm,P4gm &2,4 &P2 &\supset&P_12 &3&2&6&3&7&3\\
P6,P6mm &3,6 & & & &2&2&7&2&6&2\\
& &Pmm &\supset&P_1mm &1&1&3&1&2&1\\
P4mm &2 &Pmm &\supset&P_1mg &1&0&3&0&2&0\\
& &Pmm &\supset&P_1gg &0&0&2&0&0&0\\
& &Pmm &\supset&C_1mm &2&0&3&0&3&0\\
\ \\
& &Cmm &\supset&P_1mm &1&0&2&0&2&0\\
P6mm &3 &Cmm &\supset&P_1mg &2&0&4&0&4&0\\
& &Cmm &\supset&P_1gg &1&0&2&0&2&0\\
& &Cmm &\supset&C_1mm &0&2&2&2&0&2\\
P6,P31m,P6mm &2,4 &P3 &\supset&P_13 &0&2&2&0&0&2\\
P6mm &2 &P3m1&\supset&P_131m &0&2&0&0&0&0\\
& &P3m1&\supset&P_13m1 &0&0&2&0&0&0
\end{array}
\]
}
\begin{center}
Table 5. \textit{Number of coloured groups $G\supset G_1$ with $N$
colours corresponding to each plane group $G$ for $N=2$,\dots, 15}
\end{center}
\[
\begin{array}{lrrrrrrrrrrrrrr}
\ \ \ G &N=2& 3& 4& 5& 6& 7& 8& 9&10&11&12&13&14&15\\
\ \\
P1 & 1& 1& 2& 1& 1& 1& 2& 2& 1& 1& 2& 1& 1& 1\\
P2 & 2& 1& 3& 1& 2& 1& 4& 2& 2& 1& 3& 1& 2& 1\\
Pm & 5& 2&10& 2&11& 2&16& 3&12& 2&23& 2&13& 4\\
Pmm & 5& 1&13& 1& 9& 1&21& 2&10& 1&25& 1&10& 2\\
Pg & 2& 2& 4& 2& 5& 2& 7& 3& 6& 2&11& 2& 7& 4\\
Pmg & 5& 2&11& 2&11& 2&19& 3&12& 2&26& 2&13& 4\\
Pgg & 2& 1& 4& 1& 4& 1& 7& 2& 5& 1& 9& 1& 5& 2\\
Cm & 3& 2& 7& 2& 7& 2&13& 3& 8& 2&17& 2& 9& 4\\
Cmm & 5& 1&11& 1& 8& 1&21& 2& 9& 1&22& 1& 9& 2\\
P4 & 2& 0& 5& 1& 2& 0& 9& 1& 4& 0& 9& 1& 3& 0\\
P4mm & 5& 0&13& 0& 2& 0&29& 1& 3& 0&17& 0& 2& 0\\
P4gm & 3& 0& 7& 0& 2& 0&13& 1& 3& 0&10& 0& 2& 0\\
P3 & 0& 2& 1& 0& 1& 1& 0& 3& 0& 0& 4& 1& 0& 2\\
P6 & 1& 2& 1& 0& 5& 1& 2& 3& 0& 0& 9& 1& 2& 2\\
P31m & 1& 2& 1& 0& 5& 0& 2& 3& 0& 0& 7& 0& 2& 2\\
P3m1 & 1& 2& 1& 0& 4& 0& 1& 3& 0& 0& 7& 0& 1& 2\\
P6mm & 3& 2& 2& 0&11& 0& 4& 3& 0& 0&20& 0& 1& 2\\
\ \\
\text{Total}&46&23&96&14&90&15&170&40&75&13&221&16&82&34
\end{array}
\]
\section*{References}
% Set up for reference list
\newcommand{\hi}{\par\noindent\hangindent=1em}
\hi
\textsc{Barker, D.} (1976). \textit{Acta Cryst.}\ \textbf{A32}, 133--139.
\hi
\textsc{Hermann, C.} (1929). \textit{Z.\ Kristallogr.}\ \textbf{69}, 533.
\hi
\textsc{Loeb, A. L.} (1971). \textit{Color and Symmetry}. New York:
John Wiley.
\hi
\textsc{Schwarzenberger, R. L. E.} (1980). \textit{$N$-Dimensional
Crystallography}. London: Pitman.
\hi
\textsc{Senechal, M.} (1975). \textit{Z.\ Kristallogr.}\ \textbf{142},
1--23.
\hi
\textsc{Senechal, M.} (1979). \textit{Discrete Appl.\ Math.}\ \
textbf{1}, 51--73.
\hi
\textsc{Shubnikov, A. V. \& Belov}, N. V. (1964), \textit{Colored
Symmetry}. Oxford: Pergamon Press.
\hi
\textsc{Waerden, B. L. van der \& Burckhardt, J. J.} (1965).
\textit{Z.\ Kristallogr.} \textbf{115}, 231--234.
\hi
\textsc{Wieting, T.} (1980). To be published.
\bigskip\bigskip
\noindent From: \textit{Acta Cryst.} (1980) \textbf{A36}, 884--888
\end{document}
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