1. No category

MATHEMATICAL
COMPuTER
MODELLING
Mathematical
PERGAMON
and Computer
Modelling
38 (2003) 929-943
www.elsevier.com/locate/mcm
D-V Cells and Fundamental
Domains for
Crystallographic
Groups, Algorithms,
and Graphic Realizations
E. MOLNAR, I. PROK AND J. SZIRMAI
Department
of Geometry
Budapest University of Technology and Economics
P.O. Box 91, H-1521 Budapest, Hungary
emolnar@mail.bme.hu
<prok><szirmai>@math.bme.hu
Abstract-This
package
http:
work
is related
to graphic
CARAT, developed
by colleagues
//wwwb. math. rwth-aachende/carat/.
software
in Aachen
in progress
by our department
headed
by Plesken.
CARAT
to the computer
is available
via
Our software intends to help the applicants,
e.g.,
crystallographers,
and others
in modelling
real crystals.
Furthermore,
it, will hopefully
be developed
for visualization
of higher-dimensional
(d = 4) and non-Euclidean
(d = 2,3) investigations.
The well-known
algorithms
for DirichtetVoronoi
(D-V)
cell partition
of n points in general position
(Voronoi
diagram)
in lEd have the worst, case complexity
(11
0
dnrdj21+l
d3nldj21
logn
It becomes
more simple
for a fixed dimension
d, if we assume
a transitive
group
action
on the point,
set,
In particular,
we consider
a point
orbit
under
a (crystallographic)
space group
r in E3, and
determine
its D-V cell V and-depending
on the stabilizer
of the starting
point-u
fundamental
domain
3 for r with an appropriate
face pairing
for a set, of generators
and algebraic
presentation
of I?. This latter
algorithm
with its graphic
implementation
is our new initiative
in the topic.
In
general,
the worst, case time complexity
exponentially
increases
only by the dimension
d, but it is
completely
satisfactory
for d = 2,3,4.
@ 2003 Elsevier
Ltd. All rights
reserved.
Keywords-Crystallographic
Graphic
implementation.
groups
in d-space,
1. DESCRIPTION
A space group I? is
closed) fundamental
defined
in
space
Algorithms
for D-V
cells,
Fundamental
domains,
OF SPACE GROUPS IN IEd
IEd as a group
of
with a compact (bounded
isometrics
and
domain F’, i.e.,
U y3=lEd and
Int 3 n Int 73 = 0,
for any y E r \ {l},
il.11
-/El-
where “Int ” abbreviates interior, 73 is the y-image of 3, and 1 denotes the identity map.
Any element Q of J? associates each point X with its image crX =: Y by (d + 1)-row-column
Supported
partly
by Hungarian
0895-7177/03/$
- see front,
doi: lO.l016/SO895-7177(03)00295-4
matter
NFSR
(OTKA)
@ 2003
Elsevier
TO20498
Ltd.
(1996)
All rights
and
Project
reserved.
TE’T
- DAAD
D-4/99.
Typeset by
.~&-TI$
E. MOLNAR et al.
multiplication
as usual,
(;)
:= (;
;)
(;)
= (A;,a),
whereX(;),
Y(Y)
(1.2)
are introduced.
We write cu(a; A), as well. The d x d matrix A is called the linear part of a,
and the column vector a is its translational part. All these are expressed in a coordinate system
(0, er, . , ed) with an origin 0, and a so-called lattice basis (el, . . , ed). Here, the d-dimensional
lattice A a r, as a maximal commutative invariant subgroup of I’, consists of parallel translations
of lEd, guaranteed in I’ by the classical Schoenflies-Bieberbach theorem. A is given by a vector
set L as all integral linear combinations of its basis vectors (er , . . . , ed)
L = (1 = elll +...+edld:
(l’,...,l”)
E Zd},
(1.3)
where Z denotes the set of integers. We use R and Q for the real and rational numbers, respectively. The B-vector space of translations of lEd will be denoted by Vd; its dual will be Vd, the
set of linear forms. We know that IEd is distinguished by a positive definite symmetric bilinear
form or scalar product
(,):VdxVd4R,
(x1Y) = (Y,x> E R
(1.4)
which defines the distance metric Q of lEd(O,Vd, ( , )) by
p(X,Y)
= (FQiy2.
Thus, the Gramian matrix gij = (ei, ej) to the lattice basis will describe all the metric and
angular data of the lattice A.
Each element cr(a; A) of a space group I? leaves the lattice A invariant; i.e., the linear part A
has an integral (unimodular or Z) matrix
Aei =: ejai;
each a: E Z;
and
det
a? = fl.
( *>
(1.6)
Moreover, each cy(a; A) is an isometry of lEd; i.e, the linear part A preserves the scalar product
of vectors
i.e., ay(e,,e,)ag = aSg,sag =
(Aei, Aej) = (ei,ej),
(1.7)
gij.
Here, and later on, we use the Einstein sum convention, furthermore, the Schouten primed indices
for basis change.
We emphasize that the basis (er, . . . ,ed) of L may be not orthogonal, and an integral basis
change
eit = eie,,,i
with each ej, E Z, det (ef,) = fl,
(1.8)
will define the integral (Z) equivalence of Gramian matrices by
giljt
:=
eie$,
(
t?j4,
=
ej,(ei,ej)f$
=
W-9
ej,gij&,,
‘>
since they describe the same lattice A.
We only mention that a general afine mapping
a(f; F) : rl
+ r2 = wlwl,
a l-3WD-1,
by (a; A) H (Fa + (1 - FAF-l)
f; FAF-l)
d5,
,
(1.10)
D-V
Cells
and
Fundamental
931
Domains
expresses the equivalence (affine equivariant isomorphism) of two space groups Ii and I2 (Frobenius-Bieberbach theorem). This can be considered as an affine basis transformation
(O.ei..
,
ed) -+ (O’,elf,.
,ed,)
0 ++ O’,
00’
e, k-+e,( = Fei = e,f:
= f;
(1.11)
Here, F is a regular linear mapping, det(f$) # 0. Thus, we get finitely many equivalence
classes (types) of space groups in any lEd (Bieberbach theorem). In particular, for d = 2; 3; 4; 5; 6
we have 17(+0); 219(+11); 4783(+111); 222018(+79); 28927922(+7052) types, respectively. In
parentheses, we have given the numbers of enantiomorphic pairs when the equivalence is more
restrictive; namely, det(ft,)
> 0 is also assumed. The last five data are new. obtained by
CARAT [2,3]. For other results and respects we refer to, e.g., [4,5].
2. D-V
CELL
AND
FUNDAMENTAL
DOMAIN
Let P be a point in iEd, and consider its orbit under a space group l?
rP := {yP E IEd : y E r}
The D-V cell of the kernel point P to its F-orbit
‘~~(6’) = {X E lEd : e(P,X)
(2.1)
is
I e(yP,X),
for each Y E r> ;
(2.2)
see Figure 1. We know that D is the intersection of finitely many closed half-spaces. Each of
them is bounded by (d - 1)-plane (hyperplane) for which “=” stands between the corresponding
distances in (2.2). The equation y(D(P)) = D(yP) ex p resses that F-images of D(P) tile the
space IEd without open intersection.
Figure
1.
Figure
2
2.1.
If Stabr(P) = 1, i.e., the stabilizer subgroup in I’, fixing the point P, is trivial, then ID(P) = 3
is a fundamental domain for I’, since (1.1) is fulfilled. Then, any halving (bisector) hyperface
(facet) fr to P and yP has a pair fT-l to P and y-IP, so that a generator pair of I? can be
obtained as follows:
Y : J-y-1 +-+f”(,
3(P)
-
Y -l : fy t-i f-‘-l,
3(P)
H 3(y-lP)
3(YP)
= 73(P);
= 7+3(P).
(2.3)
We allow involutive generator 6 = 6-l (b2 = 1, 6 # 1) as well, when the facet f~ = ,fh- 1 of 3 is
paired with itself.
Note that a fundamental tiling by the images of 3 geometrically describes the space group I’
and provides a presentation by generators and defining relations. Only 3T, representing 1, with
932
E. MOLNAR
et al.
its facet pairing, denoted by Z (identifications),
and the induced Z-equivalence of (d - 2)-faces
of 3 characterizes the space group l? as we illustrate this in Figure 2.
Here the plane group l? = pl, generated by two independent translations 71 and 7-2, produces
the orbit of any point P and its D-V cell as a fundamental hexagon 3. The three side pairs
fTCl, fT,, . . , frT1, fTZ present three generators 71, 72, 7-sfor P = pl and we get two equivalence
classes of vertices. There are three vertices in the class o of VI, according to the three domains
surrounding VI, each providing the defining relation 71~7~-1 = 1 now for VI. The other three
vertices are in the class l of V2, each providing 7z7r-17s-1 = 1. We write then the presentation
Any relation of I, in general, can be described by walking a circular path from 3 through its
image domains by crossing side facets and returning to 3. Figure 3 shows the main observation
to this. If we are in the image domain a3 and cross its side facet cxfr, which is the a-image of
the facet f7 of 3 corresponding to the generator y, then we arrive at the image domain cuy3.
Figure
3.
Figure
4.
The above cx is a product of 3-generators Z (and their inverses), of course. We get so long
relation as many facets we have crossed in the path what we walk round from 3 into itself.
The special Poincare algorithm by passing round the (d-2)-faces of 3, to get defining relations
for the group I, is a very effective method in geometric (combinatorial)
group theory [6]. We
mention only that the form of a defining relation depends on the stabilizer subgroup Stabe of
the (d - 2)-face e considered. A rotational order v(e) for the equivalence class of e (edge if d = 3,
vertex if d = 2) comes into the game which is related to the facet angle sum at the (d - 2)-face
class of e [6].
We illustrate this situation in Figure 4, where the lE2-group I = p3 (no. 13 from the 17 ones)
acts on the point P. The fundamental domain 3 = D(P) is a rhombus by two pairs of sides fTF1,
f ,.1, and f,.;‘, fr, (the letter f is, and will be, omitted in the figures). We can write
P
(2.5)
D-V
Cells
and Fundamental
933
Domains
(2.5)(cont.)
with respect to the coordinate system (0; ei; es) with Gramian
(2.6)
The vertex class o of V, containing two vertices of 3 with angle sum 27r/3, provides six images
of 3 round V and a relation 1 = rlr2rlr2rlr2,
according to the fact that V is also a three-fold
rotation centre to p3. Thus, 3 involves the following presentation:
p3 = (rl,
From this a three-rotation
(or geometrically)
rT1 = qr2
r2 -
4, r3,, (rlr2)“)
can be derived by matrix
and r3 = r;‘r;’
(2.7)
multiplication
(23)
2.2.
If (I’ >) Stabr(P)
is of order p, then the D-V cell
n-(P)= u 73
(2.9)
-@tab(p)
is a union of p images of a fundamental domain 3. Namely, we take a point Q not f&d
element of Stab (P). Then, the D-V cell of Q under Stab (P)
%tab(p)(Q)
:= {Y : e(Q,Y) < e(a(Q),Y),
provides
31-
:=
%(p)
for every g E Stab(P)}
at any
(2.10)
n %ab(~)(Q),
as a fundamental domain for r.
vst&,(p)(Q)
is a corner domain with apex P which will be intersected by ‘Dr(P). If Stab(P)
is of high order, then we have some advantages in finding 3 r, but difficulties in pairing facets
of 3r may occur, too. Our example is the plane group I’ = p3, again in Figure 5. Here, the D-V
cell of 0 is a regular hexagon 2)(O). Stab (0) is the group of three-fold rotations. Choosing a
point Q
“, , q > 0, the conic domain Dst&, (p) (Q) ’is an angular sector which intersects ‘D(0) in
01
a pentagonal fundamental domain 3 in Figure 5. The arrows +, --+, -D show the pairing of 3,
respectively, where a geometric side of 3 falls into two (algebraic) parts.
The vertex class o of V, consisting of three vertices of 3, provides the relation rlr2q
= 1 for
the side pairing three-rotations rl, r2, r3, according to (2.5) and (2.8).
1
Instead of Q, we can choose a I-image
of 0, namely
1 , in order to use only one I orbit, with
0
high order stabilizer for constructing 3. We remark th& this 3 in Figure 5 is combinatorially
equivalent to 3 := D(P) in Figure 6, where the starting point P is of general position.
E.
Figure
MOLN.~
et al.
5.
Figure
6.
:$“3 is the starting point, then 3 := D(P) ’is a regular hexagon with
( 1 >
maximal inscribed circle c of radius T = a/6 and with maximal density
In particular,
if P
6 = Area(c)
r2r
Area (3) = G
n
= z’
This circle packing has a richer self symmetry group than p3; namely, three additional line
reflections can be introduced which fix any inscribed circle and leave the whole system invariant.
Thus, we obtain the extended group p6mm (no. 17), just the normalizer supergroup of p3 of
index 12. This p6mm has a translational lattice with basis bi = el (2/3) + es (l/3), bz =
ei(-l/3)
+ e2 (l/3), det(bi,bz) = det(ei,ez) . l/3.
The metric (ufine) normalizer of a space group l? in lEd is the supergroup Nrsom (I’) (ZV*ff (I’))
of isometries (affinities) of lEd which transform any r-orbit onto a corresponding r-orbit.
Note that N(r) defined before (may be a continuous group of some parameters) satisfies the
condition
(2.11)
r = ivrwl,
as usual for the normalizer and applicable for its computation on the base of formula (1.10).
Thus, we have indicated some questions related to space groups in lEd, which can further be
generalized also onto non-Euclidean space groups, not discussed here in more detail, We refer
to [4,6-g] for other references.
3. THE
ALGORITHM
FOR
2> AND
F
As we indicated in Section 1, all geometric concepts, concerning a space group l?, acting in Ed,
have to be translated onto numbers, Z (integers), Q (rationals), seldom onto R (reals), and “onto
computer”.
Any space group l? is given by a scheme as follows.
r I= {A(L,
(i) Given is a d-dimensional
of a basis (ei,...,ed).
I), . . . , Ly(a, A), 0-l
(-A-la,
A-‘)
, ,.. } .
(3.1)
lattice A with a vector set L of integral (Z) linear combination
D-V Cells and Fundamental
Domains
931;
is a positive
definite Gramian (gij) = ((ei, ej)).
Given is a finite, and so finitely generated integral (Z) matrix group {A} of linear parts,
called also arithmetic point group ro of I’, each A leaves L and (gzJ) invariant (1.6),(1.7).
(iv) Given is a vector system of “broken translations”
related to L. To each A E I’0 is
corresponded a vector class (cocycle) a + L, depending on the origin of the roordinatr
system as well, so that
(ii)
(iii)
Given
(3.2)
P(b, B) . cr(a, A) = pcr(b + Ba, BA)
holds, in general. This is for multiple products as well with the generators of I’o, yielding
the identity in the linear part. Then the translational part has to be a lattice translation,
Frobenius congruences, Zassenhaus algorithm for fixing the “broken translational parts”
up to changing the origin and the lattice basis (1.10).
(v) We may assume that the data, above, are given in Z, respectively, in Q with bounded
denominator, estimated by “small” constant times the order of arithmetic point group I’().
Now, we perform an incidence (flag) structure of a D-V ceil D and that of a fundamental
domain F for the space group I’. A Jug of a convex polyhedron in IEd consists of a (a! - l)dimensional facet f,“-’ ; then, an incident (d - 2)-face f,“-“, as intersection of two (d - l)-facets
ff-2
= f;-l
n p,
,
(3.3)
l<k<d.
(3.4)
then an incident (d - k)-face as intersection
p
=
p+1
”
$1,
Of course, any (d - 1)-facet will be given by a linear equation
0 = 21122 + . * ~+u~zd+u~+l.l=(u~
)‘..,
X
ud,ud+l)
I
0
(3.5)
in homogeneous point coordinates ( T) by embedding lEd into the projective sphere PSd [l], where
the linear form (2~1,. . . , ud,ud+l) E Vd+l \ {Od+l} is fixed up to a positive constant factor, as
usual. Then, in (3.4) there appear k independent forms for the (d - k)-face of (d - k) parameters.
If k = d, then we obtain a point (may be at the infinity (g) with direction vector y? but this
will be not our case now) as a vertex (O-face) f,” of the polyhedron. Thus, the flag
Fl := ( f,“,fi,. ,.p,p)
.
(3.6)
as a d-tuple of consecutively incident faces, or a ‘<decreasing system” of d, d - 1. . ,2: 1 independent linear equations (forms or normal vectors), respectively, will be defined for a D-V cell 2Y or
for a fundamental domain F. Of course, a (convex) polyhedron may have many flags, constituting
a flag-adjacency-structure
Cp with 0-, l-, . . . , (d - 2)-adjacencies as (d - 1) involutive operations
on the flags of + [6]. That is, a three-cube has 48 flags, a d-simplex has (d + l)! (i.e., (d + 1) factorial) flags. We mention that a flag can be characterized by its barycentric simplex with vertices
as formal midpoints of its incident (d - 1)-facet, (d - 2)-, . , l-, O-faces, respectively. Then, a
(d + I)-flag can be derived from a polyhedral space tiling where a polyhedron as d-component ff
appears in FI as a (d + 1)-tuple instead of (3.6). See, e.g., Figure 5 where the hexagon F has 12
flags aa three-tuples derived from the tiling of F-images. In the following, we restrict ourselves on
dimensions d = 2,3, although we can formulate anything more generally [lo]. For d =- 4. see [II].
936
E. MOLNAR
et al.
4. INTERSECTION
OF A CONVEX
THREE-DIMENSIONAL
POLYHEDRON
AND A HALF-SPACE
Let D be a convex closed bounded polyhedron in the three-dimensional
space S of constant
curvature, e.g., in lE3. We assume that each pair of faces lie in different planes, and each pair of
edges lie on different straight lines.
In order to describe the incidence structure of D, we give the finite sets of vertices V =
edges E = {er, ez,. . . ,ee}, faces F = {fi, fz,. . . ,fn} and cells C = {D} (the
(Wr~2,.
. ., Q},
only element of C is the polyhedron D, itself). Moreover, for each face f and each edge e we
give the subsets Ef = {ei, e2, . . . , e,} s E and V, = { wi,7~2} G V, containing their edges and
vertices, respectively. In this sense, F = FD contains the faces of the only cell 2). Knowing these
sets, e.g., we can form the subset Vf = {WI, ~2, . . . , w,} C V of vertices of an arbitrary face f,
and hence, Vf = UIcl V,,
In our data structure we correspond a variable to the components (vertex, edge, etc.) of the
polyhedron, to save their “type”, which will be defined later. The type of a component z will be
denoted by rZ. Similarly, we correspond a point rU in coordinates to each vertex v E V in order
to determine the position and the metric properties of the convex polyhedron D.
Let a closed half-space H be given by a real function Q : S --+ R, forming the signed distance
between the points of S and the plane 6’H (at least near dH), so that
e(P)
> 0,
if P E S \ H,
=O,
ifPEdH,
< 0,
if P E Int H.
Furthermore, if X E 9 \ H and Y E Int H or vice versa, let M(X, Y) denote the common point
of the segment XY and the plane dH.
Now, we sketch a natural algorithm for finding the intersection of the polyhedron 2) and
the half-space H. The result will be the incidence structure of the convex closed polyhedron
D* = Int (ZYn H) and the coordinated point set {nTvt 1 v* E V”}. The reader will be able to
generalize the method easily for higher-dimensional
cases, indicated formerly as well.
We llx a suitably small positive real number E to correct the limited precision of numbers in
the computer. In a theoretical algorithm, E must be 0.
FIRST. Taking the vertices of V, we compute their type, which describes their position related
to H, applying the following agreement:
721.‘-
fl,
if e(7rv) > +e
(3rV $ H”),
0,
if 1&7r,)j 5 E
(9rV E aH”),
-1,
if Q(T,,) < --E
(3rV E Int H”).
If r,, E (0, -1) for each vertex, then the half-space swallows the polyhedron up, the incidence
structure of 2) does not change, and the procedure comes to an end. However, if rv E (0, +l} for
each vertex, then the incidence structure must be deleted, and the procedure ends (this will not
be our case in the following).
SECOND. Taking the edges, we compute their type as follows. Considering
{vi, ~2) of a given edge e, we form the set T, = {r,, , rV2}. Then
7, :=
fl,
if + 1 E T, C (0, +l},
(“e @ H”),
0,
-1,
if T, = {0},
(“e C aHI’),
if-
(“e c IntH”),
-2,
otherwise.
1 E T, C {0,-l},
the vertices V, =
D-V
Cells
and
Fundamental
Domains
937
THIRD.
We take the faces to compute their types. Considering the edges Ef = {el, 3,. . : e,}
of a given face f, we form the set YiY’f= {T,~, Tag,. . ! T,~}. We define of in the previous manner
as at the second step, taking Tf instead of T,.
The type of the polyhedron
have ended at the first step).
FOURTH.
(as the only cell) is -2 now (otherwise the procedure would
REMARK. We can see that T,, respectively, rf is -2 if (-1, l} is a subset of T,, respectively, Tj,
or -2 E Tf, namely, if the plane dH intersects f. Furthermore, because of the convexity of V, if
rf = -2, then 0 $ Tf.
We take the edges of type -2. If e E E is the next edge in turn, we consider its vertices
‘G’,= {WI, 212). Since T, = -2, we can assume that ~~~ = -1 and 7uz = tl.
FIFTH.
(i) We compute the intersection point p = M(rTT,, , rv,) of e and dH.
(ii) We take this as a new vertex v* and join it to V, hence: V := V U {v*}.
Then p := 7r,,.
and TV. := 0.
(iii) In V, we exchange 212for w* : V, := (~1, u*}. Finally, we modify the type of c 5, := - 1.
Summing up, we cut the intersected edges, and omit the parts not belonging to N
We take the faces of type -2. When f E F is the next face in turn, we consider its edges
Ef = {el,ep?...,e,
} and their types Tf = (7,. , T,~,
, T,?}.
In the fiflh step the edges have
already been cut, so -1 E Tf C {-l,+l}
(neither -2 nor 0 belongs to Tf).
SIXTH.
(i) We consider the subset ,??f = {el, e2,. . . , et} of edges of type - 1 in Ef , where t 5 T. These
edges form the boundary of the truncated face f, but f is opened temporarily along the
cut.
(ii) In order to close f we create a new edge e* enlarging E with it: E : = E !J it?*}.
The
type TV* will be 0.
(iii) Now, we take the edges of l?f and form the subset vf := USE1 V& of vertices lrj of f.
(iv) Then, we choose the vertices of type 0 from vj, forming the set V,* := (1% 1 II E vf,
rv = 0}, which contains exactly two elements.
(v) Finally, we patch f at the cut with e*, rearranging the set of its edges: Ef := ,??f u {e*}.
After cutting, the type of will be -1.
We consider the faces F = Fv = {fl, f2,
, fn} of the polyhedron VD:and their types
TV = {Tf,, off,.
, Tf,&}. In the sixth step the faces have already been cut, so - 1 E T, C { -1, tl}
(neither -2 nor 0 belongs to TD).
SEVENTH.
(i) We consider the subset &J = {fl, f2,. . . , fu} of faces of type -1 in Fv! where u < n.
These faces form the boundary of the truncated cell ‘0, but it is opened temporarily along
the cut.
(ii) To close ?7 we create a new face f* enlarging the set F of faces with it: F :I F U {f*}.
The type Tf+ will be 0.
(iii) Now, we take the faces of & and form the subset & := Uy==, Ef, of edges ED = E.
(iv) Then, we choose the edges of type 0 from &, forming the set Ep := {e / e E I&Y, T, = 0).
(v) Finally, we patch 2) at the cut with f’, rearranging the set of faces: F = Fn := &u{f*}.
After cutting, the type TV will be -1.
EIGHTH.
We take the vertices, edges, and faces of type fl
and remove them from the incidence
structure.
As we have already indicated, the starting polyhedron 2) can be chosen as a D-V cell of the
primitive lattice orbit A,(Xo) of the space group r just considered. Then come the further orbit
points for the cutting planes, each as bisector of a segment &$X0),
y 4 Stabr(Xo)? as above.
938
E. MOLNAR
5. SOME GENERAL
et al.
ASPECTS,
EXAMPLES
The former cutting algorithm in Section 4 needs some
for a nearly optimal data base. Here, we mention some
examples. The higher-dimensional space groups bring more
Our forthcoming paper [12] will discuss some ball packing
also, [13,14]).
To 3. (i)-(ii)
(in Section
IN lE3
preparations and some estimations
aspects and give three-dimensional
difficulties into the description [ll].
optimization
problems as well (see
3)
The symmetry classification of lattices, thus the 14 Bravais lattices in E3, provides a natural
lattice basis for each space group. This may depend on some affine parameters which have to
be fixed according to some inequalities for the Gramian. This so-called Minkowski reduction
determines the number of orbit points, by the lattice hr and then by I’, being considered for a
D-V cell of the origin, say (l.lO), as kernel point. In lE3 for the triclinic (CAP) lattice, e.g., hold
the following inequalities (see, e.g., [15, p. 3971, improved here, and [IS]):
0 < 911
L 922
(remind gij := (ei, ej));
IQ33
I9117
219121
qa913
+ P923
- d912)
for every a,P E {-ho,
5 a2911
11,
(5.1)
+ P2922,
(a,P) # (O,O).
This means we choose the shortest (as possible) vectors, in each step, which can form a basis.
The other Bravais lattices, with more symmetries, provide more simple assumptions.
For
example, the tetragonal primitive (tP) lattice satisfies
gii = 922 =: a2 > 0,
gss =: c2 > 0,
For the second tetragonal-body-centred
911 =
922
=
933
(t1)-lattice
= f a2 + i c2,
and
gij = 0,
i < j.
(5.2)
the equations
g12 = ic2 - Ia2,
2
913
= 923
=
-y
hold with the same parameters a, c as before (Figures 7-10) in the conventional
1
elt = --el+
2
1
1
2e2 + 2e3,
e21
1
= -el
2
1
- -e2+
2
1
-es,
2
e3’
1
= -el
2
(ni,x)-(ni,xo+ini)<O,
2
,
(5.3)
basis
1
1
+ -e2 - -e3,
2
2
related to the primitive one (see (1.9)).
The Gramian plays also an important role in the former Section 4 in forming
vectors, distances, the cutting bisector planes, and half-spaces, respectively,
ni = aixo - x0 = ai + (Ai - 1)x0,
1
the difference
ei = (ni, ni)li2,
forX(;>
(5.4)
as variable point in the sense of (3.5).
To 3. (iii)-(iv)
By increasing distances from the kernel point X0 ( “1”) we consider, very roughly, at least 2d]I’s]
points, but at most 3d]I’a] ones as the images criX0 of the kernel point Xc in 2 x Dol\,(Xe), i.e.,
“two-times” D-V cell (in Minkowski sense) of the r-lattice Ar with the kernel in X0. This latter
would be 33 .48 = 1296 points in dimension d = 3, and 34 .1152 = 93312 points for d = 4. These
estimates increase exponentially by d (at least). However, relatively very few images of X0 will
D-V Cells and Fundamental
Domains
939
take part in forming the D-V cell ‘Dr(Xc) by Section 4. The authors do not know any other
reasonable estimate. For d = 3 the record at this time of Engel [17] is 38 for the side faces
with 70 vertices. The cases d = 3,4 do not cause any problem for a modest computer. Delone
and Sandakova [18] gave the first rough estimate for the problem as we know, in the sense above.
If the kernel point Xc has a nontrivial stabilizer Stabr(Xc)
of order p, then formula (2.10)
through another point Xi (say, a r-image of Xc), if Stabr(Xs) n Stabr(Xi)
= 1 holds, yields
(about (l/p)-times)
simplification. Then, first we take and fix LPstab(x0)(X1) as a corner domain
with apex Xc. We consider only those o-images (a E I’) of X0 where the bisector of Xoo(Xc)
intersects the domain Z&r.,(x,)(Xi).
This can be attained by multiplications with the elements
of Stab (Xc). An inverse pair cy(Xa) and cr-l(Xa) with (Y $ Stab (X0) yields another inverse
pair @(Xc), ,P’(Xc)
by the corresponding faces in Z&t,(X,)(Xi)
@(X0) = O-l(Xa) is also
permitted). The matrices by formula (1.2) accompanying the images of Xi and Xc (see Figure 5)
describe the generating point mappings, although with certain difficulties: additional vertices
edges (faces), as in Figure 5. The neighbouring images of Fr, by (2.10) now by P -+ Xc,
Q -+ Xi, under r produce as .7+(X0, Xi) with exact face pairing, its incidence and flag structure
as well.
To the Normalizers
of l?
As we have indicated at the end of Section 2, the isometric and affine normalizers of l? are
the bases of very important facts, standardization,
extremal problems, etc. The computation
of Nrsom (I?) and N*a(I’),
by (2.11) through finitely many generators, is straightforward
now
(see [19-221). Th ese may be infinite but finitely generated.
All the procedures discussed here need a standard form of any space group I’, since (1.10)
provides much freedom for the same type. Formula (2.11) is a special case of (1.10) and we
mention some applications.
The fundamental set of each N(r), which is a fundamental domain .?=J,J(~)for most space
groups I’, distinguishes point(s) of Fvcr) with maximal stabilizer(s) in N(I). One of them is to
be chosen for the origin of the coordinate system. ~~~~~~(r) and the r-stabilizers describe the
structure of orbits of l?, e.g., the locally optimal orbits for ball packings under I’. Or, varying
also the affine parameters, we get the optimal ball packings for the space group type l? [12-141.
EXAMPLES.
At the end we mention Some ezcamples in E3, mainly by figures.
Il/mmm
no. 139 (Figure 7) is illustrated by a body-centred lattice (t1) and by a fundamental
domain 3 = AA~AzB’V~B, as a prism on a rectangular equilateral triangle.
As generators
of r = 14/mmm
we have plane reflections in the prism faces, except B’VzB, where we have
a half turn about VI&.
The origin is A, and the lattice is indicated by the nodes l . The
isometric and affine normalizer of I’ (the same now) is N = Pl/mmm
with additional translation
(l/2) el + P/2) ez or additional reflection in the plane VIV~AIT. Then,
&v(I4/mmm)
= AA1TB’V2Vl
holds with plane reflections as generators.
We only mention some orbit types, e.g., that in AAIV~B’ with stabilizer m, generated by a
reflection (Figure 8). The optimal ball packing by the D-V cell of centre as kernel point depends
on the ratio of a/c in (5.3). But, we can fix the ball radius to 1, and choose a = b = 4v%, c = 4 so
that the kissing (touching) number of balls will be six. The D-V cell is a cube with an inscribed
ball with density
6 = Vol(ball)
Vol(cube)
= c M 0.5235988.
6
In this optimal case, however, the whole system has a richer symmetry group (PmSm,
than our r. Thus, we say the optimal orbit is noncharacteristic.
no. 221)
E. MOLNAR
940
Figure
7.
Figure
9.
Figure
Figure
Another example is the characteristic
ure 9). Here,
Rapt = 1,
et al
8.
10
locally optimal orbit in AB’ with stabilizer 4mm
a=b=2,
c=4+2&,
6 x 0.6134341,
and the D-V cell is an optimal type with kissing number nine.
The origin A (4mmm)
provides the absolute optimal D-V cell, a rhombic dodecahedron
ure lo), with kissing number 12, if
%,t = 1,
a=b=4,
c = 8,
x 0.7404805.
(Fig-
(5.6)
(Fig-
(5.7)
This orbit is noncharacteristic, again. Fm$m (no. 225) is the richer self symmetry group, serving
the most dense lattice-like ball packing in E3.
Pmb
no. 223 (Figure 11) is given by a primitive cubic lattice with one similarity parameter a
(the nodes l show it), a reflection cube is contained in it, and AA$AjA3 is a cube face. A
three-fold rotation axis, in the cube diagonal, and three half turn axis, through the cube centre 0
perpendicularly to OA, characterize this space group !? = Pm%. The normalizer supergroup is
N = ImSm; it can be obtained by an additional translation ((l/2) el + (l/2) ez + (l/2) es = 23)
or by the plane reflection (er w ez, es H es) in the plane through AO.
3 N(Pm3n) = AO-44
where a half turn about OF : OA3F H OAjF
= 3I,g,,
(5.8)
is a generator different from the plane reflections.
D-V Cells and Fundamental
941
Domains
In Figure 12, we have illustrated a nice D-V cell with kissing number five that provides the
locally optimal characteristic ball packing for the points in AAs (mm2). The optimal radius and
density are
a-1
R Opt = aLEa-
2 1 + Jz
2
’
6 = 2~ (5 &i - 7) M 0.44653223.
(5.9)
The ball does not touch the six %ide” deltoids; however, it touches the base octagon and the
four “roof” deltoids.
Figure 11.
Figure 12.
Our computer animation shows the space group 14 no. 79 with tetragonal
(tl) lattice where a four-rotation
IMPLEMENTATION.
body-centred
100;
0
-1
0
0
and the additional
0
0
translation
0
1
0
;
001;
01
0001
1
1. tridinic (anorthic)
I 1 primiti~
2. moaoeli~
6 1. primitive (cP)
6.2. face-centred (cF)
6 3. body-centred (~1)
1
-
3 D-Vcell
orbit
I
foragenerd
:: il%%aiRi
nontrivial stabilizer.
6 Kissing number
Figure 13.
2.2:
(5.10)
942
E. MOLNAR
et al.
are expressed in the primitive (W) lattice by (1.2). The normalizer of 14 is of infinite index
(denoted by Z14/mmm
in [19,20]) and has a plane fundamental set (a rectangular equilateral
triangle z = 0, 0 5 z 5 y, z + y _< l/2) not detailed more.
Varying the affine parameters a and c in (5.2),(5.3) we get three types of fundamental domains,
only for the (tl) lattice. The equation c = aa yields the rhombic dodecahedron that divorces
the other two combinatorial types (cube-octahedron with six quadrangles, eight hexagons for
0 < c < a & and a dodecahedron with four hexagons in the middle, moreover, 4-4 rhombuses
on the top and on the bottom for c > a a).
Then, varying the orbits by the normalizer, we get various D-V cells, fundamental domains for
16orbits. We have experienced many faces, more than 20, in the algorithm of Section 4. Some
polyhedra are very surprising. However, the cube is also a fundamental D-V cell for 14.
Finally, our table in Figure 13 indicates the scheme of our planned software for specialists, not
detailed more.
REFERENCES
1. D. Avis and B.K. Bhattacharya,
Algorithms
for computing
d-dimensional
Voronoi
diagrams
and their duals,
Computational
Geometry
1, 159-180,
(1983).
2. W. Plesken
and T. Schulz,
Counting
crystallographic
groups
in low dimensions,
Experiment.
Math.
9 (3),
407-411,
(2000).
3. B. Souvignier,
Enantiomorphism
of crystallographic
groups
in higher
dimensions,
Acta Cryst.
A59, 210-220,
(2003).
4. E. MolnBr,
Some old and new aspects
on the crystallographic
groups,
Periodica
Polytechnica,
Ser. Mech.
Eng. 36 (3/4),
191-218,
(1992).
5. J. Opgenorth,
W. Plesken
and T. Schulz,
Crystallographic
algorithms
and tables,
Acta Cryst.
A54, 517-538,
(1998).
6. E. MolnBr,
Polyhedron
complexes
with simply
transitive
group
actions
and their realizations,
Acta Math.
Hung.
59 (l/2),
175-216,
(1992).
7. H. Brown,
R. Billow,
J. Neubiiser,
H. Wondratschek
and H. Zassenhaus,
Crystallographic
Groups
of FourDimensional
Space, pp. 77-84,
A. Wiley-Interscience,
(1978).
8. E.B. Vinberg
and O.V. Shvartsman,
Discrete
transformation
groups
of spaces of constant
curvature,
VINITI,
Ztogi Nauki
i Techniki,
Sour. Probl.
Math.
&nd.
Napr.,
(in Russian)
29, 147-259,
(1989);
In Geometry
II.
Encyclopedia
of Math.
SC%, English
translation,
Springer,
Berlin,
(1993).
9. N.F.M.
Henry
and K. Lonsdale,
International
Tables
for X-Ray
Crystallography,
In Symmetry
Groups,
(Edited
by N.F.M.
Henry
and K. Lonsdale),
Kynoch,
Birmingham,
(1969);
New Edition
by T. Hahn,
Znternational
Tables for X-Ray
Crystallography,
Volume
A, Reidel Co., Dordrecht,
(1983);
Fifth Edition,
Kluwer
Academic,
Dordrecht
(2002).
10. I. Prok, Data structures
and procedures
for a polyhedron
algorithm,
Periodica
Polytechnica,
Ser. Mech. Eng.
36 (3/4),
299-316,
(1992).
11. L. Acs and E. Molnir,
Algorithm
for D-V cells and fundamental
domains
of lEd space groups
(to decagonal
and icosahedral
families
in E4), PU.M.A.
13 (l/2),
l-20,
(2002).
12. E. Moln&r,
I. Prok and J. Szirmai,
Bestimmung
der transitiven
optimalen
Kugelpackungen
fiir die 29 Raumgruppen,
die Coxetersche
Spiegelungsuntergruppen
enthalten,
Sludia Sci. Math. Hungar.
39,443-483,
(2002).
13. J. Szirmai,
Ein Computeralgorithmus
fiir die Bestimmung
der einfach
transitiven
optimalen
Kugelpackungen
unter zum Wiirfelsystem
gehsrigen
Raumgruppen,
In Proceedings
of 3 rd International
Conference
on Applied
Infownatics,
Eger-Noszvaj,
Hungary,
pp. 285-301,
(1997).
14. Cs. M&t6 and J. Szirmai,
A kockarendszerhez
tartoz6
t&csoportok
egyszeresen
tranzitiv
g6mbkit6ltbeinek
meghat&rozla
sz&mit6g6ppel,
Alkalmazott
Matematikai
Lapok 19, 87-111,
(1999).
15. J.H. Conway
and N.J.A.
Sloane, Sphere Packings,
Lattices
and Groups,
Springer-Verlag,
New York,
(1988).
16. A.G. Horv&h,
On the Dirichlet-Voronoi
cells of the unimodular
lattices,
Geometriae
De&da
63, 183-191,
(1996).
17. P. Engel,
ober
Wirkungsbereichsteilungen
von kubischer
Symmetrie,
Zeitschrift
fiir Ktitallographie
154,
199-215,
(1981).
18. B.N. Delone
and N.N. Sandakova,
The theory
of stereohedra,
tidy
Mat. Inst.
AN SSSR, (in Russian)
64,
28-51,
(1961).
19. E. Koch and W. Fischer,
Euclidean
and afTme normalizers
of space groups
and their use in crystallography,
In
International
Tables for X-Ray
Crystallography,
Volume
A, Second
Rev. Edition,
Chapter
15, pp. 855-869,
(1987).
20. E. Koch
and W. Fischer,
DID0
95 and VOID
95-programs
for the calculation
of Dirichlet
domains
and
coordination
polyhedra,
Zeitschtift
fiir Kristallographie
211, 251-253,
(1996).
D-V Cells and Fundamental
943
Domains
21. J. Opgenorth, Normalisatoren
und Bravaismannigfaltigkeiten
endlicher unimodularer
Beitriige zur Mathematik 16 (Dissertation),
(1996).
22. J. Opgenorth, Dual cones and the Voronoi algorithm, Experiment. Math. 10. (2001).
Grupprn.
Aachener