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 [email protected] <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
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