planigon tessellation cellular automata

Planigon Tessellation Cellular Automata

ALEXANDER KOROBOVKharkov University, P.O.Box 10313, 310023 Kharkov, Ukraine,

e-mail: [email protected]

Received March 17, 1999; accepted June 23, 1999.

Cellular automata (CA) do not find as wide a use in chemical complexity determined by a crystal structure as

in homogeneous reacting systems. A reason for this is discussed, and a superposition of planigons and paral-

lelogons or Wigner-Seitz tessellations is suggested that is derived from a crystallographic plane and is capable

of representing chemical complexity at a single crystal face in terms of CA. The interplay between growth and

form of two-dimensional negative crystals is discussed in these terms as a particular example. q 1999 John

Wiley & Sons, Inc.

Key Words: chemical complexity, crystallographic plane, discrete dynamics, growth and form, planigon

tessellations

INTRODUCTION

T wo points gave rise to this communication. (1) Thoughthe analogy between the crystal lattice and the latticesof cellular automata (CA) sites was explicitly under-

lined by Wolfram [1] in one of his early works, CA havefound much wider applications in homogeneous chemistrydespite the lack of natural discrete lattices. The balanceseems to remain practically invariant since Kapral’s review[2]. (2) When a crystalline solid is (partly) responsible for theemergence of a complex behavior, two essentially differentsituations are possible: a solid as a substrate and a solid asa reagent. The former is relatively easier to describe than thelatter. As a result, many manifestations of complex behaviorthat would be worthy of study remain in the shadow. Thesemanifestations are patterns observed in the surface science,

especially when superlattices at single crystal faces interactwith chemically active media, etched crystal surfaces [3],thin films, and complicated patterns of solid-state reactions[4]. In all cases, the dynamics of the origination and evolu-tion of these patterns (the advance of the reaction front, inchemistry terminology) is of special interest.

In thinking this over, it comes to mind that the analogybetween lattices is direct only if the space crystal lattice, inthe strict sense, is implied. Generally, among sites of a crys-tal structure there are translationally nonequivalent sites,which means nonuniformity from the standpoint of CA.This becomes more explicit in terms of planigon tessella-tions [5,6] (see Glossary). These tessellations provide an al-ternative way of representing crystal structures in two di-mensions. In comparison with the habitual language of

© 1999 John Wiley & Sons, Inc., Vol. 4, No. 6 C O M P L E X I T Y 31CCC 1076-2787/99/06031-08

crystal lattices, the language of planigons possesses a num-

ber of advantages in the present context. Three of the main

advantages are the following:

1. Both the combinatorial-topological structure and sym-

metry of a crystallographic plane are described in terms

of planigons and symmetry. Because of this, as many as

46 types of planigon tessellations correspond to only 17

two-dimensional Fedorov groups [5], that is, the classifi-

cation is considerably more detailed. Figure 1(a) shows

as an example one of four possible tessellations for the p6

symmetry group.

2. The combinatorial-topological structure enters the

mathematical theory of planigons via the adjacency sym-

bol, the components of which are generators of the cor-

responding symmetry group. This means, in particular,

that there is the possibility of operating naturally withsuch notions as adjacency and neighborhoods.

3. Closely connected with the theory of planigons is theso-called “local theorem” [7], on the basis of which theconsistent geometrical crystallography may be devel-oped [8]. According to this theorem, the geometricalregularity is determined locally. In particular, a planigontessellation is determined completely by its finite part,that is, the first belt [5].

SUPERPOSITIONS OF TESSELLATIONS AS CAThe theory of planigons was developed irrespectively of thetheory of CA. Nevertheless, one may notice a number ofsimilar notions and terms. In particular, planigons are ad-equate for describing local chemical interactions betweenindividuals. “Individuals” in this context are real atoms situ-ated in the center of action of the planigons. For simplicity,we will talk further about planigons as individuals. Butwhen one thinks about the way in which these interactionsare transformed into involved spatiotemporal patterns, thefollowing peculiarity must be taken into account: Planigontessellations are always regular but, generally, are not uni-form. The required uniformity is inherent in two different(slightly larger scale) types of tessellations: parallelogon tes-sellations [9] and Wigner-Seitz tessellations [10] (see Glos-sary). The latter completely characterize the translationalsymmetry of a crystal. In both cases, cells are either hexa-gons [as in Figure 1(b)] or quadrangles. In describing thedynamics of a chemical process, they may be treated as thecells of a CA. Wigner-Seitz cells and parallelogons may co-incide (as in Figure 1) or may not coincide (see below).

The superposition of these tessellations may be consid-ered as CA, each cell of which consists of planigons [as isshown, for example, for the central cell of Figure 1(b)]. SuchCA will be referred to as planigon tessellation CA (PTCA).The only exception is the p1 symmetry group, which servesin our context as a link with conventional CA. It is repre-sented by two tessellations, hexagonal and rectangular,each cell of which is a planigon, parallelogon, and Wigner-Seitz cell simultaneously. Note that, in terms of crystal lat-tices, the hexagonal tessellation is a more general case, withrespect to which the rectangular tessellation is a particularcase corresponding to perpendicular lattice vectors.

It is worth noting that this construction is not definedformally as some arbitrary cells on an abstract plane but isderived from a crystallographic plane and preserves theone-to-one correspondence with it.

Among a wide variety of possible rules, we will be inter-ested first of all in rules that take into account and makeexplicit distinctions between the translational and non-translational (point) symmetry of a crystal lattice. Keep inmind that in many chemical interactions we will talk aboutplanigons (or, more accurately, atoms represented by pla-nigons) entering a reaction. From this standpoint, one or

FIGURE 1

An example of CA derived from a single crystal face with p6symmetry group. (a) The planigon tessellation; one planigon isshown by hatching. (b) An interaction is propagated first towardplanigons forming the Wigner-Seitz cell. (c) An interaction ispropagated toward boundary planigons belonging to adjacentcells. (d-f) Further evolution according to rule R1.

32 C O M P L E X I T Y © 1999 John Wiley & Sons, Inc.

another rule describes the transition (or propagation) of aninteraction from one planigon to others or (in terms ofsolid-state chemistry) the advance of a reaction front. At agiven step an interaction is transmitted to planigons thathave equal probabilities of entering a reaction.

Consider, for example, the planigon tessellation for p6symmetry group shown in Figure 1. Planigons in this caseare quadrangles [5]. Each planigon has four symmetry axessituated on its boundary [Figure 1(a)]. Wigner-Seitz cellsand parallelogons coincide, each containing six planigons[Figure 1(b)]. One of the simplest possible rules may beformulated as follows.

Rule R1

1. Two values, p = 0 and p = 1, are possible for a planigon,depending on whether it has entered a reaction. Pla-nigons with p = 1 are shown by hatching.

2. Only two values, 0 and 1, are admitted for a hexagonalcell.

3. The value c of a cell is determined by the values p ofcomponent planigons, that is,

c = p1 % p2 % p3 % p4 % p5 % p6

where % denotes the addition of modulo 6.4. If a planigon or a cell has taken on the value 1, no further

changes are possible for it (monotony condition).5. If a cell with the value 0 has in its neighborhood specified

as a set of displacement vectors in the form

N = HS00D;S2

0D;S11D;S−1

1 D;S−20 D;S−1

−1D;S 1−1D;J

at least one cell with the value 1, the reaction front isadvanced toward the boundary planigons [Figure 1(c)].Planigons that have entered a reaction at the last step areshown by more dense hatching.

Note that this rule keeps the reaction front inside a celluntil all planigons of this cell have entered a reaction [Figure1(b) and (d)]. As a result, we get two alternating patterns[Figures 1(e) and (f)].One of them is a regular hexagon; an-other one has the regular hexagonal convex hull. The convexhull will be identified in the present context with the habit.

In addition to conventional ways of complicating the dy-namics of cellular configurations (more than two states maybe associated with each cell or planigon, as discussed later,for example), an essential novelty arises in declaring variousneighborhoods when parallelogons do not coincide withWigner-Seitz cells. The planigon tessellation for the p2 sym-metry group, shown in Figure 2, is a suitable example. Thequadrangle parallelogon formed by two triangular pla-nigons does not coincide with the hexagonal Wigner-Seitz

cell determining the translational symmetry of the corre-sponding crystal structure. This may be taken into accountby defining the neighborhood

N = HS00D;S1

0D;S01D;S−1

0 D;S 0−1D;S 1

−1D;S−11 DJ

which is, in a sense, the “intermediate” between von Neu-mann and Moore neighborhoods. In Figure 2(a), this isshown by dots. Preserving this rule, except when addingmodulo 2 instead of modulo 6 according to the lattice sym-metry, we get the pattern shown in Figure 2(b).

GROWTH AND FORM OF TWO-DIMENSIONALNEGATIVE CRYSTALSTo illustrate some features of PTCA that are not inherent inconventional CA, consider one important aspect of the“growth and form” problem. Since its exposition by Thom-son [12], it becomes a many-sided interdisciplinary prob-lem. A good idea of its wide spectrum is provided in Refer-ence [13].

A related point concerns various figures (thermal decom-position figures and etching figures, for example) that areformed within single crystals as a result of one chemicalreaction or another. Numerous examples of these figuresmay be found, for instance, in References [3,4]. The generalterm accepted for these figures in heterogeneous chemicalkinetics is “localization forms.” The clear understandingthat the sites at which such a chemical process originatedand the directions in which it evolves in some cases areconnected with the symmetry of a crystal, whereas in someother cases they are not, goes back to the past century [14].But even nowadays the interplay of numerous factors re-sponsible for localization forms is far from being sufficiently

FIGURE 2

(a) An example of PTCA for which the parallelogon (solid hatch-ing) formed by two planigons (line hatching) does not coincidewith the Wigner-Seitz cell (hexagon). This determines the neigh-borhood that is shown by dots. Filled circles denote 2-fold sym-metry axes. (b) The first steps of the evolution according to therule described in the text.

© 1999 John Wiley & Sons, Inc. C O M P L E X I T Y 33

understood. A peculiar feature is that localization formsmay vary with the variation of external conditions. One ex-ample, among numerous ones, is that the dehydration fig-ures at (100) faces of Na3P3O96H2O single crystals may behexagons, rhombi, circles, or irregular in form, dependingon particular conditions under which the dehydration reac-tion proceeds [4].

Various localization forms appear through the formationand growth of nuclei (seeds, germs). In all cases, this in-volves the disintegration of the crystal structure, which maybe considered as the growth of negative crystals. In describ-ing chemical reactions in which single crystals participate,there are good grounds to treat the growth of a negativecrystal along a crystal face and into the bulk separately [15],and we will discuss further the growth of two-dimensionalnegative crystals within a crystallographic plane.

Nuclei are formed according to the laws of chance. Thequestion is in what way regular negative crystals that fre-quently are oriented in the same manner grow from randomnuclei. This is one of the less-studied related issues.

In the course of the processes under discussion, nucleiare formed as a result of chemical reactions rather than of achange of physical conditions. The peculiarities of studyingnucleation of this type are determined by the fact that anextremely small amount of a substance is formed. It is prac-tically impossible to register this by chemical methods. Onthe other hand, the interpretation of physical characteristicsis usually ambiguous [16]. As a result, main regularities ofchemical nucleation are speculated on proceeding from theobserved kinetic behavior of a system, that is, the intercon-nection between nucleation, growth, and impingement pro-cesses. This means, in particular, that mechanistic studiesare essentially based on kinetic studies. In this context, theproblem of orientation acquires additional importance be-cause the geometric-probabilistic approach that is capableof interrelating processes of nucleation, growth, and im-pingements is efficient only when all growing nuclei areoriented in the same manner (see, e.g., Reference [11] forreview).

In a broader context, we are dealing with ordered spa-tiotemporal structures on a macroscopic scale, the phe-nomenology of which is difficult to predict proceeding di-rectly from microscopic considerations or from the proper-ties of simpler subsystems. It is hardly an overstatement tosay that, within heterogeneous chemical kinetics, evennowadays there is no approach capable of exploring theproblem from this angle.

In terms of PTCA, it is logical to start with the followingquestions:

● Can a regular, symmetrical negative crystal (pattern) beobtained from a nonsymmetrical nucleus?

● Can negative crystals with one and the same habit beobtained from different random nuclei?

● In what way may the difference in the number of “atoms”forming nuclei be felt in terms of PTCA?

● Is it is possible to subdivide nuclei into compatible andincompatible categories with the requirement that theyhave the same habit and orientation of negative crystals(after a sufficient number of steps)?

Consider as an example another variety of planigon tes-sellations for the p6 symmetry group shown in Figure 3. Inthat case, planigons are regular triangles with symmetryaxes as shown in Figure 3(a). Any two adjacent planigonsform the parallelogon that is a rhombus. Parallelogons differ

FIGURE 3

Another planigon tessellation for the p6 symmetry group. Theevolution according to rule R2 is followed. (a)-(d) Planigons withp = 1 are shown by dots, and planigons with p = 2 are shown bysmall crosses.


in the situation of symmetry axes. In this respect, there are

two types of parallelogons: one with a 2-fold symmetry axis

in the center [Figure 3(b)] and one with two 2-fold symme-

try axes on the boundary [B in Figure 4(a)]. Wigner-Seitz

cells are regular hexagons organized around 6-fold symme-

try axes. Rule R1 will work in this case in the same way.

According to Rule R1, a set of alternating patterns possess-

ing a 6-fold symmetry axis will be obtained, the convex hulls

of which are regular hexagons. Along with this, another rule

may be suggested.

Rule R2

1. Three values are admitted for a planigon: p = 0 means

that the planigon has not entered a reaction, p = 1 means

that it has entered a reaction (dotted hatching in Figures

3 and 4), and p = 2 means that both planigons joined by

a 2-fold symmetry axis have entered a reaction (cross

hatching in Figures 3 and 4).

2. The value p = 2 is assigned to nuclei.

3. If a planigon with p = 1 or p = 2 has in its neighborhood

a planigon with p = 0 separated by the 2-fold symmetry

axis, both of them take on the value p = 2 [Figure 3(b)]. As

a result, a parallelogon with one 2-fold symmetry axis in

the center is formed. Parallelogons of this type will be

further described as A-parallelogons to distinguish them

from B-parallelogons.

4. If both planigons of an A-parallelogon have p = 0 and

if each of them has at least one neighbor with p = 2,

both the planigons take on the value p = 2 (i.e., an A-

parallelogon may be formed at one step).

5. If five planigons of a Wigner-Seitz cell have values of p Þ

0 and if one planigon has the value p = 0, it takes on the

value p = 1.

6. When the boundary of a growing pattern has no 2-fold

symmetry axes [as in Figures 3(a) and (d)], all boundary

planigons with p = 0 take on the value p = 1.

As a result, one gets a set of convex hulls with alternating

patterns that are elongated hexagons. Some of them appear

star-like, as in Figure 3(e), and others are close in shape to

conventional convex polygons. This rule will be illustrated

later in more detail.

In comparing the two rules, it may be said that rule R1

gives preference to 6-fold symmetry axes, whereas rule R2

gives preference to 2-fold symmetry axes. According to the

former, an interaction is propagated easily through bound-

aries without symmetry axes; according to the latter, an in-

teraction is propagated first through boundaries bearing

2-fold symmetry axes. Rule R2 is more unconventional and,

at first glance, may seem less logical from the perspective of

the p6 symmetry group, and the more so, in the case of a

single-planigon nucleus, because it leads to elongated

rather than regular patterns. But in discussing nuclei with

many planigons, we see that the introduction of this rule in

the present context is justified.

Further we will talk about n-nuclei, where n is the num-

ber of planigons forming a nucleus. In the present article,

we will restrict ourselves to nuclei in which each of the

planigons has at least one common edge with some other

planigon. In other words, two planigons that have only a

common vertex are not considered here as a nucleus. For

this reason no attention will be paid to 3-fold symmetry

axes, which violate in this case the adjacency relation, and

accordingly these axes are not shown in the figures.

All 1-nuclei planigons are equivalent. All 2-nuclei, which

are parallelogons, fall into two types, as mentioned previ-

ously [A and B in Figure 4(a)]. according to both rules, A-

parallelogons grow into elongated patterns similar to those

FIGURE 4

Two types of 2-nuclei are possible (A and B). Step-by-step evo-lution of nucleus B according to rule R2 is followed in detail.


shown in Figure 3. To the contrary, B-parallelogons alwaysgrow into apparently regular hexagons, which justify intro-ducing rule R2 in the present context. The way in which aB-parallelogon evolves, according to rule R2, is followedstep by step in Figure 4. Step 1 is based on item 3 of rule R2:An interaction is propagated through boundaries bearing2-fold symmetry axes. Step 2 illustrates item 4; step 3 cor-responds to item 6; step 4 is again based on item 3; step 5 isbased on item 6, and so on. Other patterns are drawn on asmaller scale. Note that geometrically similar patterns like 7and 10 in Figure 4(b) may evolve in different ways depend-ing on the situation of symmetry axes. Pattern 10 transformsinto pattern 11 according to item 4, whereas pattern 7 trans-forms into pattern 8 according to item 6; the transition frompattern 9 to pattern 10 illustrates item 5.

Note that all patterns (negative crystals) shown in Figure4, including patterns 2, 3, 6, and 11, actually possess 3-foldsymmetry axes. At the same time, the convex hull of each ofthese patterns is a regular hexagon. To underline this, wewill talk about apparently regular hexagons. When a B-parallelogon evolves according to rule R1, it leads to trueregular hexagons possessing 6-fold symmetry axes.

An interesting novelty arises in passing in relation to3-nuclei. In this case, also there are two types of nuclei[Figure 5(a)], but their interplay with rules is essentially dif-ferent (Table 1). We see that, compared to 2-nuclei, eachtype of negative crystal may generally be obtained fromeach type of nucleus. In other words, if nuclei will be al-lowed to “make their own choice” and if nuclei of type A willchoose rule R1 whereas nuclei of type B will chose rule R2,the result will be the formation of negative crystals with thesame habit, oriented in the same manner. (Nuclei of type Bevolve according to rule R2 into apparently regular hexa-gons in a way similar to that shown in Figure 4).

This means that there is a possibility of discussing someaspects of the interplay between symmetry and adaptability,in terms of PTCA, in the nucleation and evolution of nega-

tive crystals. All 3-nuclei of both types have the same formbut differ in the terms of symmetry axes. As a result, theevolution of each nucleus according to different rules is dif-ferent. But if an adaptive choice is allowed, there is thepossibility that all negative nuclei will be of the same habit,although some of them will have 6-fold symmetry axes andothers will have 3-fold symmetry axes.

When we turn to discussing 4-nuclei, the picture is asfollows. There are five types of 4-nuclei [Figure 5(b)]. Onlyone of them is “adaptive” (A). Nuclei of two types (C and D)may evolve only into elongated hexagons, and nuclei of twoother types (B and E) may evolve only into regular or ap-parently regular hexagons. But when considering 5-nuclei,the number of adaptive nuclei increase. And although thehope that all 6-nuclei will be adaptive cannot realized, mostof them do adapt. According to a preliminary study, thenumber of adaptive 6-nuclei is at least two times greaterthan the number of nonadaptive nuclei. Examples of adap-tive 6-nuclei are shown in Figure 6(a). It is worth noting thatnuclei that evolve into negative crystals with a regular hex-agonal habit according to rule R2 usually prevail. Most ofthem actually possess 3-fold symmetry axes. This is in linewith the general idea that sometimes the rigorous symmetryhas to partly yield to adaptability, thus providing flexibleenough conditions of growth [8]. It is interesting to note thattwo adaptive 6-nuclei (E and F) have the same form andevolve into regular hexagons but according to differentrules. Also, many adaptive 6-nuclei have “twins” that pos-sess the same form but are not adaptive because of thedifferent symmetry axes. Four examples are shown in Figure6(b). These nuclei may evolve only into elongated hexagons.

We started to discuss the problem of growth and form oftwo-dimensional negative crystals from four questions.Now we may answer these questions. (1) In terms of PTCA,negative crystals with the same habit may be obtained fromrandom nuclei that differ in both form and number of pla-nigons. (2) A stable symmetrical evolution can be obtainedproceeding from a casual nonsymmetrical nucleus. (3) Nu-clei formed by different numbers of planigons (atoms) dif-

FIGURE 5

All possible types of 3-nuclei (a) and 4-nuclei (b).

TABLE 1

Habit of two-dimensional negative crystals obtained from differentnuclei according to different rules

Type of Nuclei Rule R1 Rule R2

2-nuclei (Figure 4(a))A Elongated hexagons Elongated hexagonsB Regular hexagons Regular hexagons

3-nuclei (Figure 5(a))A Regular hexagons Elongated hexagonsB Elongated hexagons Regular hexagons


fer, in particular, in the percentage of adaptive nuclei. (4)

When the set of rules is given, nuclei may be subdivided into

compatible and incompatible categories, with the require-

ment that they have the same habit and orientation as nega-

tive crystals, an important fact in solid-state reaction kinet-

ics. And although a number of new questions arise in this

connection, it may be argued that the language of PTCA

provides a fresh insight into some essential aspects of the

growth and form of two-dimensional negative crystals con-

cerning the evolution of regular patterns from random nu-

clei. In the present article, we restrict ourselves to critical

nuclei, in the sense that nuclei were only allowed to grow.

This forms the basis for a further step when atoms (pla-

nigons) are allowed both to join and to disjoin subcritical

nuclei, depending on local conditions. In this case, one ar-

rives at a wide variety of rules and structures that are much

more intriguing but are much more involved and deserve a

separate discussion.

A large-scale pattern is the result of local forces [8].

Mechanisms adhering to local rules that lead to involved

symmetrical patterns are of considerable interest in many

fields, not just in crystallography or solid-state chemistry.

This problem is one of the essential issues of modern sym-

metry theory. The use of tessellations, which are directly

connected with the geometry of crystal space, to explore

this problem seems to be promising.

4. CONCLUDING REMARKSThe problem of growth and form is not the only problem

that may be approached from the new angle of PTCA. Note

in this connection that, in comparison to conventional CA,

both the form of a PTCA cell and all its dimensions are

meaningful since they are determined uniquely by a parent

crystallographic plane. Thus, the cells of the PTCA shown in

Figure 2 are parallelograms. Where conventional CA are

concerned, no distinctions are made between square, rect-

angular, and parallelogram cells. In the case of PTCA, the

situation is different: squares represent a different type of

planigon tessellation [5,6]. The meaning of forms and di-

mensions is that a measure may be naturally associated

with PTCA cells. As a result, topological, metrical, and sym-

metrical aspects appear to be closely interrelated on the

physical basis. In addition, planigons represent the symme-

try of not only atomic nuclei positions, but also the average

two-dimensional cross sections of electron density distribu-

tion [9]. These features may be useful for, among other

things, considering the general issue of the interplay be-

tween discrete and continual models. In particular, due to

their origin and features, PTCA are promising for modeling

the dynamics of various surface and bulk solid-state chemi-

cal reactions [11]. Finally, irrespective of their origin, PTCA

may be of independent interest as a mathematical construc-

tion that is worthy of study and is connected with various

tessellations of the Euclidean plane, which have been

springing surprises on geometers since the time of Kepler.

GLOSSARYPlanigon: A convex polygon such that congruent planigons

form a regular tessellation of the plane. Any planigon is the

fundamental region of one of 17 two-dimensional Fedorov

groups. Accordingly, symmetry elements may be situated at

planigon boundaries and not inside a planigon. A planigon

may have three to six edges. Each planigon has an atom of

a two-dimensional crystal structure inside. The situation in

which it exists as the Dirichlet domain of this atom is of

special interest in the present context. The complete math-

ematical theory of planigons was developed by Delone, Dol-

bilin, and Shtogrin, [5] and by Grunbaum and Shephard [6].

The term “planigon” was introduced by Fedorov [17] and is

widely used in the context of crystallographic problems.

When purely mathematical aspects are concerned, the term

“tile” is also used. But this term is not specific: it is used to

discuss various other tessellations. Note that, despite many

years of effort, until now there has been no analogy for three

dimensions to the planigon theory.

FIGURE 6

Examples of adaptive (a) and nonadaptive (b) 6-nuclei. Eachnucleus shown in (b) has a “twin” in (a). For each nucleus in (a),the rule is specified according to which it evolves into negativecrystals with regular hexagonal convex hulls.


Parallelogon: A convex polygon formed by planigons ac-cording to their symmetry and such that a tessellation of theplane may be obtained by translating it in two directions. Aparallelogon may be either hexagon or quadrangle [9].

Wigner-Seitz cell: The Dirichlet domain of a space latticepoint. It is one of the possible and widely used elementarycells possessing the symmetry of the corresponding Bravaislattice. Accordingly, it absolutely represents the transla-tional symmetry of a crystal structure. A Wigner-Seitz cellmay be either a hexagon or a quadrangle, and each type ofthe translationally nonequivalent points of a crystal struc-ture is represented by one point inside it [10].

Dirichlet domain: The DA«of some point A belonging to a

discrete system of points « is the convex geometric locus,the points of which are closer to any A than to any otherpoint «.

Negative crystal: The domain of a crystal structure withinwhich this structure is disintegrated for one or another rea-son and that has the form of a crystal.

ACKNOWLEDGMENTSThis work was undertaken at the Chemistry Department ofKharkov University. I would like to thank Professor V.D.Orlov and Dr. Yu. V. Kholin for their encouragement.

REFERENCES1. Wolfram, S. Rev Mod Phys 1983, 55, 601–644.2. Kapral, R. J Math Chem 1991, 6, 113–163.3. Sangwal, K. Etching of crystals; Elsevier: Amsterdam, The Netherlands, 1987.4. Prodan, E.A. Topochemistry of crystals (in Russian); Nauka and Technica: Minsk, Bielorus, 1990.5. Delone, B.N.; Dolbilin, N.P.; Shtogrin, M.I. Proc Math Inst Acad Sci USSR 1978, 148, 109–140.6. Grunbaum, B.; Shephard, G.C. Tillings and patterns; Freeman: New York, 1987.7. Delone, B.N.; Dolbilin, N.P.; Shtogrin, M.I.; Galiulin, R.V. Dokl Acad Nauk SSSR 1976, 227, 19–31.8. Galiulin, R.V.; Senchal, M. Recent results and unresolved problems in the symmetry theory in patterns of symmetry. Senchal, M.; Fleck, G., eds. University

of Massachusetts Press: Amherst, MA, 1977.9. Shubnikov, A.V; Kopcik, V.A. Symmetry in science (in Russian); Nauka: Moscow, USSR, 1973.10. Ashkroft, N.W.; Mermin, N.D. Solid state physics; Holt, Rinehart and Winston: New York, 1976.11. Korobov, A. J Math Chem 1998, 24, 261–290.12. Thomson, D’A.W. On growth and form; Cambridge University Press: Cambridge, UK, 1917.13. Ishizaka, S. (ed.). Proceedings of the first international symposium for science on form; KTK Scientific: Tokyo, 1986.14. Lehmann, O. Molecularphysic; Leipzig, 1889.15. Korobov, A. Thermochimica Acta 1995, 254, 1–10.16. Delmon, B. Introduction a la cinetique heterogene; Editions Technip: Paris, France, 1969.17. Fedorov, E.S. Basic doctrine of figures: Classics of science series (in Russian); USSR Academy of Science: Moscow, USSR, 1953.


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