Symmetry analysis of neolithic painted pottery from the republic of macedonia

8/12/2019 Symmetry analysis of neolithic painted pottery from the republic of macedonia

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Archeologia e Calcolatori21, 2010, 261-280

SYMMETRY ANALYSIS OF NEOLITHIC PAINTED POTTERYFROM THE REPUBLIC OF MACEDONIA

1. I

Neolithic pottery from Southeast Europe and Anatolia is well knownfor its remarkable and specic decoration. Very often the visual features ofthese objects are used for determining the relative chronology of the excavatedsites, without considering their potential for mathematical observation. The

repertoire of patterns used for developing the compositions painted on thevessels provides abundant data for such analysis. Almost all of the fragmentsdiscovered so far as well as the completely preserved painted pots from theseregions were decorated following several visual principles, in order to createa precise disposition of the patterns onto the spherical surface of the vessel.This decorative approach was used in all the standards of Neolithic geometry,employing both symmetry and principles of visual entropy.

The painted vessels from Early and Middle Neolithic settlements discov-ered in the Republic of Macedonia, for example, provide a variety of infor-mation about the organization and structures incorporated on the decorated

pottery. In the earlier phases, these painted compositions were mostly based onprinciples of the Four Rigid Motions of pattern disposition on one-dimensionalformat, while later, besides using this concept, the principle of antisymmetrywas implemented in order to compose a two-dimensional image.

This paper aims to detect all possible forms of plane symmetry, patternsand compositions applied in the decoration of Neolithic vessels from Macedo-nia, as well as to promote the study of geometric symmetry as a possibility forthe reconstruction of decorated fragments. Consequently, methods of math-ematical observation will be proposed as a means for detailed examination ofthe vessels visual potential and will be further implemented in tracing the ele-ments of visual identities of communities inhabiting the Neolithic Balkans.

2. N R M

The beginning of the Neolithic in Macedonia has been dated to the sec-ond half of the seventh millennium BC (Fig. 1). The process of Neolithizationstarted rapidly after the rst wave in the Balkan Peninsula and introduceddomestication, economy, architecture and new rituals and social relationsamong communities. Newly established communities mainly spread acrossthe plains, close to rivers where they built their dwellings made of wattle anddaub. The abundance of vegetation and animals insured a steady development


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Fig. 1 Position of the Republic of Macedonia in Europe.

of the settlements and increased economic cohesion with those in the surround-ing area, as well as to those located far from the region. The importation ofexotic materials in the settlements occurs in the course of this process.

Newly established cultures were settling different regions and generating

local identities, thus creating regional features, which later would be easilyrecognized by archaeologists. In this way, they conrmed the existence ofseveral Neolithic cultural groups, which were dispersed within geographicallyauthentic and isolated areas (plains, lakes, rivers, etc.), but also emerging indifferent periods. According to tradition, among the Early Neolithic culturalgroups, those of Amzabegovo-Vrnik, Veluina-Porodin and Zlastrana groupshave been identied: the rst one is located in the Ove Pole plain, the sec-ond in the Pelagonian plain, while the third was in the territory of the Ohridregion (G 1979; S 2004). During the Middle Neolithic in thePelagonian plain a new Trn cultural group emerged, whilst Veluina-Porodin


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and Amzabegovo-Vrnik entered into their second phases. In the Late Neolithicthe territory held by the Amzabegovo-Vrnik group demonstrated new culturalmanifestations to the Angelci-Zelenikovo group, whereas in the Ohrid regiona distinct Ustie na Drim group has been conrmed (B 1979; G1979; B 1989; S 1995).

All of these cultures displayed their own features in their pottery pro-duction, but also shared similarities along with other groups, mainly on thelevel of architecture, economy and religion, thus including similar stone andbone tools, gurines, altars, ornaments, as well as dwellings (with the ex-ception of those in Ustie na Drim which are palatte or stilt houses). As forthe vessels, along with the Neolithic package pottery production was alsoestablished, which enabled the development of a variety of vessels, as well asa wide range of techniques for their manufacture and decoration. Most ofthe vessels were made of coarse clay, and mainly used for storing raw goodsand liquids, but they also made vessels for the preparation of food, whichincluded pots, jars, plates, askoi and lids (F 2009). Those vesselsbelonging to the category of ne pottery show a different approach towardsmodeling and conceptualization. Due to their polished surface and ne fabric,these vessels were not intended for cooking, thus allowing more concentrationon their appearance and decoration. Most of these vessels include cups andamphora-like pots. Painting with white, black or brown pigments on the redpolished surface was the preferred technique for their decoration. The creativeapproach towards the styles of patterns applied to these vessels resulted inthe visual identication of the communitys identity with the visual principlesimplemented on the decorated pottery.

3. S N

The most remarkable and representative attribute of the Neolithicpainted vessels is their rm and structured decoration. These vessels representthe earliest attempts to employ this type of decoration found in the territoryof the present the Republic of Macedonia. The globular shape of the ves-sel itself facilitated this kind of organized decorative visualisation, and thecyclical movement and intertwining of the forms. The organised placementof patterns, incorporated in the compositions placed in symmetry with eachother, demonstrates that the Neolithic communities maintained rational andlogically developed perception of space. The potters attempted to establishall possible symmetrical principles, to depict them and to create visuallyreminiscent compositions. The visual approach, in which a composition istransposed onto the surface of the vessel, suggests another innovation intro-duced in the Neolithic era. Namely, it refers to rmly dened principles ofarranged imagery forms, which were established and repeated over several


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Fig. 2 Early and Middle Neolithic painted pottery from the Republic of Macedonia. 1-3: Amza-begovo; 4-6: Veluka Tumba; 7-9: Madjari (photos by G. Naumov).


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generations. This suggests the existence of a Neolithic design, represented byestablished combinations of patterns which, on the one hand correspondedto the imagined idea or concept, but on the other, also represented the visualentity through which a cultural group identied itself (Fig. 2).

In various periods and regions there have been communities whereimagery rather than the written language was used as the major means ofcommunication. Diverse imagery forms, both representational and nonrep-resentational, embodied visually projected ideas fundamental to the crucialconcepts of those cultures. These ideas were usually visualized through exactprinciples conceived along the lines of concrete geometric accuracy. In thissense, the basic mathematical values might be concerned and used in under-standing these regularities. This provides the profound depiction of patternsin its most elementary level and consequently the elaboration of its featuresdened by a certain model. Symmetry analysis enables us to determine theregularities within the visual structure of a repeated pattern, which was usedin various aspects of developing compositional structure or certain exactimagery forms. As today, symmetry was also identied in the past as a vastlyeffective principle for structuring visual and social relationships among thecultural groups and their essential imagery forms. Considering that theserelationships were based among illiterate populations, they were metaphori-cally embedded through proper symmetrical standards that structure patternscreated onto visible surfaces of many forms of material culture (W,C 2004).

Analyses of the patterns morphology and structure demonstrate thatthey are executed very accurately, thus following a scheme determined be-forehand and used in the positioning of the patterns in a symmetrical ratio.Symmetry was used as a fundamental principle which places the patternsin organized compositions, establishing the concept of eurhythmy, i.e. theharmony of the parts into one entirety. In spite of the fact that some authorsconsider that these patterns and their arrangement are of entoptic origin(L-W, P 2005, 120-122; B 2004, 62), the geometricalexibility of these patterns should be considered, as well as their capabilityto be mathematically arranged on one surface, i.e. composition (N2005, 71; N 2009a). The linear structures that build the compositionare often referred to as modules, where the symmetrical arrangement of thecomposition is constructed by their permutation and transposing.

These principles of modularity, generative grammar, symmetry andasymmetry have been analyzed by several authors who stress the mathemati-cal approach towards pottery decoration (H 1985; J 1989;

J 1995; W 1999; H 2003, 61-74). Consequently, aEuclidean geometry was employed, thus providing the structure of patterns(designs composed of regularly repeated elements). This way, compositions


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are developed by the symmetries, i.e. rigid motions that generate the patternsdue to four elementary constructive movements: translation, rotation, mirrorreection and glide reection (W 1999, 549). This geometrical ap-proach implies that a Neolithic community developed a strict psychologicaldenition of space and its organization. The mental potential for deningorganized visual structures onto pottery was also reected and intertwinedwith the other elements of social life: settlement and habitat organization,denition of the hierarchical system, exchange of goods, as well as with theelements of religious practice and symbolic communication.

Symmetry is one of the most crucial principles in the formation ofdesign practiced worldwide. Success of the theory of groups of symmetry incrystallography, which proposes the use of discrete groups of isometries of the

Euclidean plane in the study of the planar patterns, inuenced mathematiciansanalyzing the Moorish decorations in the Alhambra as well as the study ofthe Southwest American Indian pottery. This meant that exact mathematicalmethods were used in order to be applied to ethnographic and anthropologicalresearch (G 2004). The study of classication and analysis of patternsbased on symmetries was enriched by the contributions of different authors(S 1948; B 1956; W 1977; C 1986; J 1995;G 2002; G, S 2002; D 2004; F 2004;R 2004). Instead of analogies, these authors used a more precise geometric-crystallographic terminology and the theory of symmetry, which thus became

potent means for researching pattern principles. Gradually, symmetry analysisof patterns was developed as a consistent method, used mainly to study ancientdecorative arts or that of tribal populations (J 1995, 4).

According to D.K. W (1977, 12, 13, 23): the denition ofsymmetry focuses upon the symmetrical gure as an end product of a seriesof motions across a plane or about a point. In this perspective, a symmetryoperation can be dened as a process by which the basic asymmetric parts ofa gure pass through specied motions on specied axial loci until they reachidentity with themselves or superposition with the next gure. All symmetri-cal patterns are based upon one or more of the following four motions:

Translation: involves the simple movement of fundamental part or partsalong the line axis. Reection: requires the fundamental parts to be reected across line axisin a mirror image relationship. Rotation: requires the fundamental parts to be moved about a point axis. Theycan change orientation any number of N-fold times within 360 degrees arc. Glide reection: combines motions or mirror reections across a line fol-lowed by translation along that axis into the succeeding position. This motionproduces a gure that resembles the alternating left-right movement involvedin human locomotion.


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Fig. 3 The Four Rigid Motions in the Plane: translation, rotation, mirror re-ection and glide reection (after W 1999, g. 1).

Symmetry operations producing both single color and two color designunits result in three major design categories for plane pattern designs: nite,one dimensional and two-dimensional (Fig. 3). Finite patterns are single gures


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Fig. 4 Illustrations of a) classes of the one-dimensional design; b)most common forms of purenite design; c) generation of two-dimensional innite patterns (after W 1977, gs. 16,18, 24, 27, 30, 34, 35).

Fig. 5 Illustrations of a) 21 counterchanged classes of one-dimensional design; b) most commonforms of counterchanged nite designs; c)two-dimensional counterchanged design (after W1977, gs. 17, 19, 55).


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generated around a single point axis through which line reection axes mayor may not pass. One-dimensional are generated along a single midline axis.Two-dimensional patterns are generated along both horizontal and verticalaxes (Fig. 4). Finite designs can possess only the motions of reection androtation, whereas those that are one- and two- dimensional can be generatedby all four of the basic motions: translation, reection, rotation and glide reec-tion. The majority of pure nite classes, as well as one- and two-dimensionaldesigns have corresponding counterchanged forms (Fig. 5).

In his work, D. C (2004, 9) asserts that each of the four rigidmotions may be present, or absent, in given 1 or 2 dimensional pattern. Oftenthe four-symbol notation is given, so that the presence of certain movements isemphasized. Thus, the rst symbol isp signaling the existence of translation.The second is m used wherever there is vertical reection and lotherwise.The third symbol is also m if there is a horizontal reection, than a ifthere is glide reection and lotherwise. The fourth symbol is 2 if there is ahalf turn and lotherwise.

4. P R M

As has been emphasized by other researchers, painted pottery in Mace-donia can be classied into two cultural groups (G 1979; S1995): Amzabegovo-Vrnik (consisting of Ove Pole, Polog and Skopje regions)and Veluina-Porodin (Pelagonia and Ohrid regions), although some of thepatterns are found in both regions of Ove Pole and Pelagonia (Fig. 6). As forthe earlier phases of the Amzabegovo-Vrnik group (Fig. 2, 1-3; Fig. 7), mostof the patterns consisted of stair-like triangles, inclined stripes with triangles,netted stripes and doubled triangles, dots placed beside or on the line, zigzaglines, bent and vertical lines, angled lines positioned vertically or oppositelywith the peaks to the front. Due to the fragmentary state of the pottery, onlya few complete Early Neolithic compositional structures can be determinedincluding compositions constructed of zigzag lines where one zigzag line wasused as a pattern which is replicated all over the body of the pot (Fig. 7, 1).

Other compositions of this cultural group were created by using the hori-zontal arrangement of patterns where one of the patterns, consisting of a verticalstripe of angled lines, essentially serves as a frame in order to create two largerareas in which the central pattern is positioned, and duplicated or modied onthe other side of the pot as well (Fig. 7, 2-3). Rarely, vertical belt arrangementis present, although this manner of arranging the patterns is common for theMiddle Neolithic of the Amzabegovo-Vrnik group (N 2005; N2009a, 123-125). These compositions consisted of two belts with different pat-terns in each, mostly including triangles in the upper rst belt, while in the largerbelt below netted angled stripes or vertical lines are positioned (Fig. 7, 6-7).


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Fig. 6 Regional cultural groups developed in Middle Neolithic. I: Veluina-Porodin; II: Amzabe-govo-Vrnik.

In Pelagonia, a major part of the Veluina-Porodin cultural group,white painted pottery is present in both Early and Middle Neolithic with pat-terns mostly shaped as: triangles with lozenges, stair-like triangles and lines,stretched triangles, C and patterns, lozenges in negative, multiple crosses,as well as the so called drops (Fig. 2, 4-6). Within compositions, the beltdivision is rarely present and is only used for the spatial organization of pat-terns in groups. A vertical arrangement of patterns is most often employedby multiple horizontal stripes of one pattern, vertically positioned all overthe vessel, or only on the rim, the end of the neck and around the handles.Despite this approach, there is much more precise positioning of patterns inthe more complex and apparently abstract compositions. These compoundlarge patterns are actually duplicated on the other side of the vessel thuscreating entities on the whole surface (Fig. 2, 5 and Fig. 8).


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Fig. 7 The compositional principles on the white painted vessels. 1-6: Amzabegovo (after G1979, T. XIII: 1, 3, 6; T2006, gs. 3, 8; K 1971, 135, g. 1); 7, 8: Govrlevo (drawingby G. Naumov); 9: Nemanjica (drawing by G. Naumov).

Fig. 8 Reconstruction of white painted composition from Veluka Tumba (drawing by G. Nau-mov).


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Fig. 9 Geometric principles on the painted pot-tery from Podgorie, Giannitsa and Kuruay (afterT2007, g. 7).

5. N B A

The principles of Neolithic geometry were implemented among all cul-tures in Southeast Europe, which produced pottery in that period. An almostidentical approach toward decoration of vessels was practiced in Greece,Bulgaria, Serbia, Albania, Romania and Croatia, as well as in Turkey, wherein earlier periods some of the most recognizable patterns originated (Fig. 9).These cultures used similar mathematical arrangements, but incorporateddifferent patterns which in specic local styles were developed in authenticcompositions typical for each region. V. N (2002) and D.K. W


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Fig. 10 Geometrical approach towards squar-

ing the circle (after T2007, 107).

(1984) made important contributions towards a denition of visual structureswhich were painted on Early Neolithic vessels from Bulgaria and Greece.

In his attempt to decipher the mathematical approach in the decorationof vessels, N. Tasi deduced the geometrical concept of squaring the circleimplemented within painted compositions on vessels from Balkans and Ana-tolia (T2007, 2009). Analyzing pots from Giannitsa, Rakitovo, Kuruayand Hailar he noticed that the height of the pattern is almost identical to the

radius of the pot (Fig. 9), suggesting that Neolithic potters were using principlesof squaring the circle in order to precisely arrange 6 or more metopes alongthe vessels surface, without having an empty space between them (Fig. 10).Sometimes vertical lines were used, which have helped in applying the geometricequitation related with the incorporation of square patterns onto sphericalshell. On other vessels from Podgorie, the potters were calculating the size ofeach triangle (or a square made of two triangles) relative to the perimeter ofthe vessel. This calculation had to be in relation with the diameter of the pot,as well as with the sphere and the diameter of the base (T2007, 107).

6. C NMorphological analysis of pattern displays that painted compositions in

Macedonia were a result of rmly established conventions. The establishmentof such principles indicates that in one settlement or culture these patternswere repeated for a long period of time. During the Early Neolithic, someof the compositions were applied for at least 200 years, while in the MiddleNeolithic identical painted patterns were preserved for about 400 years. Someresearchers suggest that, during a continuity of 450-500 years, there was achange of 27-30 generations that implemented painted decoration as a systemof communication over this period (N, K 2003).


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Fig. 11 Regional Neolithic communities and their visual identities manifested through whitepainted pottery (design by G. Naumov). Skopje region (1-5: Govrlevo, 6-8: Zelenikovo); Polog (9-15: Stene, 16: Dolno Palite); Ove Pole (17-21: Amzabegovo, 22: Nemanjica); Pelagonia (23-26:Veluka Tumba, 27-28: Vrbjanska uka).

The extensive elaboration of painted patterns and compositions sug-gests that they were used as an essential element for preserving the traditionof the community. Hence, the tradition became a result of the creation ofvisual memory which strengthened the communitys identity. Consequently,painted vessels were used as an element of symbolic communication amongthe members of one community, but also they embodied the visual identityof that community. Analyses of compositional styles asserted the afnity of acertain community in a region towards concrete patterns and compositions.Considering that Neolithization in the territory of Macedonia was a rapidprocess, it can be observed that this quick demographic or cultural activitydeveloped new local identities (N 2009b). This cultural colonizationwas probably performed by a group or several individuals who inhabited acertain area and, independently or in contact with the indigenous population,created new settlements in which the elements of local identity were developed(T 1999; Z 2001; P 2003). Although these settlementswere based on general Neolithic acquisitions (agriculture, cattle breeding,architecture and pottery), in the domain of mobile mediums authentic localfeatures were developed.

Pelagonian design was developed according to its own principles andit was specic only to that region: complex compositions with bent lines andtriangles, and patterns, lozenges, dots, angled stripes, replicated cross, etc.(Fig. 11, 23-28). Although some of these patterns were used in Ove Pole aswell, they were developed in completely different compositions. The disposi-tion and invention of new patterns, as well as the horizontal organization


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of the compositions with zigzag and oral patterns and stair-like trianglesdemonstrate that the population of this region had a sense for visual isola-tion (Fig. 11, 17-22).

In contrast to these two regions, completely new patterns and composi-tions appear in Skopje Plain, including bordered triangles and netted stripes(Fig. 11, 1-8). It is still too early to tell if t they were synchronic with thosefrom Ove Pole or if they belong to Middle Neolithic painted vessels as anEarly Neolithic tradition. Similar patterns were also painted in Polog, includ-ing elongated triangles and stripes with triangles resembling those both onPelagonia and Ove Pole pottery (Fig. 11, 9-16).

7. M N The Early Neolithic traditions developed authentic visual features in the

communities of each region in the Republic of Macedonia. These communitiesexisted synchronically in different regions, and therefore intended to emphasizethe elements of visual identication. There are noticeable differences in theselection of patterns and in the approach towards compositional structure, i.e.what is typical for Ove Pole is not present in the Skopje Plain and vice versa.In the domain of painted decoration at the beginning of the Middle Neolithicthere are apparently rapid changes. In all sites from the Amzabegovo-Vrnikgroup, where previously different Early Neolithic designs were present, inthe sixth millennium BC new unied compositions in brown and black pig-ments began to be painted (Fig. 2, 7-9). These compositions often consistedof curved patterns which were developed by the multiplication of a Y motifcontinuously rotated within the complete composition, thus constructing avisual unit (Fig. 12).

Unlike these regions, Pelagonian Middle Neolithic was not affected bythe design changes. The communities from this region maintained white paint-ing and changed the repertoire of patterns and compositions only slightly. Still,despite the generalization of new compositions in the Amzabegovo-Vrnikgroup and preserving the decorative traditions in Pelagonia, the settlementsin these regions shared certain visual and symbolic types of communicationin the domain of patterns, beliefs and gurine production. At the beginning ofthe Middle Neolithic, specic white painted patterns, as well as clay modelsof anthropomorphic houses were produced in both of the above mentionedregions. Although some regional and typological characteristics can be noticedin the decorative and sculptural structures (N 2006, 66; N 2007,258; C 2007), they integrate identical morphological and semanticfeatures. This demonstrates that in the Middle Neolithic there was a newlyestablished communication, which further strengthened the relations betweencommunities in different regions.


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Fig. 13 Reconstruction of a rectilinear and curved composition from Zelenikovo and Govrlevo(drawings by G. Naumov).

Fig. 12 Curved compositions painted on the Middle Neolithic cups (drawings by G. Naumov).

8. S

Symmetry analysis makes it possible to reconstruct the painted compo-sitions where the denite canons and their independent and group-orderedrelations with other patterns have been established. By gathering data providedby a number of preserved compositions, the compositional entireties can beidentied even on shards on which elements of the dened styles are beingpreserved. On the fragments where parts of elaborated patterns are still vis-ible and where it is noticeable that they are organized in an already denedcompositional entity, the rest of the composition which is not preserved canbe envisaged. That way, the feasible reconstruction would be approached withcertain probability, mostly because one composition is not always identicallyrepeated.

Two examples are given below, where options for the reconstruction ofcompositions are illustrated. Each of them, if we do not consider the dimen-sions, possesses enough elements for reconstruction, especially those that aremost typical and which belong to the style with alternated lines in a form ofa triangle, and those belonging to the style with wave-like lines in negative(Fig. 13). After the reconstruction of the fragments has been completed, we


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can attempt to apply the reconstruction to the surface of the pot. When thefragments are consistently transformed into the reconstruction of a pot, anidea of the quantity and variety of ceramic production in the Neolithic canbe obtained.

9. T

Stylistic analyses of the design demonstrate that symmetry is the basicconcept of the composition construction. It is used for the arrangements ofthe patterns and the building of the horizontal and vertical structure of thecomposition. Symmetry is a basic element in the perception of nature, andvarious populations used these visual principles whenever constructed visualunit. The patterns and symmetry are also employed as visual metaphor in therepresentation of the essential principles of a culture. Consequently, a cultureemploys the symmetrical structure of patterns as an element of metaphorictransposition of the ideological principles through an artistic medium, i.e.pottery.

The function of the visual metaphor is fundamental for preservingthe basic codex and crucial concepts of the culture. Specic symmetryused for metaphoric visualization of the cultures permanent values dif-fers between cultures, depending upon the different approach toward theconceptualization of the essential life advancements and relations (E-S 1980, 121-160; C 1994, 39-67; W 1995, 14,51; W 1999, 548-553; G 2003, 259-269; C 2005,16-129). Therefore, durable mediums are usually used as an archive ofthese cognitive elements. This knowledge about the ones own culture andthe world in which it exists is most often in relation with identity and re-ligion in general.

The Neolithic painted pottery from the Republic of Macedonia alsogenerates metaphoric cognizance of the cultures and their surrounding (Fig.2). Symmetry analysis reveals that this visual approach has specic localfeatures in every region, indicating that several authentic concepts existedin Pelagonia, Ove Pole and Skopje Plain. It is still difcult to say whetherthese concepts were related to a common idea, but considering the identicalelements of socio-economic existence, it can be assumed that all these cul-tures intended to represent similar perpetual processes. This was followedby the incorporation of geometric principles which except for their aestheticor decorative utility, appeared along side the most complex cognitive pro-cesses in the Neolithic. Moreover, they demonstrate that Neolithic commu-nities developed absolutely rational perception of space and employed itsstructural manifestation in the production of material culture and the ideasincorporated within.


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Acknowledgments

I would like to thank M. Katalin Bir for accepting to include this paper on Neolithicsymmetry in this publication, as well as for her understanding and patience during the nal-ization process.

G NInstitute for History of Art and Archaeology

University of SkopjeRepublic of Macedonia

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ABSTRACT

Neolithic pottery from the Balkans and Anatolia is well known for its remarkable andunique decoration. Very often the visual features of these objects are used for determining therelative chronology of excavated sites, without considering its potential for mathematical ob-servation. The repertoire of patterns used for developing the compositional structures paintedon the vessels provide abundant data for such analysis. Almost all of the fragments discovered

so far as well as the completely preserved painted pots from these regions were decorated fol-lowing several visual principles which made it possible to achieve a precise disposition of thepatterns on the spherical surface of the vessel. This decorative approach was established on thebasis of the standards of Neolithic geometry which employ both symmetry and the principlesof visual entropy. For this reason, the painted vessels from Early and Middle Neolithic settle-ments discovered in the Republic of Macedonia provide a variety of information about theorganization and structures incorporated on the decorated pottery. In the earlier phases thesepainted compositions were mostly based on The Four Rigid Motions of pattern dispositionon a one dimensional format, while later, besides using this concept, the principle of asym-metry was implemented in order to compose a two-dimensional image. The aim of this paperis to identify all possible forms of plane symmetry, patterns and compositions applied in thedecoration of Neolithic vessels from the Republic of Macedonia, as well to promote the useof geometric symmetry as a possibility for the reconstruction of decorated fragments.

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Symmetry analysis of neolithic painted pottery from the republic of macedonia