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Quantitative Analysis of the Symmetry of Artefacts: Lower Paleolithic Handaxes
Journal of Archaeological Science (1998) 25, 817–825 Article No. as970265
Quantitative Analysis of the Symmetry of Artefacts: Lower Paleolithic Handaxes Idit Saragusti* and Ilan Sharon Institute of Archaeology, The Hebrew University of Jerusalem, Jerusalem 91905, Israel
Omer Katzenelson and David Avnir Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (Received 22 September 1997, revised manuscript accepted 11 November 1997) The ability to perceive and produce symmetrical shapes is usually taken to be a major step in the development of human cognition. Despite the importance of the concept of symmetry to studies of early human development, its current use is usually based on loose qualitative assessment. A more informative approach would be to use a continuous scale of this shape property rather than the strict current language of ‘‘more’’ or ‘‘less’’. A symmetry measurement tool has been developed, which is based on evaluation of the minimal distances that the vertices of a structure have to move in order to attain the required symmetry. Using this Continuous Symmetry Measure (CSM) method, it is possible to evaluate quantitatively how much symmetry exists in a non-symmetrical configuration and how the nearest symmetrical shape looks. We test and demonstrate the feasibility and versatility of this approach on handaxe samples from three Lower Paleolithic sites in Israel, Ubeidiya, Gesher Benot Ya’aqov and Ma’ayan Barukh, representing various stages in the Acheulian Techno-complex of the Levant. We provide, to the best of our knowledge for the first time, quantitative demonstration that the overall symmetry of handaxes generally increases, and that the variability decreases over time. 1998 Academic Press
Keywords: ACHEULIAN, LOWER PALEOLITHIC, HANDAXES, SYMMETRY.
Background: The Near Symmetry of Handaxes
ability to perceive bilateral symmetry, as manifested in typical Late Acheulian handaxes, as a major argument for relating a higher degree of intelligence to the makers of these tools compared with the degree of intelligence demonstrated by the earlier Oldowan industrial tradition. He argued that the spatial concepts required to manufacture Oldowan tools are rather simple, topological in their nature, whereas the manufacture of typical symmetric Late Acheulian handaxes (as well as other tools) had required more sophisticated Euclidean concepts. Using the Piagetian theory of developmental psychology, Wynn argued that this difference in spatial concept requirements is indicative of the different developmental stages of the makers of these two industrial complexes. He also proposed that the ‘‘minimum necessary competence’’ manifested in the Oldowan repertoire is that of preoperational intelligence (Wynn, 1985), and is similar to that recognized for modern apes (Wynn, 1993: 306). This degree of intelligence is inferior to the one ascribed by Wynn to the makers of the Late Acheulian assemblages, the operational intelligence, the characteristic intelligence of modern adults. In effect, Wynn proposed qualitatively a correlation between the degree
T
he concept of symmetry has attracted virtually all domains of intellectual activity and has strongly influenced the sciences and the arts, archaeology being no exception (Hargittai, 1986, 1989; Gruber & Iachello, 1988 and earlier volumes in this series; Toth, 1990; Heilbronner & Dunitz, 1993). It has functioned as a condensed language for the description and classification of shapes and structures, as an identifier of inherent correlation between structure and physical properties of matter, and as a guiding property to artistic and practical aesthetic design. Because symmetry is one of the main shape characteristics of handaxes from Acheulian assemblages, it has been at the focus of many Lower Paleolithic studies. Scholars investigating this period have often argued that the ability to perceive symmetrical shapes, as expressed in the production of many handaxes, is meaningful in attempts to reconstruct the evolution of human cognitive, behavioural and technological capacities. Wynn (1985: 37), for instance, used the
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of intelligence and the degree of symmetry of the artefacts. Our main proposition in this report is that the degree of symmetry is a measurable trait. In fact the intuitive feeling that symmetry and intelligence or cultural manifestation are linked, appears in many other studies as well. Thus, Isaac (1986), though partly questioning Wynn’s conclusions, also saw the ability to conceive and produce symmetrical shapes as one of the indications of the higher complexity of the Acheulian industrial complex, compared to earlier traditions. In yet another study, Mithen (1994) argued that the ‘‘nearly perfect symmetry’’ of many handaxes from Acheulian sites in southeastern England as well as other properties which in his view have no functional significance, is the result of intense social learning, as opposed to individual learning, ascribed to the Clactonian and High Lodge assemblages. We draw attention to Mithen’s use of the adjective ‘‘nearly’’, signalling the actual need to provide a conceptual and practical tool that will allow one to assess this shape property on a quantitative level. Such scaling of symmetry would allow one to answer questions such as ‘‘how much symmetry is there in a given handaxe?’’; ‘‘by how much is one handaxe more symmetrical than another?’’; and so on. Indeed, despite the central importance of the concept of symmetry in these and many other studies, it has usually been treated in a qualitative rather than quantitative way, and sentences like ‘‘. . . these early handaxes are roughly symmetrical — none approaches the fine symmetry of later example . . .’’ (Wynn, 1985: 40 [our emphasis]) are often found in the literature. This relative and subjective description, not exclusive to archaeology, is by its nature imprecise. The need for objective means for symmetry measurement (Crompton & Gowlett, 1993) is the main motivation for this study. To conclude this introduction we recall that the traditional methods for morphometric description and analysis of handaxes have been based on various size parameters such as length, width and location of maximum width. These methods include those of Bordes (1961), Roe (1964, 1968), Isaac (1977) and others. These parameters were used to define various shape indices. The ‘‘index of refinement’’, for example, which expresses the degree of flatness in cross-section, is calculated by dividing the maximum thickness of the tool by its maximum width (Roe, 1968). In general these methods assume implicitly or explicitly that handaxes are symmetrical. As a consequence these methods have an inherent difficulty in treating tools with irregular forms, including those which largely depart from bilateral symmetry, thus ignoring a potential source for inter and intra assemblages variability (Wynn & Tierson, 1990). Bordes, for example, who used various size ratios as criteria for typological classification of handaxes, resorted to the term ‘‘varia’’ for irregular handaxes (Bordes, 1961). Being aware of this problem, Wynn & Teirson (1990) suggested an
alternative method based upon polar coordinates and which does not assume symmetry a priori. To the best of our knowledge, none of the proposed morphometric methods addresses the concept of quantitative symmetry and the goal of measuring the degree of symmetry. In a recent series of publications, the concept and the methodology of symmetry content measurement were developed (Zabrodsky & Avnir, 1995 and references cited therein; Avnir et al., 1996 — a review) and applied successfully to a variety of symmetry-related problems at the molecular and supra molecular levels (Buch et al., 1995; Kanis et al., 1995; Katzenelson, Zabrodsky Hel-Or & Avnir, 1996; Keinan, Zabrodsky Hel-Or & Avnir, 1996; Pinto, Zabrodsky Hel-Or & Avnir, 1996). This approach of Continuous Symmetry Measures (CSM), described in the next section, is general and not limited to the molecular scale. Having this quantitative tool at hand, the main aims of this study were, first, to introduce the concept of quantitative measurement of symmetry to archeological artefact studies; second, to test whether the CSM method is sensitive enough to detect visible differences in the degree of symmetry of artefacts; and third, to obtain preliminary indications regarding possible quantitative evaluation of changes in symmetry over time. In general then, we are testing here whether quantitative analysis of symmetry is applicable to archaeological research as an objective tool for assessment of shape through symmetry. Thus in the next section we describe the method for quantitative evaluation of symmetry. We then apply it to the analysis of handaxe samples from three Lower Paleolithic assemblages from Israel, as a test case for this method.
The Continuous Symmetry Measure (CSM) Approach As the examples mentioned above indicate, at present, symmetry serves as an approximate descriptive attribute of reality, lacking the necessary quantitative dimension. This problem is not unique to archaeology, but is common to the natural sciences and many other areas of study. As already mentioned, a general conceptual approach and a practical solution to the problem of symmetry measurement have been developed recently (by one of us (D. A.) and his colleagues (Zabrodsky & Avnir, 1995; Avnir et al., 1996)). It is briefly described here, with special attention given to the details needed for the current report. The CSM approach is based on the notion that quite often it is important to evaluate ‘‘how much’’ of a given symmetry there is in a structure rather than treating it as a ‘‘more’’ or ‘‘less’’ property. Consequently, this implies treating symmetry as a structural property of a continuous behaviour, which should be evaluated by a measurement tool. The design of a measurement tool involves some degree of arbitrariness, in the sense that one has to decide on issues such
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^ P0
P0 P'0 Normalize P2
Symmetry transform P'2
P1
(a)
P'1 (b)
^ P2
^ P1
(c)
^ P0 P'0
P'2
^ P2
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Figure 1. The basic features of the Continuous Symmetry Measure (CSM): In order to evaluate how much bilaterality (mirror symmetry) there is in the triangle (a), its size is normalized (b), and the mirror symmetrical structure (c) which is nearest to (b) is found using the symmetry transform described in the text and in Appendix 1. (d): The S(G) value is calculated from the minimal distance between (b) and (c), using equation 1. In this case, S(G)=4·96.
as how should the zero-reference level be set, what should be the maximal value, what should be the actual measurement and what normalization procedures should one employ. Any such decision is open to debate. Keeping this in mind, the symmetry measure is based on a definition that is as minimalistic as one could practically get. Our proposed answer (Zabrodsky & Avnir, 1995) to the question ‘‘How much of a given symmetry is there in a given structure?’’ has been: ‘‘Find the minimal distances that the vertices of a shape have to undergo, in order for the shape to attain the desired symmetry’’. In a formal way, given n vertices of the original configuration, located at Pi, and given a symmetry point group G, the amount, S(G), of this symmetry in this configuration 100 S(G)= n
n
( i=1
// P 1P| // i
i
2
(1)
where P | i are the corresponding points in the nearest Gsymmetric configuration. Equation 1 is general and allows one to evaluate the symmetry measure of any shape relative to any symmetry group or element. In order to avoid size effects, the size of the original structure is normalized to the distance from the centre of mass of the structure, (which is placed at the origin) to the farthest vertex. This distance is 1 and each of the other Pi’s is divided by the same maximal Pi. The nearest set of P | i’s, i.e. the set of coordinates describing the nearest symmetrical shape, also obtained in terms of the normalized coordinates, is computed with an algorithm (see example in Appendix 1), described in detail in Zabrodsky & Avnir (1995). The algorithm
thus also provides one with the shape of the nearest symmetrical object (Figure 1). The factor 100 is used for convenience of expanding the scale which is bound by 0cS(G)c100. If a shape has the desired symmetry, S(G)=0. A shape’s symmetry measure increases as the shape departs from G-symmetry and it reaches a maximal value (not necessarily 100). Square values are taken so that the function is isotropic, continuous and differentiable. In the archaeological literature the term ‘‘symmetry’’ has usually been limited to bilateral symmetry, namely to symmetry around a mirror that bisects the body. In the natural sciences this symmetry is also known as achirality: a chiral object is one that lacks a symmetry mirror (denoted ó), and its main property is that it cannot be superimposed with its mirror image (leftright hands being a classical example). Many other symmetry operations (e.g. symmetry of rotation, tetrahedral symmetry and so on, which may be of relevance for other archaeological problems as well) are treatable by equation 1, but we limit our discussion here to bilateral symmetry, referred to in this report as simply ‘‘symmetry’’. The meaning of equation 1 is then the minimal distance of a given shape from exact bilateral symmetry. We shall also now make the distinction between three-dimensional (3D) symmetry and twodimensional (2D) symmetry: the former refers to the symmetry of an object which occupies volume, and the latter to an object confined to a plane. An example for a 2D object is the contour of a projection of a handaxe onto a drawing paper, as commonly used in archaeology: 2D presentations of plan view and various crosssections of the 3D handaxes are standard reliable
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S(σ) = 2.65 G.B.Y. N Haifa Ubeidiya Jordan River
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(d) Figure 2. Demonstration of the CSM method: the mirror symmetry values of a series of polygons vertices, the nearest mirror symmetric shapes and the nearest mirror lines.
representations of symmetry features. For the 2D contours, the raw materials of this study, the symmetry mirror is a line that bisects the shape in-plane. An important feature of the CSM approach is that no pre-selected specific reference shape is assumed at the beginning of the analysis, though it is obtained, as mentioned above, as an end product (Figure 2). An additional important feature is that the S(ó) values are on the same scale, so that one can ask, for example which member of a set of polygons is the closest or farthest from being symmetrical. In our example we find that the polygon (Figure 2(b)), having the lowest S(ó) value, is the most symmetrical, i.e. it is the closest to bilaterality, and the triangle (Figure 2(a)) is the most asymmetrical. The main practical problem is then how to find the nearest symmetrical shape to a given one, namely how to locate the specific set of P | i’s that will minimize S(ó). A detailed solution to this problem was provided, which is based on what we termed the ‘‘folding-unfolding’’
Figure 3. The location of the studied sites, Ubeidiya, Gesher Benot Ya’aqov (G.B.Y.) and Ma’ayan Barukh (top right).
rationale (Zabrodsky & Avnir, 1995). It was also proved that the algorithm (Appendix 1) indeed provides a minimal solution to the symmetry problem.
Bilateral Symmetry Analysis of Handaxes The handaxes Samples of handaxes from three Lower Paleolithic sites in Israel (Figure 3) were used as a test-case. These are: Ubeidiya (N=20), Gesher Benot Ya’aqov (N=16) and Ma’ayan Barukh (N=8), all located in the Jordan Valley (a segment of the Dead Sea Rift) representing various stages in the Acheulian Techno-complex of the Levant. Ubeidiya, a Lower Pleistocene site which is located c. 3·5 km south of the Sea of Galilee, is currently dated to 1·4 million years ago (mya) (Bar-Yosef & Goren-Inbar, 1993). The sites of Gesher Benot Ya’aqov and Ma’ayan Barukh have no absolute dating yet, but based on geological, biostratigraphical and archaeological considerations, they are tentatively
Quantitative Analysis of the Symmetry of Artefacts
(a1)
(a2)
(a3) S(σ) = 1.84
(b1)
(b2)
(b3) S(σ) = 0.77
(c1)
(c2)
(c3) S(σ) = 0.29
Figure 4. Handaxe samples from the three sites: (a) Ubeidiya (Bar-Yosef & Goren-Inbar, 1993); (b) Gesher Benot Ya’aqov; and (c) Ma’ayan Barukh (Stekelis & Gilead, 1966); Original drawings: a1, b1 & c1. Outlines redrawn, using x-, y-coordinates: a2, b2 & c2. The nearest symmetrical shapes, their S(ó) values and the mirror lines: a3, b3 & c3.
dated to the Middle Pleistocene (i.e. between c. 0·78 and 0·13 mya). Gesher Benot Ya’aqov, located c. 15 km north of the Sea of Galilee, is tentatively dated to the early part of the period (Goren-Inbar, et al., 1992), while Ma’ayan Barukh, located further north, is somewhat later (Stekelis & Gilead, 1966; Gilead, 1970). Underlying the choice of these sites, was the generally accepted hypothesis, mentioned in the introduction, that the overall symmetry of handaxes generally increases over time. Indeed, as illustrated in Figure 4, the assemblages from these sites agree with this intuitive assumption: handaxes from Ubeidiya, the earliest site in this study, look less symmetrical than those from Gesher Benot Ya’aqov, which look less symmetrical than the ones found at Ma’ayan Barukh, the latest site. Can the CSM method translate it to quantitative terms?
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Data acquisition and analysis CSM analysis requires that the shape of the object to be analysed be represented by a set of n vertices or boundary points (see Appendix 1). For the purpose of this pilot study, we produced images of the handaxes by hand-drawing their outlines in plan-view projection (i.e. the longitudinal axis of the handaxe is used as the y-axis and the latitudinal axis (the longest straight line orthogonal to the longitude) as the x-axis; the thickness of the object is ignored). This view is the one most commonly used for shape evaluation of handaxes, although the degree of symmetry of longitude and latitude cross-sections may also be of interest. In a different facet of the same research programme, we are experimenting with automating data acquisition, using a digital scanning camera. This will enable the more rapid processing of the much larger samples needed for the next stages of research. These drawings were scanned into a personal computer. We found that scanning in a resolution of 300 dots per inch is optimal; a higher resolution is unnecessary and a lower one may result in loss of important information. The scanned outlines were then automatically traced, resulting in sets of x-, y-coordinates, each set of which served as the desired boundary points. The number of points was automatically determined in this procedure and it depended on the nature of the original outline: two points are sufficient to describe a straight line segment, whereas more points are needed for a curved one. In any event, as described above, the resulting symmetry measure is normalized to the number of points. In the analysed sample the number of coordinate points (n in equation 1) ranges between 133 and 343. As shown in Figure 4, redrawing of the outlines with these coordinates preserves the information and the redrawn outlines are very close to the originals. The symmetry values of each of the outlines was computed according to equation 1, using the sets of coordinates. The output is the minimal S(ó) value, namely, the minimal distance between the original outline and the nearest mirror-symmetrical shape. An additional output is, as explained above, the coordinates of the symmetrized shape and the position and inclination of the mirror line, ó (Figure 4). In most cases analysed we found that this mirror line is longitudinal (base to tip axis of the handaxe). Yet, in a very few cases the transversal and longitudinal axes provide two close minima of S(ó), with the former being slightly lower (e.g. Figure 4(a)). However, in order to make the symmetry comparison of all handaxes meaningful, the S(ó) value of the longitudal axis was always taken. Underlying this decision is the axiomatic assumption that handaxes have a ‘‘natural’’ direction defined through their ‘‘base’’ and opposite ‘‘tip’’. Results The frequency distribution of the S(ó) values by sites is represented in Figure 5. The S(ó) mean values for the
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0.3
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1.3
1.5
1.7
1.9
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2.3
2.5
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S(σ) values
Figure 5. Distribution of S(ó) values, the degree of bilateral symmetry, of the sampled handaxes, per site. UB, Ubeidiya; GBY, Gesher Benot Ya’aqov; MB, Ma’ayan Barukh.
samples are: Ubeidiya: 0·91 (N=20); Gesher Benot Ya’aqov: 0·77 (N=16); and Ma’ayan Barukh: 0·27 (N=8). Although ranges and standard deviations for small samples are not reliable, we cite them here just as an additional comparative value which translates what is clearly seen in Figure 5. These values are: for Ubeidiya, ..=0·87; range=0·1–3·3; Gesher Benot Ya’aqov, ..=0·54; range=0·3–2·3; and for Ma’ayan Barukh, ..=0·07 and the range=0·2–0·4.
Discussion Although this feasibility study is based on a relatively small number of samples, below the routine requirements for statistical analysis, some interesting points can already be made at this feasibility study stage: (1) The CSM methodology seems to translate correctly the qualitative impression of degree of symmetry of handaxes into quantitative values. (2) As seen clearly in Figure 5, the S(ó) values generally tend to decrease over time, i.e. handaxes become generally more symmetrical. This trend is also reflected by the mean S(ó) values. The hypothesis that symmetry and time correlate inversely seems, therefore, to be corroborated. Needless to say, one cannot argue that time is the only factor or even the prime factor which affects the degree of symmetry of
handaxes. Many other factors, for instance the nature of the raw material, must have played an important role in determining the shape of these tools. Yet the analysis of these factors is beyond the focus of this report. (3) Another pertinent observation is that the spread of S(ó) values decreases over time. Keeping in mind the limitation of rigorous statistical interpretation in this case, this observation is nevertheless consistent with another well accepted assumption, namely that interassemblage standardization increases over time (Isaac, 1972, 1986; Belfer-Cohen & Goren-Inbar, 1994). This phenomenon is expressed either by the metrical variables mentioned above (Gowlett, 1984) or in qualitative terms. Isaac (1986), for instance, refers to: ‘‘the imposition of a standard, arbitrary form, which is repeated in numerous cases’’ and to ‘‘the presence in assemblages of an increasing number of distinct, arbitrary forms, each manifest by a series of standardized, repeating items of like forms’’. The preliminary results of our study may serve as a quantitative support for this view. The relatively wide spread of S(ó) values in the earliest assemblage (Ubeidiya) may indicate that the occasional appearance of nearly symmetrical handaxes at the earliest phases of the Acheulian is random. However the decrease of the spread and the mean S(ó) values over time indicates that the emerging prevalence of near symmetry in the later phases of the
Quantitative Analysis of the Symmetry of Artefacts
Acheulian is a more consistent phenomenon. This trend may reflect the increasing ability to perceive and impose specific (symmetrical) forms. (4) Although the degree of symmetry increases over time and the variability narrows, these tools do not reach absolute perfection, and none of the handaxes gave a S(ó) value of zero (ideally symmetrical), not even in the latest assemblage from Ma’ayan Barukh in which the S(ó) values concentrate very close to zero, but not at it. This result may reflect an inherent difficulty associated with the very manufacturing method of these and other tools which are individually hand-made and not industrial products. (5) It would be an over simplification to argue at this stage that the increase in the degree of symmetry and standardization over time simply reflects an increase in the abilities to perceive, produce and impose symmetrical shapes. Other interpretations are possible. The sporadic occurrence of nearly symmetrical handaxes in very early assemblages, for example, may suggest that the ability to perceive symmetry already existed at this phase, but for some unknown reason did not become fully manifested until the later phases. The correlation between the degree of symmetry and other morphometric and technological variables must be studied and analysed before a more thorough interpretation is possible.
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consideration other contributing factors, as well as basing such research on a much wider chronological, geographical and cultural perspective. Yet, an in-depth study of this crucial issue requires a quantitative measure for symmetry of artefacts, as outlined in this report.
Acknowledgements We thank Dr Hagit Zabrodsky Hel-Or for close assistance in the computational aspects of this project, Professor N. Goren-Inbar for encouraging this study and for allowing us to use the material from G.B.Y. and Professor O. Bar-Yosef for permitting us to use the material from Ubeidiya. We gratefully acknowledge the financial support of the following Foundations for supporting various portions of this study: The Israel Science Foundation (to both D.A and I.S.), the Robert Szald Institute for Applied Science of the P.E.F Israel Endowment Fund, the Horowitz Foundation, the Minerva Foundation, and the National Center for Cooperation between the Sciences and Archaeology. I.S. is grateful to S. and G. Itai, A. Belfer-Cohen and Y. Cohen for their comments. Figure 3 was drawn by G. Hivroni.
References Conclusions Our starting point was the prevailing situation where symmetry often serves as an approximate descriptive ‘‘language’’ in archaeological studies. We proposed that while it is true that an imprecise language helps grasp complex situations and identify first order trends, there is always a danger of missing the full picture because of vagueness. Thus, it has been our aim here to provide a method for a quantitative evaluation of symmetry in archaeology. This method, when used to evaluate the degree of symmetry of handaxes, was found to be sensitive enough for measuring observed differences in symmetry. The CSM method may, therefore, be applied as a means to achieve a quantitative, objective and more accurate description of an important global shape feature of artefacts in archaeological research. As such, this method provides a new tool for discussing issues like the effects of raw materials on the shape of artefacts (Jones, 1979), regional comparisons between assemblages (Wynn & Tierson, 1990), allometry (Crompton & Gowlett, 1993) (size related shapes), chronological changes, and other topics of the archaeological research in which symmetry plays an important role. Possible correlation between the mental abilities of Early Man and the tools he made (as suggested in the studies cited above), is a far reaching and complex issue. As mentioned above, it would be an over simplification to argue that increased symmetry directly reflects increased cognitive ability, without taking into
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Heibronner, E. & Dunitz, J. D. (1993). Reflections on Symmetry in Chemistry and Elsewhere. Basel: VCH. Isaac, G. L. (1972). Chronology and the tempo of cultural change during the Pleistocene. In (W. W. Bishop & J. Miller, Eds) Calibration of Hominid Evolution. Edinburgh: Scottish Academic Press, pp. 381–430. Isaac, G. L. (1977). Olorgesailie, Archaeological Studies of a Middle Pleistocene Lake Basin In Kenya. Chicago: The University of Chicago Press. Isaac, G. L. (1986). Foundation stone: early artefacts as indicators of activities and abilities. In (G. N. Bailey & P. Callow, Eds) Stone Age Prehistory. Cambridge: Cambridge University Press, pp. 221– 241. Jones, P. R. (1979). Effects of raw materials on biface manufacture. Science 204, 835–836. Kanis, D. R., Wong, J. S., Marks, T. J., Ratner, M. A., Zabrodsky, H., Keinan, S. & Avnir, D. (1995). Continuous symmetry analysis of hyperpolarizabilities. Characterization of second order nonlinear optical response of distorted benzene. Journal of Physical Chemistry 99, 11061–11066. Katzenelson, O., Zabrodsky Hel-Or, H. & Avnir, D. (1996). Chirality of random supramolecular structures. Chemistry European Journal 2, 174–181. Keinan, S., Zabrodsky Hel-Or, H. & Avnir, D. (1996). How much chirality was there in Pasteur’s tartrate? The quantitative evaluation of the chirality of crystals. Enantiomer 1, 351–357. Mezey, P. (1997). Fuzzy measures of molecular shape and size. In (D. H. Rouvray, Ed.) Fuzzy Logic in Chemistry. Sand Diego: Academic Press, pp. 139–223. Mithen, S. (1994). Technology and society during the Middle Pleistocene: hominid group size, social learning and industrial variability. Cambridge Archaeological Journal 4, 3–32. Pinto, Y., Zabrodsky Hel-Or, H. & Avnir, D. J. (1996). Continuous chirality analysis of interconversion pathways of the water-trimer enantiomers. Journal of the Chemical Society, Faraday Transaction 92, 2523–2527. Roe, D. A. (1964). The British Lower and Middle Palaeolithic: some problems, methods of study and preliminary results. Proceedings of the Prehistoric Society 30, 245–267. Roe, D. A. (1968). British Lower and Middle Palaeolithic handaxe groups. Proceedings of the Prehistoric Society 34, 1–82. Stekelis, S. & Gilead, D. (1966). Ma’ayan Barukh—a Lower Palaeolithic Site in Upper Galilee. Jerusalem: Israel Prehistoric Society. Toth, N. (1990). The prehistoric roots of a human concept of symmetry. Symmetry: Culture & Science 1, 257–281. Wynn, T. (1985). Piaget, stone tools and the evolution of human intelligence. World Archaeology 17, 32–43. Wynn, T. (1993). Two developments in the mind of early Homo. Journal of Anthropological Archaeology 12, 229–322. Wynn, T. & Tierson, F. (1990). Regional comparison of the shapes of Later Acheulean handaxes. American Anthropologist 92, 73–84. Zabrodsky, H. & Avnir, D. (1995). Continuous symmetry measures. Journal of American Chemical Society 117, 462–473.
Appendix 1: The Folding/Unfolding Method The ‘‘folding-unfolding’’ algorithm is one of the methods for finding the nearest symmetrical shape. Algorithms, proofs and variants on the theme described below, can be found in the references cited above. The main features of the folding-unfolding method are described here for bilateral symmetry. The example we use demonstrates how to measure the bilateral symmetry of a general boundary line (Figure 6). The object to be symmetrized (Figure 6) is converted to a necklace of an even number of boundary points, Pi, as dense as required (N=30 points in this case). Where the object is already represented by vertices, this
(a)
(b)
Y
Y P2
P2
X
~ P2
X ~ P1 = P1
P1 (c)
(d)
Y
Y
^ P2 ~ P2
X ^ P1 ~ P1
(e)
X ^ P1
(f)
Figure 6. The folding-unfolding method explained for 2D bilaterality (chirality): (a) A contour composed of 30 points, Pi. The size is scaled to 1 as shown by the arrow. (b) The nearest S(ó)-symmetric structure, P | i, to (a). (c)–(f): Evaluation of the S(ó) value for a pair of points, P1, P2, (c), with respect to a given mirror line. (d) P1, P2 are folded to P x 1, P x 2, averaged (e) to P | 1 and unfolded (f) into the S(ó) symmetric pair, P | 1, P | 2.
step is not needed. Its centre of mass is then determined from the averaged sum of coordinates and placed at the origin, and the distance from this centre to the farthest Pi is scaled to 1 (Figure 6(a)). The aim is to find the nearest set of P | i’s which is ó-symmetrical, namely, to find that reflection line that will cause the set of Pi’s to move minimally to the set P | i’s (Figure 6(b)). In the symmetrized object, each P | i must have a ó-symmetric counterpart P | i–j across the reflection line (or it must be located on the reflection line). The full set of Pi’s is divided into subsets of two points, and all possible divisions are tested. (Points on the reflection line are duplicated). Here is how a pair of points, P1, P2, are ó-symmetrized with respect to a given reflection line ó (Figures 6(c)–6(f)): (1) The identity element, E, operates on P1 and it x 1); ó operates on P2 forming remains in place (P1P the reflected P x 2; a pair of adjacent points, P x 1, P x 2 is obtained (Figure 6(d)). We termed the step of applying the elements on the vertices the folding step. The
Quantitative Analysis of the Symmetry of Artefacts
essence of our methodology is to minimize the P x 1, P x2 distance. (2) The folded points, P x 1, P x 2, are averaged to P |1 (Figure 6(e)), and P | 1 is unfolded (Figure 6(f)) by applying to it the elements of the group: E leaves it in place and ó forms P | 2 across the reflection line; the pair {P | 1, P | 2} is ó-symmetrical, and the sum of distances | 1//2 +//P2 P | 2//2 is calculated. Minimization is //P1 P
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performed by screening over all possible divisions into opposite pairs, namely over all inclinations of ó, passing through these pairs. Other algorithms for quantitative evaluation of chirality have been described elsewhere (Avnir et al., 1996; Gilat, 1996; Mezey, 1997). The later references have evaluated the CSM approach as the most practical and general.
Quantitative Analysis of the Symmetry of Artefacts: Lower Paleolithic Handaxes Idit Saragusti* and Ilan Sharon Institute of Archaeology, The Hebrew University of Jerusalem, Jerusalem 91905, Israel
Omer Katzenelson and David Avnir Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (Received 22 September 1997, revised manuscript accepted 11 November 1997) The ability to perceive and produce symmetrical shapes is usually taken to be a major step in the development of human cognition. Despite the importance of the concept of symmetry to studies of early human development, its current use is usually based on loose qualitative assessment. A more informative approach would be to use a continuous scale of this shape property rather than the strict current language of ‘‘more’’ or ‘‘less’’. A symmetry measurement tool has been developed, which is based on evaluation of the minimal distances that the vertices of a structure have to move in order to attain the required symmetry. Using this Continuous Symmetry Measure (CSM) method, it is possible to evaluate quantitatively how much symmetry exists in a non-symmetrical configuration and how the nearest symmetrical shape looks. We test and demonstrate the feasibility and versatility of this approach on handaxe samples from three Lower Paleolithic sites in Israel, Ubeidiya, Gesher Benot Ya’aqov and Ma’ayan Barukh, representing various stages in the Acheulian Techno-complex of the Levant. We provide, to the best of our knowledge for the first time, quantitative demonstration that the overall symmetry of handaxes generally increases, and that the variability decreases over time. 1998 Academic Press
Keywords: ACHEULIAN, LOWER PALEOLITHIC, HANDAXES, SYMMETRY.
Background: The Near Symmetry of Handaxes
ability to perceive bilateral symmetry, as manifested in typical Late Acheulian handaxes, as a major argument for relating a higher degree of intelligence to the makers of these tools compared with the degree of intelligence demonstrated by the earlier Oldowan industrial tradition. He argued that the spatial concepts required to manufacture Oldowan tools are rather simple, topological in their nature, whereas the manufacture of typical symmetric Late Acheulian handaxes (as well as other tools) had required more sophisticated Euclidean concepts. Using the Piagetian theory of developmental psychology, Wynn argued that this difference in spatial concept requirements is indicative of the different developmental stages of the makers of these two industrial complexes. He also proposed that the ‘‘minimum necessary competence’’ manifested in the Oldowan repertoire is that of preoperational intelligence (Wynn, 1985), and is similar to that recognized for modern apes (Wynn, 1993: 306). This degree of intelligence is inferior to the one ascribed by Wynn to the makers of the Late Acheulian assemblages, the operational intelligence, the characteristic intelligence of modern adults. In effect, Wynn proposed qualitatively a correlation between the degree
T
he concept of symmetry has attracted virtually all domains of intellectual activity and has strongly influenced the sciences and the arts, archaeology being no exception (Hargittai, 1986, 1989; Gruber & Iachello, 1988 and earlier volumes in this series; Toth, 1990; Heilbronner & Dunitz, 1993). It has functioned as a condensed language for the description and classification of shapes and structures, as an identifier of inherent correlation between structure and physical properties of matter, and as a guiding property to artistic and practical aesthetic design. Because symmetry is one of the main shape characteristics of handaxes from Acheulian assemblages, it has been at the focus of many Lower Paleolithic studies. Scholars investigating this period have often argued that the ability to perceive symmetrical shapes, as expressed in the production of many handaxes, is meaningful in attempts to reconstruct the evolution of human cognitive, behavioural and technological capacities. Wynn (1985: 37), for instance, used the
*For correspondence: E-mail: [email protected]
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of intelligence and the degree of symmetry of the artefacts. Our main proposition in this report is that the degree of symmetry is a measurable trait. In fact the intuitive feeling that symmetry and intelligence or cultural manifestation are linked, appears in many other studies as well. Thus, Isaac (1986), though partly questioning Wynn’s conclusions, also saw the ability to conceive and produce symmetrical shapes as one of the indications of the higher complexity of the Acheulian industrial complex, compared to earlier traditions. In yet another study, Mithen (1994) argued that the ‘‘nearly perfect symmetry’’ of many handaxes from Acheulian sites in southeastern England as well as other properties which in his view have no functional significance, is the result of intense social learning, as opposed to individual learning, ascribed to the Clactonian and High Lodge assemblages. We draw attention to Mithen’s use of the adjective ‘‘nearly’’, signalling the actual need to provide a conceptual and practical tool that will allow one to assess this shape property on a quantitative level. Such scaling of symmetry would allow one to answer questions such as ‘‘how much symmetry is there in a given handaxe?’’; ‘‘by how much is one handaxe more symmetrical than another?’’; and so on. Indeed, despite the central importance of the concept of symmetry in these and many other studies, it has usually been treated in a qualitative rather than quantitative way, and sentences like ‘‘. . . these early handaxes are roughly symmetrical — none approaches the fine symmetry of later example . . .’’ (Wynn, 1985: 40 [our emphasis]) are often found in the literature. This relative and subjective description, not exclusive to archaeology, is by its nature imprecise. The need for objective means for symmetry measurement (Crompton & Gowlett, 1993) is the main motivation for this study. To conclude this introduction we recall that the traditional methods for morphometric description and analysis of handaxes have been based on various size parameters such as length, width and location of maximum width. These methods include those of Bordes (1961), Roe (1964, 1968), Isaac (1977) and others. These parameters were used to define various shape indices. The ‘‘index of refinement’’, for example, which expresses the degree of flatness in cross-section, is calculated by dividing the maximum thickness of the tool by its maximum width (Roe, 1968). In general these methods assume implicitly or explicitly that handaxes are symmetrical. As a consequence these methods have an inherent difficulty in treating tools with irregular forms, including those which largely depart from bilateral symmetry, thus ignoring a potential source for inter and intra assemblages variability (Wynn & Tierson, 1990). Bordes, for example, who used various size ratios as criteria for typological classification of handaxes, resorted to the term ‘‘varia’’ for irregular handaxes (Bordes, 1961). Being aware of this problem, Wynn & Teirson (1990) suggested an
alternative method based upon polar coordinates and which does not assume symmetry a priori. To the best of our knowledge, none of the proposed morphometric methods addresses the concept of quantitative symmetry and the goal of measuring the degree of symmetry. In a recent series of publications, the concept and the methodology of symmetry content measurement were developed (Zabrodsky & Avnir, 1995 and references cited therein; Avnir et al., 1996 — a review) and applied successfully to a variety of symmetry-related problems at the molecular and supra molecular levels (Buch et al., 1995; Kanis et al., 1995; Katzenelson, Zabrodsky Hel-Or & Avnir, 1996; Keinan, Zabrodsky Hel-Or & Avnir, 1996; Pinto, Zabrodsky Hel-Or & Avnir, 1996). This approach of Continuous Symmetry Measures (CSM), described in the next section, is general and not limited to the molecular scale. Having this quantitative tool at hand, the main aims of this study were, first, to introduce the concept of quantitative measurement of symmetry to archeological artefact studies; second, to test whether the CSM method is sensitive enough to detect visible differences in the degree of symmetry of artefacts; and third, to obtain preliminary indications regarding possible quantitative evaluation of changes in symmetry over time. In general then, we are testing here whether quantitative analysis of symmetry is applicable to archaeological research as an objective tool for assessment of shape through symmetry. Thus in the next section we describe the method for quantitative evaluation of symmetry. We then apply it to the analysis of handaxe samples from three Lower Paleolithic assemblages from Israel, as a test case for this method.
The Continuous Symmetry Measure (CSM) Approach As the examples mentioned above indicate, at present, symmetry serves as an approximate descriptive attribute of reality, lacking the necessary quantitative dimension. This problem is not unique to archaeology, but is common to the natural sciences and many other areas of study. As already mentioned, a general conceptual approach and a practical solution to the problem of symmetry measurement have been developed recently (by one of us (D. A.) and his colleagues (Zabrodsky & Avnir, 1995; Avnir et al., 1996)). It is briefly described here, with special attention given to the details needed for the current report. The CSM approach is based on the notion that quite often it is important to evaluate ‘‘how much’’ of a given symmetry there is in a structure rather than treating it as a ‘‘more’’ or ‘‘less’’ property. Consequently, this implies treating symmetry as a structural property of a continuous behaviour, which should be evaluated by a measurement tool. The design of a measurement tool involves some degree of arbitrariness, in the sense that one has to decide on issues such
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^ P0
P0 P'0 Normalize P2
Symmetry transform P'2
P1
(a)
P'1 (b)
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P'2
^ P2
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Figure 1. The basic features of the Continuous Symmetry Measure (CSM): In order to evaluate how much bilaterality (mirror symmetry) there is in the triangle (a), its size is normalized (b), and the mirror symmetrical structure (c) which is nearest to (b) is found using the symmetry transform described in the text and in Appendix 1. (d): The S(G) value is calculated from the minimal distance between (b) and (c), using equation 1. In this case, S(G)=4·96.
as how should the zero-reference level be set, what should be the maximal value, what should be the actual measurement and what normalization procedures should one employ. Any such decision is open to debate. Keeping this in mind, the symmetry measure is based on a definition that is as minimalistic as one could practically get. Our proposed answer (Zabrodsky & Avnir, 1995) to the question ‘‘How much of a given symmetry is there in a given structure?’’ has been: ‘‘Find the minimal distances that the vertices of a shape have to undergo, in order for the shape to attain the desired symmetry’’. In a formal way, given n vertices of the original configuration, located at Pi, and given a symmetry point group G, the amount, S(G), of this symmetry in this configuration 100 S(G)= n
n
( i=1
// P 1P| // i
i
2
(1)
where P | i are the corresponding points in the nearest Gsymmetric configuration. Equation 1 is general and allows one to evaluate the symmetry measure of any shape relative to any symmetry group or element. In order to avoid size effects, the size of the original structure is normalized to the distance from the centre of mass of the structure, (which is placed at the origin) to the farthest vertex. This distance is 1 and each of the other Pi’s is divided by the same maximal Pi. The nearest set of P | i’s, i.e. the set of coordinates describing the nearest symmetrical shape, also obtained in terms of the normalized coordinates, is computed with an algorithm (see example in Appendix 1), described in detail in Zabrodsky & Avnir (1995). The algorithm
thus also provides one with the shape of the nearest symmetrical object (Figure 1). The factor 100 is used for convenience of expanding the scale which is bound by 0cS(G)c100. If a shape has the desired symmetry, S(G)=0. A shape’s symmetry measure increases as the shape departs from G-symmetry and it reaches a maximal value (not necessarily 100). Square values are taken so that the function is isotropic, continuous and differentiable. In the archaeological literature the term ‘‘symmetry’’ has usually been limited to bilateral symmetry, namely to symmetry around a mirror that bisects the body. In the natural sciences this symmetry is also known as achirality: a chiral object is one that lacks a symmetry mirror (denoted ó), and its main property is that it cannot be superimposed with its mirror image (leftright hands being a classical example). Many other symmetry operations (e.g. symmetry of rotation, tetrahedral symmetry and so on, which may be of relevance for other archaeological problems as well) are treatable by equation 1, but we limit our discussion here to bilateral symmetry, referred to in this report as simply ‘‘symmetry’’. The meaning of equation 1 is then the minimal distance of a given shape from exact bilateral symmetry. We shall also now make the distinction between three-dimensional (3D) symmetry and twodimensional (2D) symmetry: the former refers to the symmetry of an object which occupies volume, and the latter to an object confined to a plane. An example for a 2D object is the contour of a projection of a handaxe onto a drawing paper, as commonly used in archaeology: 2D presentations of plan view and various crosssections of the 3D handaxes are standard reliable
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S(σ) = 2.65 G.B.Y. N Haifa Ubeidiya Jordan River
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(d) Figure 2. Demonstration of the CSM method: the mirror symmetry values of a series of polygons vertices, the nearest mirror symmetric shapes and the nearest mirror lines.
representations of symmetry features. For the 2D contours, the raw materials of this study, the symmetry mirror is a line that bisects the shape in-plane. An important feature of the CSM approach is that no pre-selected specific reference shape is assumed at the beginning of the analysis, though it is obtained, as mentioned above, as an end product (Figure 2). An additional important feature is that the S(ó) values are on the same scale, so that one can ask, for example which member of a set of polygons is the closest or farthest from being symmetrical. In our example we find that the polygon (Figure 2(b)), having the lowest S(ó) value, is the most symmetrical, i.e. it is the closest to bilaterality, and the triangle (Figure 2(a)) is the most asymmetrical. The main practical problem is then how to find the nearest symmetrical shape to a given one, namely how to locate the specific set of P | i’s that will minimize S(ó). A detailed solution to this problem was provided, which is based on what we termed the ‘‘folding-unfolding’’
Figure 3. The location of the studied sites, Ubeidiya, Gesher Benot Ya’aqov (G.B.Y.) and Ma’ayan Barukh (top right).
rationale (Zabrodsky & Avnir, 1995). It was also proved that the algorithm (Appendix 1) indeed provides a minimal solution to the symmetry problem.
Bilateral Symmetry Analysis of Handaxes The handaxes Samples of handaxes from three Lower Paleolithic sites in Israel (Figure 3) were used as a test-case. These are: Ubeidiya (N=20), Gesher Benot Ya’aqov (N=16) and Ma’ayan Barukh (N=8), all located in the Jordan Valley (a segment of the Dead Sea Rift) representing various stages in the Acheulian Techno-complex of the Levant. Ubeidiya, a Lower Pleistocene site which is located c. 3·5 km south of the Sea of Galilee, is currently dated to 1·4 million years ago (mya) (Bar-Yosef & Goren-Inbar, 1993). The sites of Gesher Benot Ya’aqov and Ma’ayan Barukh have no absolute dating yet, but based on geological, biostratigraphical and archaeological considerations, they are tentatively
Quantitative Analysis of the Symmetry of Artefacts
(a1)
(a2)
(a3) S(σ) = 1.84
(b1)
(b2)
(b3) S(σ) = 0.77
(c1)
(c2)
(c3) S(σ) = 0.29
Figure 4. Handaxe samples from the three sites: (a) Ubeidiya (Bar-Yosef & Goren-Inbar, 1993); (b) Gesher Benot Ya’aqov; and (c) Ma’ayan Barukh (Stekelis & Gilead, 1966); Original drawings: a1, b1 & c1. Outlines redrawn, using x-, y-coordinates: a2, b2 & c2. The nearest symmetrical shapes, their S(ó) values and the mirror lines: a3, b3 & c3.
dated to the Middle Pleistocene (i.e. between c. 0·78 and 0·13 mya). Gesher Benot Ya’aqov, located c. 15 km north of the Sea of Galilee, is tentatively dated to the early part of the period (Goren-Inbar, et al., 1992), while Ma’ayan Barukh, located further north, is somewhat later (Stekelis & Gilead, 1966; Gilead, 1970). Underlying the choice of these sites, was the generally accepted hypothesis, mentioned in the introduction, that the overall symmetry of handaxes generally increases over time. Indeed, as illustrated in Figure 4, the assemblages from these sites agree with this intuitive assumption: handaxes from Ubeidiya, the earliest site in this study, look less symmetrical than those from Gesher Benot Ya’aqov, which look less symmetrical than the ones found at Ma’ayan Barukh, the latest site. Can the CSM method translate it to quantitative terms?
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Data acquisition and analysis CSM analysis requires that the shape of the object to be analysed be represented by a set of n vertices or boundary points (see Appendix 1). For the purpose of this pilot study, we produced images of the handaxes by hand-drawing their outlines in plan-view projection (i.e. the longitudinal axis of the handaxe is used as the y-axis and the latitudinal axis (the longest straight line orthogonal to the longitude) as the x-axis; the thickness of the object is ignored). This view is the one most commonly used for shape evaluation of handaxes, although the degree of symmetry of longitude and latitude cross-sections may also be of interest. In a different facet of the same research programme, we are experimenting with automating data acquisition, using a digital scanning camera. This will enable the more rapid processing of the much larger samples needed for the next stages of research. These drawings were scanned into a personal computer. We found that scanning in a resolution of 300 dots per inch is optimal; a higher resolution is unnecessary and a lower one may result in loss of important information. The scanned outlines were then automatically traced, resulting in sets of x-, y-coordinates, each set of which served as the desired boundary points. The number of points was automatically determined in this procedure and it depended on the nature of the original outline: two points are sufficient to describe a straight line segment, whereas more points are needed for a curved one. In any event, as described above, the resulting symmetry measure is normalized to the number of points. In the analysed sample the number of coordinate points (n in equation 1) ranges between 133 and 343. As shown in Figure 4, redrawing of the outlines with these coordinates preserves the information and the redrawn outlines are very close to the originals. The symmetry values of each of the outlines was computed according to equation 1, using the sets of coordinates. The output is the minimal S(ó) value, namely, the minimal distance between the original outline and the nearest mirror-symmetrical shape. An additional output is, as explained above, the coordinates of the symmetrized shape and the position and inclination of the mirror line, ó (Figure 4). In most cases analysed we found that this mirror line is longitudinal (base to tip axis of the handaxe). Yet, in a very few cases the transversal and longitudinal axes provide two close minima of S(ó), with the former being slightly lower (e.g. Figure 4(a)). However, in order to make the symmetry comparison of all handaxes meaningful, the S(ó) value of the longitudal axis was always taken. Underlying this decision is the axiomatic assumption that handaxes have a ‘‘natural’’ direction defined through their ‘‘base’’ and opposite ‘‘tip’’. Results The frequency distribution of the S(ó) values by sites is represented in Figure 5. The S(ó) mean values for the
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Figure 5. Distribution of S(ó) values, the degree of bilateral symmetry, of the sampled handaxes, per site. UB, Ubeidiya; GBY, Gesher Benot Ya’aqov; MB, Ma’ayan Barukh.
samples are: Ubeidiya: 0·91 (N=20); Gesher Benot Ya’aqov: 0·77 (N=16); and Ma’ayan Barukh: 0·27 (N=8). Although ranges and standard deviations for small samples are not reliable, we cite them here just as an additional comparative value which translates what is clearly seen in Figure 5. These values are: for Ubeidiya, ..=0·87; range=0·1–3·3; Gesher Benot Ya’aqov, ..=0·54; range=0·3–2·3; and for Ma’ayan Barukh, ..=0·07 and the range=0·2–0·4.
Discussion Although this feasibility study is based on a relatively small number of samples, below the routine requirements for statistical analysis, some interesting points can already be made at this feasibility study stage: (1) The CSM methodology seems to translate correctly the qualitative impression of degree of symmetry of handaxes into quantitative values. (2) As seen clearly in Figure 5, the S(ó) values generally tend to decrease over time, i.e. handaxes become generally more symmetrical. This trend is also reflected by the mean S(ó) values. The hypothesis that symmetry and time correlate inversely seems, therefore, to be corroborated. Needless to say, one cannot argue that time is the only factor or even the prime factor which affects the degree of symmetry of
handaxes. Many other factors, for instance the nature of the raw material, must have played an important role in determining the shape of these tools. Yet the analysis of these factors is beyond the focus of this report. (3) Another pertinent observation is that the spread of S(ó) values decreases over time. Keeping in mind the limitation of rigorous statistical interpretation in this case, this observation is nevertheless consistent with another well accepted assumption, namely that interassemblage standardization increases over time (Isaac, 1972, 1986; Belfer-Cohen & Goren-Inbar, 1994). This phenomenon is expressed either by the metrical variables mentioned above (Gowlett, 1984) or in qualitative terms. Isaac (1986), for instance, refers to: ‘‘the imposition of a standard, arbitrary form, which is repeated in numerous cases’’ and to ‘‘the presence in assemblages of an increasing number of distinct, arbitrary forms, each manifest by a series of standardized, repeating items of like forms’’. The preliminary results of our study may serve as a quantitative support for this view. The relatively wide spread of S(ó) values in the earliest assemblage (Ubeidiya) may indicate that the occasional appearance of nearly symmetrical handaxes at the earliest phases of the Acheulian is random. However the decrease of the spread and the mean S(ó) values over time indicates that the emerging prevalence of near symmetry in the later phases of the
Quantitative Analysis of the Symmetry of Artefacts
Acheulian is a more consistent phenomenon. This trend may reflect the increasing ability to perceive and impose specific (symmetrical) forms. (4) Although the degree of symmetry increases over time and the variability narrows, these tools do not reach absolute perfection, and none of the handaxes gave a S(ó) value of zero (ideally symmetrical), not even in the latest assemblage from Ma’ayan Barukh in which the S(ó) values concentrate very close to zero, but not at it. This result may reflect an inherent difficulty associated with the very manufacturing method of these and other tools which are individually hand-made and not industrial products. (5) It would be an over simplification to argue at this stage that the increase in the degree of symmetry and standardization over time simply reflects an increase in the abilities to perceive, produce and impose symmetrical shapes. Other interpretations are possible. The sporadic occurrence of nearly symmetrical handaxes in very early assemblages, for example, may suggest that the ability to perceive symmetry already existed at this phase, but for some unknown reason did not become fully manifested until the later phases. The correlation between the degree of symmetry and other morphometric and technological variables must be studied and analysed before a more thorough interpretation is possible.
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consideration other contributing factors, as well as basing such research on a much wider chronological, geographical and cultural perspective. Yet, an in-depth study of this crucial issue requires a quantitative measure for symmetry of artefacts, as outlined in this report.
Acknowledgements We thank Dr Hagit Zabrodsky Hel-Or for close assistance in the computational aspects of this project, Professor N. Goren-Inbar for encouraging this study and for allowing us to use the material from G.B.Y. and Professor O. Bar-Yosef for permitting us to use the material from Ubeidiya. We gratefully acknowledge the financial support of the following Foundations for supporting various portions of this study: The Israel Science Foundation (to both D.A and I.S.), the Robert Szald Institute for Applied Science of the P.E.F Israel Endowment Fund, the Horowitz Foundation, the Minerva Foundation, and the National Center for Cooperation between the Sciences and Archaeology. I.S. is grateful to S. and G. Itai, A. Belfer-Cohen and Y. Cohen for their comments. Figure 3 was drawn by G. Hivroni.
References Conclusions Our starting point was the prevailing situation where symmetry often serves as an approximate descriptive ‘‘language’’ in archaeological studies. We proposed that while it is true that an imprecise language helps grasp complex situations and identify first order trends, there is always a danger of missing the full picture because of vagueness. Thus, it has been our aim here to provide a method for a quantitative evaluation of symmetry in archaeology. This method, when used to evaluate the degree of symmetry of handaxes, was found to be sensitive enough for measuring observed differences in symmetry. The CSM method may, therefore, be applied as a means to achieve a quantitative, objective and more accurate description of an important global shape feature of artefacts in archaeological research. As such, this method provides a new tool for discussing issues like the effects of raw materials on the shape of artefacts (Jones, 1979), regional comparisons between assemblages (Wynn & Tierson, 1990), allometry (Crompton & Gowlett, 1993) (size related shapes), chronological changes, and other topics of the archaeological research in which symmetry plays an important role. Possible correlation between the mental abilities of Early Man and the tools he made (as suggested in the studies cited above), is a far reaching and complex issue. As mentioned above, it would be an over simplification to argue that increased symmetry directly reflects increased cognitive ability, without taking into
Avnir, D., Katzenelson, O., Keinan, S., Pinsky, M., Pinto, Y., Salomon, Y. & Zabrodsky Hel-Or, H. (1996). The measurement of symmetry and chirality. In (D. H. Rouvray, ed.) Concepts in Chemistry. Somerset: Research Studies Press, pp. 283–324. Bar-Yosef, O. & Goren-Inbar, N. (1993). The Lithic Assemblages of ‘Ubeidiya, A Lower Palaeolithic Site in the Jordan Valley. Jerusalem: The Hebrew University of Jerusalem. Belfer-Cohen, A. & Goren-Inbar, N. (1994). Cognition and communication in the Levantine Lower Palaeolithic. World Archaeology 26, 144–157. Bordes, F. (1961) Typologie du Pale´olithique Ancien et Moyen. Bordeaux: Impremeries Delmas. Buch, V., Greshgoren, E., Zabrodsky Hel-Or, H. & Avnir, D. (1995) Symmetry loss as a criterion for cluster melting, with application to (D2)13. Chemical Physics Letters 247, 149–153. Crompton, R. H. & Gowlett, J. A. J. (1993). Allometry and multidimensional form in Acheulean bifaces from Kilombe, Kenya. Journal of Human Evolution 25, 175–199. Gilat, G. (1996). The concept of structural chirality. In (D. H. Rouvray, Ed.) Concepts in Chemistry. Somerset: Research Studies Press, pp. 325–352. Gilead, D. (1970). Early Paleolithic Culture in Israel and the Near East. Ph.D. Thesis. The Hebrew University of Jerusalem. Goren-Inbar, N., Belitzky, S., Verosub, K., Werker, E., Kislev, M. E., Rosenfeld, A., Heimann, A. & Carmi, I. (1992). New discoveries at the Middle Pleistocene Gesher Benot Ya’aqov Acheulian site. Quaternary Research 38, 117–128. Gowlett, J. A. J. (1984). Mental abilities of early man: a look at some hard evidence. In (R. Foley, Ed.) Hominid Evolution and Community Ecology. London: Academic Press, pp. 166–192. Gruber, B. & Iachello, L. (Eds) (1988). Symmetries in Science. III. New York: Plenum Press. Hargittai, I. (Ed) (1986). Symmetry: Unifying Human Understanding. New York: Pergamon Press. Hargittai, I. (Ed) (1989). Symmetry 2: Unifying Human Understanding. Oxford: Pergamon Press.
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Appendix 1: The Folding/Unfolding Method The ‘‘folding-unfolding’’ algorithm is one of the methods for finding the nearest symmetrical shape. Algorithms, proofs and variants on the theme described below, can be found in the references cited above. The main features of the folding-unfolding method are described here for bilateral symmetry. The example we use demonstrates how to measure the bilateral symmetry of a general boundary line (Figure 6). The object to be symmetrized (Figure 6) is converted to a necklace of an even number of boundary points, Pi, as dense as required (N=30 points in this case). Where the object is already represented by vertices, this
(a)
(b)
Y
Y P2
P2
X
~ P2
X ~ P1 = P1
P1 (c)
(d)
Y
Y
^ P2 ~ P2
X ^ P1 ~ P1
(e)
X ^ P1
(f)
Figure 6. The folding-unfolding method explained for 2D bilaterality (chirality): (a) A contour composed of 30 points, Pi. The size is scaled to 1 as shown by the arrow. (b) The nearest S(ó)-symmetric structure, P | i, to (a). (c)–(f): Evaluation of the S(ó) value for a pair of points, P1, P2, (c), with respect to a given mirror line. (d) P1, P2 are folded to P x 1, P x 2, averaged (e) to P | 1 and unfolded (f) into the S(ó) symmetric pair, P | 1, P | 2.
step is not needed. Its centre of mass is then determined from the averaged sum of coordinates and placed at the origin, and the distance from this centre to the farthest Pi is scaled to 1 (Figure 6(a)). The aim is to find the nearest set of P | i’s which is ó-symmetrical, namely, to find that reflection line that will cause the set of Pi’s to move minimally to the set P | i’s (Figure 6(b)). In the symmetrized object, each P | i must have a ó-symmetric counterpart P | i–j across the reflection line (or it must be located on the reflection line). The full set of Pi’s is divided into subsets of two points, and all possible divisions are tested. (Points on the reflection line are duplicated). Here is how a pair of points, P1, P2, are ó-symmetrized with respect to a given reflection line ó (Figures 6(c)–6(f)): (1) The identity element, E, operates on P1 and it x 1); ó operates on P2 forming remains in place (P1P the reflected P x 2; a pair of adjacent points, P x 1, P x 2 is obtained (Figure 6(d)). We termed the step of applying the elements on the vertices the folding step. The
Quantitative Analysis of the Symmetry of Artefacts
essence of our methodology is to minimize the P x 1, P x2 distance. (2) The folded points, P x 1, P x 2, are averaged to P |1 (Figure 6(e)), and P | 1 is unfolded (Figure 6(f)) by applying to it the elements of the group: E leaves it in place and ó forms P | 2 across the reflection line; the pair {P | 1, P | 2} is ó-symmetrical, and the sum of distances | 1//2 +//P2 P | 2//2 is calculated. Minimization is //P1 P
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performed by screening over all possible divisions into opposite pairs, namely over all inclinations of ó, passing through these pairs. Other algorithms for quantitative evaluation of chirality have been described elsewhere (Avnir et al., 1996; Gilat, 1996; Mezey, 1997). The later references have evaluated the CSM approach as the most practical and general.