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On the Coding of Geometrical Shapes And Other Representations, with Reference to Archaeological Documents

J.C.GARDIN

1. Introduction

The object of the present paper is to describe a method of analyzing various representations, geometrical or otherwise, related to their storage and retrieval.

By “representation” we mean either abstract designs or figurative constructions, such as may be found in scientific or technical documentation. They may be two or three dimensional, and they may constitute either the very subject-matter of research, or only a graphic rendering of the various data under consideration.

2. Method

The method consists in breaking down the representation into various elements, and in expressing in different ways the relations according to which those elements are assembled.

The individuality of a representation is thus made explicit through a particular combination of terms, chosen from a limited set, to indicate both constituent elements and relations. Those terms are derived by making an analysis of the components of the representations being studied. They have no intrinsic value from the point of view of scientific understanding or, if so, only incidentally; their sole purpose is to provide a means whereby a certain type of document, namely graphical, can be analyzed and so be made accessible to search.

The criteria which are used for the selection of such terms are therefore purely operational. The objective is to devise the most economical system of

J.C.GARDIN Centre National de la Recherche Scientifique, Paris.



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--> On the Coding of Geometrical Shapes And Other Representations, with Reference to Archaeological Documents J.C.GARDIN 1. Introduction The object of the present paper is to describe a method of analyzing various representations, geometrical or otherwise, related to their storage and retrieval. By “representation” we mean either abstract designs or figurative constructions, such as may be found in scientific or technical documentation. They may be two or three dimensional, and they may constitute either the very subject-matter of research, or only a graphic rendering of the various data under consideration. 2. Method The method consists in breaking down the representation into various elements, and in expressing in different ways the relations according to which those elements are assembled. The individuality of a representation is thus made explicit through a particular combination of terms, chosen from a limited set, to indicate both constituent elements and relations. Those terms are derived by making an analysis of the components of the representations being studied. They have no intrinsic value from the point of view of scientific understanding or, if so, only incidentally; their sole purpose is to provide a means whereby a certain type of document, namely graphical, can be analyzed and so be made accessible to search. The criteria which are used for the selection of such terms are therefore purely operational. The objective is to devise the most economical system of J.C.GARDIN Centre National de la Recherche Scientifique, Paris.

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--> elements and relations through the various combinations of which specific representations can be defined with an acceptable degree of approximation, from the documentation point of view. A compact code is thus constituted; it provides a way of expressing with a relatively small set of elementary, non-ambiguous features, a very large number of aggregate percepts, which have names too vague or no names at all in common usage. The formulas which are obtained by analyzing representations in terms of that code may be considered as names, in so far as they can be made pronounceable through a reasoned attribution of phonemes (or morphemes) to each term in the code. The merits of such designations tend to be threefold: objective, i.e., conform to fixed standards of description, irrespective of personal appreciations; international, i.e., independent of national differences in the process of naming identical entities; analytical, i.e., capable of being broken down into several terms, which makes for a more compact storage, and a more flexible retrieval of information. 3. Developments on different levels The method which has been outlined is applicable on several different levels, according to the degree of complexity of the representations under study. 3.1. ABSTRACT DESIGN Abstract design on a plane surface is the simplest case. The code is then essentially composed of terms borrowed from two-dimensional geometry, some designating, for instance, the most common figures (circle, triangle, etc.), while others refer to various ways of combining them in different patterns (intersecting, interlocking, radial, etc.). A specific design is thus considered as the result of n successive “operations,” carried out in a given order on one or several elementary “signs.” Operations and signs are to be determined, not through deductive reasoning, as in the case of pure geometry, but through an empirical study of the particular kind of design at hand so as to produce the most economical system of analysis. An example is given under “Abstract Ornament” (4.1). 3.2. THREE-DIMENSIONAL CONSTRUCTIONS These can be analyzed in the same way, following, however, different procedures according to the kind of data under investigation. 3.21. In the simplest case one will have only to add a new term to the two-

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--> dimensional formula, so as to transform the corresponding plane design into a regular three-dimensional construct. 3.22. For more complex shapes it may prove convenient to choose three-dimensional figures as the fundamental units in the code, to be used in connection with a reduced set of operations. 3.23. Lastly, for constructions which cannot easily be reduced to such simple transformations, a good procedure consists in analyzing successively various cross-sections, either for the whole or for some predetermined parts of the object. An example of that method is given under “Artifacts” (4.2). 3.3. ICONOGRAPHICAL COMPOSITIONS As representations grow more elaborate, and as they tend to develop into truly iconographical compositions, it becomes convenient to shift again the level of reference: the units of description are no longer geometrical elements—two or three dimensional—but figurative motives which are usually well understood under their common designations: a man, a building, a tree, etc. 3.31. Each motive may appear under different forms, which do not always receive specific names. The “building,” for instance, is sometimes termed a hut or a skyscraper, but many times also a four-floor house, a flat roof house, etc., according to its shape, its height, etc. The analyst has therefore to consider the various manifestations of each motive; and his first task is to try to reduce the total set of variations, for all the motives taken into consideration, to a limited number of abstract terms, which take up different meanings according to the particular motive with which they are combined. The vocabulary is thus made of two sections: a lexicon, where elementary motives are entered under their common name, and a morphology, where a list is given of the various complementary terms which serve to specify the form of each motive in the representation. (See the example in Section 4.31). 3.32. One has also to indicate the relations which prevail between the various constituents. Relations are of two kinds, logical and topographical. By logical relations we mean the sort of interactions which bind different beings to one another in the same picture: A kills B, A sits in front of C, etc. Such relations are not made wholly explicit through verbs, nor through any other terms referring to an action, since the combination of three elements—two nouns A and B, and a verb ×—is usually ambiguous: A×B, or B×A (e.g., A kills B, or B kills A). In all cases where A and B can be substituted for each other with reference to ×, one has therefore to specify in addition to × the part respectively played by A and B in the corresponding action. Declensions provide in this respect a convenient solution: A (nominative, or subject)×B (accusative, or object), as opposed to A (object)×B (subject).

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--> An example of a code with inflected terms is given under “Logical Relations” (4.32), for the analysis of iconographical documents in the field of archaeology. The second kind of relations to be specified, topographical, concerns the relative position of the various beings which are associated in a given representation. Here again, the grammatical terms which usually convey that information (mainly prepositions) would not entirely satisfy the purpose, for they could often combine with any of the surrounding elements to provide entirely different meanings. To obviate the difficulty a few more grammatical “cases” have to be used, such as the locative and the instrumental, which, associated with definite nouns, play the same part as prepositions, without leaving any ambiguity as to the way in which the representation is actually organized. Sometimes, for the more common disposition of subjects and objects, a single term may convey all the information which is needed as to the relative disposition of the various elements: Term p S→O←S. Ex. Two heroes (S) striking a lion (O) standing between them. Term q O←S→O. Ex. A hero (S) grasping to his left and to his right two lions (O) Term r S2→S→O. Ex. A hero (S2) striking a lion (S) which is attacking a lamb (O). S and O indicate respectively any subject or object in an iconographical sentence. 3.33. The more complex figurative representations can thus be analyzed in the same way as geometrical designs or functional objects. By combining terms pertaining to three different categories, lexicon, morphology, and grammar, the analyst is able to describe a very large number of iconographical themes, with a degree of approximation which he can set high or low, according to the range of his investigation. Yet the total number of terms, in the whole code, remains relatively low (cf.parag.43.3). 4. Applications to a specific field, archaeology The theoretical developments which form the subject of the preceding paragraphs have found an application in a systematic experiment carried out under the auspices of the French Centre National de la Recherche Scientifique for a specific field, archaeology. The aim was to show that archaeological data, which are now scattered in countless publications and collections, could be made readily accessible to search through an appropriate coding of the information, independent of dif-

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--> ferences in national languages or private idiosyncrasies, and through the recording of the coded information on punched cards (Peek-a-boo system). Various codes have been devised, each corresponding to one of the three types of representations which have been previously discussed: abstract ornament (cf. 3.1, two-dimensional design), artifacts (cf. 3.2, three-dimensional constructions), paintings, engravings, etc. (cf. 3.3, Iconographical compositions). A brief account will now be given of their general structure, so as to provide a concrete illustration of our method. 4.1. ABSTRACT ORNAMENT Individual motives, on the one hand, are defined by the combination of a root, indicating an elementary sign (d for dash, s for an S, b for the loop, etc.) with one or several affixes, indicating various geometrical arrangements (r for radial, li for linear, m for symmetrical, etc.) (see Fig. 1). The distinction be FIGURE 1. (a) Elementary signs, examples. tween both kinds of morphemes is purely operational; it is based on considerations of economy and substantiated by statistical observations. Each elementary sign could actually be regarded as a specific arrangement of one single fundamental sign, the dot. The linguistic system built on that convention would, however, be very unwieldy for the purposes of documentation. With only twenty elementary signs (roots), actually distributed in two parallel series of ten signs each, rectilinear and curvilinear, and, on the other hand, ten sorts of geometrical arrangements (affixes), subdivided into about thirty variants indicated by vowel gradations for each affix, one determines a first series of primary ornaments (about six hundred). After undergoing a second treatment of the same kind, each of these generates a secondary ornament (i.e., a root+two affixes, about 18,000 in number), which in turn can develop into a ternary sign (i.e., a root+three affixes; 500,000 in number), etc. At the fourth degree, the number of motives reaches fifteen millions, each of them being named in terms of the very condensed language mentioned above, with less than thirty morphemes (see Fig. 1).

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--> FIGURE 1. (b) Combinations of elementary signs (roots) in various arrangements (affixes).

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--> Through various conventions, one can analyze according to the same notions the most complex ornamental compositions, where several motives of the 1st, 2nd, 3rd, and 4th degrees come into play. The relative position of the various motives is expressed by a third set of terms, most of which are the same as roots and affixes, but are written in block letters to indicate their syntactical function (Fig. 2). FIGURE 2. (a) “spi F,” several spirals (spi), arranged on a curve (F). (b) “suti KA,” several “suti” (three S forming a triangle) arranged in square (KA). In a longer formula, the position which any symbol holds is naturally as important as its absolute meaning when considered separately. Provision has therefore to be made to indicate, through different devices (index numbers, for instance) the interrelations of the various terms in the formula. 4.2. ARTIFACTS 4.21. Five different parts of pottery shapes are considered: base, main body, neck, handle, and spout. A series of conventions fixes the dividing lines between base and body, neck and body, etc., so as to channel individual appreciations into a unique pattern, close to common usage. Each part is then de- FIGURE 3. Pottery, main body: conventional division between upper part and lower part.

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--> scribed through a combinatorial analysis, which always proceeds in the same way: (a) Conventional division into various subparts: The main body, for instance, is examined successively under two different cross-sections, horizontal and vertical. In turn, the vertical section is divided into two conventional parts, which it has proved convenient always to distinguish on each side of a horizontal plane imagined at one of the following levels (Fig. 3): maximum (a) or minimum (b) width, change of curve (c), or, in all other cases, medium height (d). (b) Morphological description of each subpart, in terms of a mnemotechnical code such as the following: d, straight v, concave x, convex i, divergent u, parallel o, convergent Figure 4 shows how the combinations of the six terms, under two different FIGURE 4. Pottery, main body: 9 elementary profiles. FIGURE 5. Pottery, main body: examples of code words for different shapes.

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--> headings, shape (d,v,x) and relative inclination of the sides (i,u,o), provide nine elementary profiles for each subpart, i.e., 81 shapes for the whole body. (See examples, Fig. 5.) (c) Description of the liaison between the various subparts: Figure 6 illustrates, for instance, three additional terms (c, curve; 1, angle; q, moulding), which, combined with the 81 previous formulas, determine 243 types of profiles for the main body. At this stage the purely morphological description of any of those 243 shapes is effected with only five terms, chosen among a total vocabulary of nine. FIGURE 6. Pottery, main body: liaison between upper and lower part. (d) Relative dimensions: In addition to morphology, the analysis also has to consider dimensions, both absolute (a simple matter of determining successive bands, from x to y, for the most significant measures, height, width, etc.) and relative. Figure 7 shows the three ratios which have been taken into FIGURE 7. Pottery, main body: relative dimensions: 1 x<1/2 6 y<1/2 z<1/2 2 1/2<x<1 7 1/2<y<1 z>1/2 3 x#1 8 y#1   4 1<x<2 9 1<y<2   5 x>2 0 y>2   consideration in the case of the main body: height of the upper part to height of the lower part, total height to maximum or minimum width, width of the opening (either on the neck or on the base side, according to the morphological type at hand), to maximum or minimum width. Simple fractions or integers are used to determine a suitable scale of variations—five steps for each of the first two ratios, and two for the third one. The delineation of the main body is thus determined to a sufficient degree of approximation by

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--> adding three dimensional terms to the five morphological terms previously mentioned. The number of shapes which can be differentiated in this way amounts to 12,150, with the use of eight terms in each case, chosen among a total of only 27. Actually, among those 27 terms, eleven recur twice (d,v,x; i,u,o; and each of the five arithmetical expressions), so that the total number of descriptive features amounts in fact to only sixteen distributed under six headings. The procedure is the same for analyzing other parts or subparts of a pottery; the foregoing features of description recur several times in different contexts. For the whole analysis the proposed code does not include more than about thirty terms, each of which corresponds to a simple notion which can be expressed without ambiguity in any language. 4.22. Tools and weapons. A combinatorial method has been developed in the same fashion to describe metal implements. In fact, only the preliminary carving up (i.e., the establishing of conventional divisions) differs; for each part and subpart, the successive steps of analysis and the terms which are used in the process remain unchanged, whether the artifact be a container, a sickle, or a spear. 4.3. ICONOGRAPHICAL MONUMENTS Two codes have been constituted, one for Greek coins, where iconographical themes are relatively simple, and the other for oriental seals, where, on the contrary, the number of elements brought into play is very high. Others are being developed for other fields, such as Egyptian sculpture and Indian reliefs; however, they all exhibit the same general structure, which can be summarized in the following way: 4.31. First comes a vocabulary, in which the various nominal features of representation are distributed under ten main headings: Animate beings, with their individual attitudes (posture, gestures) and angle of representation Humans (i.e., human-shaped) Animals Hybrids Inanimates Clothing Building and furniture Containers Instruments (tools, weapons, etc.) Emblems Nature (sky, earth, plants) Ornament

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--> A first kind of coding takes place within each of the ten sections. Instead of developing a formal analysis which would prove unwieldy, it is more convenient to treat the various elements in a given class (houses, axes, stars, etc.) as entities, each of which is to be defined by a drawing, and named by a conventional symbol (a figure or a letter) placed after the general designation (Ex.: house, type b; axe, type M7). Ambiguities or long periphrases are thereby avoided. Moreover, whenever two elements are found to be mutually exclusive in a given iconographical field, their respective variations can be specified with the same set of symbols, as illustrated in Fig. 8. For example, in Greek numismatics, temples and FIGURE 8. Variants of three different motives, designated with the same set of symbols a, b, c, etc. houses are never found together on the same coin; similarly, chariots and boats are mutually exclusive, etc. That procedure has two merits: it reduces the number of terms in the code, and it provides a level of generalization (temples, houses, etc.), where phenomenological subtleties can be provisionally ignored, if the scope of a given investigation makes it desirable. 4.32. Another section in the code deals with the logical relations which are explicit in the picture. Actions. The manifold variety of actions represented on the iconographical documents which we have considered has been analyzed in terms of only two notions, negative and positive, specific meanings being provided by the context. The negative refers to actions which are detrimental to the object, whereas the positive embraces all other actions, whether they be beneficient from the

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--> point of view of the object, or simply neutral That simple dichotomy is generally all that is needed to differentiate iconographical themes such as the following: man, ewe, +positive action=to tend, to shear, to feed, to milk, etc. _____ +negative action=to slaughter, to immolate, to sacrifice, etc. man, building, +positive action=to build, to repair, etc. _____ +negative action=to demolish, to plunder, etc. There remains, admittedly, some ambiguity; but the meaning is specified by other terms in the formula: man, ewe, positive action+container=(probably) to milk ______ “+instrument=______ to shear _____ “+plants=______ to fodder Where this is not so, the ambiguity generally comes from the representation itself, not from the way in which it has been analyzed: the killing of a lamb may be related to magic (divination), religion (sacrifice), alimentation (slaughter), etc., without any change in the purely graphical expression of the event, from one case to the other. The lack of precision is here an accepted consequence of the objectivity which must be achieved in the coding process. Cases. There remains however another kind of ambiguity, due to the different roles which a given element may play in the picture (see logical relations in Section 3.32). That problem is solved by affecting to each nominal term a grammatical case, chosen among the following: subject Ex. A coin adorned with a tree. object Ex. A man cutting a tree. qualificative Ex. A man seated, holding a branch. instrumental Ex. A hero killing a lion with a branch. locative Ex. A woman seated in a tree. Various logical devices, supported by statistical observations, make it possible to abstract those grammatical features themselves, for certain classes of iconographical elements (humans, animals, plants, etc.), so as not to raise unduly the amount of terms in the code. In this way, the ratio of the number of inflected terms to the number of corresponding radicals can be reduced to a low value, usually smaller than two. 4.33. Iconographical codes are thus constituted with a relatively small number of combinable elements, chosen at different levels of analysis: nominal components, with their morphological variants, grammatical cases, and verbal relations. None of those codes exceeds for the present a thousand terms; yet, the number of representations which can be defined with such terms is very

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--> large, from simple motives to the most complex iconographical themes, with a high degree of concision and objectivity, and at any level of generalization which may be required in the course of research. 5. General remarks It is likely that the method which has been outlined is adaptable to the analysis of other kinds of graphical representations. One might, for instance, consider its application to the tabulation of crystallographic and organic bodies, for documentation purposes in mineralogy and chemistry. The indexing of trademarks, of standardized but variegated pieces of equipment in specific technical fields could also be opened to similar procedures. Photographic or schematic recordings of data yet “unexplained” in terms of a scientific theory could perhaps also be indexed in the same way. The sole condition is that such representations should exhibit a certain degree of homogeneity, or orderliness. A code can then be proposed, which is but one of several interpretations of the observed order. Formulations of that kind, undertaken for practical purposes (the storage and retrieval of information) are not guided by the care for logical elegance only. Like all linguistic systems, they tend to establish a compromise between the conflicting needs of economy and rapidity of communication. They are not directly concerned either with the meaning of the various notions to which they lead, from a scientific point of view; their main purpose is to substitute combinations of well-defined terms for synthetical descriptions which usually convey information either too vague or insufficient.

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