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Colloque "3O ans de sémiologie graphique"

The third choice

La troisième voie
Jan Ketil Rød


Il y a 30 ans Jacques Bertin publiait Sémiologie Graphique. Cette théorie s’est imposée comme primordiale, tant dans le domaine de l’enseignement de la cartographie, que dans celui de la cartographie appliquée. Cet article ne reviendra pas sur la fructueuse utilisation de la Sémiologie. Nous entendons montrer la contradiction entre les niveaux d’organisations proposés par Bertin, et ceux utilisés par les staticiens et les cartographes anglo-saxons. En effet, si Bertin identifie trois niveaux d’organisation les cartographes anglo-saxons en utilisent quatre. Dans une optique d’élaboration de règles de base pour un système expert destiné à l’élaboration de cartes thématiques. Ce floue dans la définition des niveaux d’organisation est un problème fondamental. Nous analyserons les principales contradictions entre les deux systèmes de mesures et proposerons des solutions. Cette réflexion aboutit à un schéma de synthèse utilisable comme règle de base. Ce schéma s’applique aux deux fonctions de la carte : outil de visualisation et moyen de communication.

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Texte intégral


"Reality" and its representation

1A map is supposed to portray some aspects of reality, an ambitious expectation owing to the fact that reality is extremely complex and detailed. Maps are therefore often denoted as models in which symbols stands for spatial phenomena in the "real" world :

… the features on the map represent symbolic versions of reality, not reality itself (Muehrcke 1976 : 124).

2It is an underlying conception that the map model is to reflect as accurately as possible aspects of reality, and thus work as its representation. In this respect, the contemporary cartographer should ask some ontological and epistemological questions. What is "reality", how can it be represented, and how is it interpreted by the map user  ? Several are asking such questions and are hence broadening their horizons.

The normative, positivistic approach of many cartographers must be supplemented by an understanding of different epistemological approaches such as those of the humanist and the structural/realist. This will affect not only what we choose to represent, but how we choose to represent it (Taylor 1996 : 16).

3This view corresponds with the obvious claim that ‘cartography is about representation’ (MacEachren 1995 :1). Fortifying links towards other disciplines concerned with representation have an influence on cartography. Representation is among the main concepts in postmodernistic writing. Postmodernism can be described as a movement in philosophy characterised by scepticism towards the grand claims, the grand truths, and the grand theories. In this philosophic movement, representation is closely linked with the postmodern method, deconstruction.

Deconstruction is a principal strategy, a mode of critical interpretation which seeks to demonstrate how the (multiple) positioning of an author (or a reader) in terms of culture, class, gender, etc. has influenced the writing (and reading) of a text (Ley 1994 : 466).

4If the word text is exchanged by map in the last citation, the citation expresses a concurrent view with the American cartographer Harley (1989, 1990). His critical interpretation has lead to a changing philosophical perspective,

… voiced by an increasing number of scholars, that does not accept the concept of maps as "objective" representations of reality and therefore discounts the idea that objective research is possible (MacEachren 1995 : 6).

5In accordance with the point of view, that no map represents reality free from personal prejudice or bias, I take the position that, as cartographers we should show professional modesty when we are claiming to portray aspects of reality. We are not able to portray reality, only our generalised representation of it, but still, a generalised representation of reality will continue to be of immense importance for the map user in order for her to understand the complex and detailed human existence.

The end product of the cartographic process is an ordered conception of reality which for certain purposes is asserted to actually serve the map user better than reality itself (Muehrcke 1976 : 123).

6Although these ontological and epistemological questions might not provide fundamental questions about how we should select symbolisation strategies, they will influence these strategies because of the changing understanding of the map-reader from a passive receptor to an active and knowledgeable person. In order to illuminate and suggest answers for cartographic symbolisation, we have to begin with the basis of all mapmaking, generalisation.

Information processing and graphics

7Generalisation is the fundamental part of all mapmaking, which results in ‘an ordered conception of reality’. Robinson et al (1984 :125) found it convenient to group cartographic generalisation into four elements as follow :

  • Simplification

  • Classification

  • Symbolisation

  • Induction

8These fundamental and complex operations are all consciously accomplished by the cartographer in the practice of mapmaking, but each map provides a different set of requirements. The "mix" of the processes of generalisation will, therefore, vary from map to map (Robinson et al 1984 : 125-126).

9The generalisation process will hence vary from map to map. This paper’s interest is thematic mapping, or more precise, statistical mapping. In the process of mapping statistical data, a form of generalisation is accomplished by partitioning the range of the data. Baudouin (1987 : 323) recognised three groups of decisions necessary for this particular generalisation process. These are :

  • The choice of the number of classes,

  • The choice of the method applied for determining class limits, and

  • The choice of graphic representation

10These three decisions correspond with the first three elements of generalisation recognised by Robinson et al (1984). While the fourth element of generalisation, induction is linked to effective map perception.

Induction occurs when we make inferences from interrelationships among features on the map (Robinson et al 1995 : 451).

11With effective map perception, I mean the users’ ability to obtain spatial information from a map. This is recognised as the purpose of all maps. However, the spatial information obtained from topographic map differs from the one obtained from statistical map. While a topographic map tries to depict perceptible objects at the earth’s surface with accurate position, a statistical map tries to visualise the distribution of some kind of phenomena with accurate characteristics.

The most widespread and serious error, because it leads to wrong decisions, consists of mistaking not the geographical position but the characteristics. To represent the inherent order of quantities by a visual non-order or disorder of the signs is obviously a mistake and therefore gives a false image – in other words, false information (Bertin 1983 : 70).

12Therefore, if an inherent order is represented by non-order or disorder, that is an inaccurate representation of characteristics, this lead to false information leading to mistakes and wrong decisions. What does it mean to obtain information and when is this information spatial  ? According to Bertin :

Information is a relationship between two elements or between two sets of elements. In graphics, it is the answer to a question (Bertin 1981 : 180).

13Thus, to obtain information is to obtain answers to posed questions and the information becomes spatial if it reveals understanding of the relationships that exist among elements, subset or sets (Bertin 1981 : 12).

A graphic is not "drawn" once and for all ; it is "constructed" and reconstructed until it reveals all the relationships constituted by the interplay of the data (Bertin 1981 : 16).

14Spatial information is revealed relationship and graphics can be used as a tool for revealing it. Revealing information is a process of construction and reconstruction since it may generate new questions rather than demonstrating conclusions, therefore, a graphic is not drawn once and for all.

15Cartographers have adopted the semilogy of graphics mainly as a method for cartographic symbolisation. This is, however, to confine the concept of graphics. Bertin used the term in the general sense as the Greek origin grafein indicates. Grafein means to draw and thus the map is only one of several possibilities of "drawing". Although the present paper elaborate graphics applied for statistical mapping, I will not delimit the achievements of graphics only to mapping. I would take the position, as stated by MacEachren, that the map represents only one of many potential representations of phenomena in space that a user may draw upon as a source of information or an aid to decision making and behavior in space (1995 : 12).

16While maps often are considered as static end product, Bertin (1981 : 16) viewed maps both as a tool for exploration and as an end product serving as a communication device. This duality of mapping purposes is found in his distinction between graphic information processing and graphic communication. Bertin (1981) uses the term graphic information processing to denote the discovery of the information needed for making decisions. Applied to cartography, it denotes the exploratory tool a map might be, using the map to reveal the unknown. Bertin distinguished this term from graphic communication that denotes communication of the discovered and simplified data (1981 : 16).

Graphic communication involves transcribing and telling others what you have discovered. Its aim : rapid perception and, potentially, memorization of the overall information. Its imperative : simplicity (Bertin 1981 : 22).

17Bertin amplifies that graphic information processing and graphic communication should not be mixed.

We should avoid confusing graphic processing and graphic communication, drawing simplified maps when comprehensive studies are necessary, or superimposing comprehensive documents in a vain attempt to increase the information value (Bertin 1981 : 22).

Purposes of maps

18In the following of the present paper, two different intentions of map use are based on Bertin’s distinction between graphic information processing and graphic communication. These two purposes are expressed by the matching concepts borrowed from MacEachren and Ganter (1990) :

  1. Maps as a visualisation tool

  2. Maps as a communication device

19There are several concepts used to characterise these two general mapping purposes and some of them are summarised in the following table :

Bertin (1981)

Muehrcke (1990)

MacEachren (1995)

Graphic information processing

Geographic thinking

Visualisation tool

Graphic communication

Geographic illustration

Communication device

Table 1. The two purposes of maps expressed by different concepts.

20These two purposes of maps are used in order to distinguish between ‘cartography as communication science’ paradigm and the visualisation perspective.

The most basic tenet of the cartographic communication model is that the goal of cartography is to effectively communicate a particular message. There is an assumption not only that the message is known, but that there is an optimal map for each message, and that our objective as cartographer is to identify it. For cartographic visualization the message is unknown and, therefore, there is no optimal map ! The goal is to assist an analyst in discovering patterns and relationships in the data (MacEachren and Ganter 1990 : 65).

21The term paradigm comes from Kuhn (1962) who argued that two paradigms could not be in co-existence due to their incompatible nature. When MacEachren, however, denotes the cartographic research approach in the 1970s as the communication paradigm but the present approach as the visualisation perspective, he is not omitting the view of cartography from the communication paradigm to co-exist with visualisation perspectives.

This view of cartography does not discount the importance of communication-oriented research (MacEachren 1995 : 1).

22At the contrary, although visualisation and communication denotes two different sorts of map use, an absolutely distinction between them is impossible.

All visualization with maps involves some communication and all communication with maps involves some visualization (MacEachren 1995 : 357).

23The distinction between map used as visualisation tool or as a communication device is in a state of flux. MacEachren (1995), however, gives some important characteristics to what he considers as prototypes for visualisation and communication. These might be used to identify intended map use.



Revealing unknowns

Presenting knowns

High human-map interaction

Low human-map interaction

The map is tailored to an individual (private map use)

The map is designed for a wide audience (public map use)

Table 2. Characteristics of visualisation and communication (from MacEachren 1995 : 357)

24I would in this respect put forward the view that both the communication and the visualisation perspective co-existed in the writings of Bertin. Due to the narrow perspective of the communication paradigm, however, his theory of graphic semiology was interpreted and applied unilaterally for the purpose of map as a communication device. The aim of graphics is not restricted to communicate a simplified message known by the cartographer, it is also an exploratory tool to reveal information, generate new ideas and construct knowledge. Information is not something fixed or ready, but something that is constructed by the aid of graphics.

The goal of map use is to stimulate a hypothesis rather than to communicate a message. Information is instead "constructed" by the user, from the spatial representation of the world provided by the cartographer (MacEachren 1995 : 7).

Addressing the question

25In practical cartographic work, none of the elements of generalisation are treated individually. As indicated by the title of this paper, the main subject is to elaborate the third issue of generalisation, symbolisation or the choice of graphic representation. This might be the most difficult task for the cartographer. To be able to perform symbolisation, it is necessary to recognise its two components :

  1. the choice of the level of measurement,

The symbolization process includes selecting the level of measurement (nominal, ordinal, interval, or ratio) to use in the feature’s visualization (Robinson et al 1995 : 475).

  1. the choice of the visual variables,

choosing appropriate visual variables is the crux of cartographic symbolization. The success or failure of a map depends on this process (Robinson et al 1995 : 478).

26and to understand the relationship between these two components :

Cartographers must learn the level of measurement connoted by each visual variable and the efficiency with which each can symbolize a conceived feature dimension. They can then link their desired conception of the feature to a visual variable and create an acceptable symbol (Robinson et al 1995 : 476).

27Thus it seems to be of importance that the level of measurement correspond with a symbolising schema that guide the use of the visual variables.

Symbolisation and the level of measurement

Systems of organising the data

28The term levels of measurement stems from the achievements of certain psychologists, among them Stevens (1951), who defined the hierarchical, four level measurement scheme : nominal, ordinal, interval and ratio. Stevens realised that the qualitative/quantitative dichotomy placed restrictions on scientific advancement. In order to avoid these restrictions he created a more detailed measurement system, the four-levelled measurement system, that has since been adopted by several disciplines including geography and cartography. Thus, among the main textbooks in cartography (i.e. Robinson et al 1984) this four-levelled measurement system is applied.

29Between the conventional four-levelled measurement system and the old two-levelled dichotomy, we find the three-levelled organisation system used by Bertin. By identifying the visual variables and systemising their properties, Bertin established a relationship between data collected from the "real world" and their graphical representation (the signs) in the "map world". This relationship has to do with the meanings these signs (the visual variables) represent or signify. According to Bertin, there are three organisation levels (des niveaux d’organisation) or signifiers (signifiés) in graphics in which the map-reader might organise the visual variables.

Une composante sera donc qualitative, ordonnée ou quantitative. Ce sont les trois niveaux d’organisation de composantes (Bertin 1967 : 34).

Resemblance, order and proportions are the three signifieds in graphics. These signifieds are transcribed by visual variables having the same signifying properties (Bertin 1981 :177).

30As recognised by Board (198 : 61), Bertin’s three levels of organisation (qualitative, order and quantitative) cannot be equated simply with the conventional scales of measurement for data.

Image 1000020000000207000000994F957043CE64A413.png

Figure 1. The inconsistency existing between the two measurement systems.

31The problem of consistency becomes obvious when the cartographic literature, operating with the four-levelled measurement system, tries to make symbolisation rules with basis in the visual variables adapted for a three-levelled organisation system. The inconsistency appears in the distinction between interval and ratio and how these levels should be equalised with Bertin’s terms quantitative. If Bertin’s graphic semiology is to be applied with other measurement levels than those for which it was constructed, the inconsistencies must be resolved.

32As a point of departure for the case of the four-levelled system, let us examine how Bertin is denoting ordered and quantitative.

Une variable est ORDONNÉE (O) lorsque le classement visuel de ses catégories, de ses paliers est spontané et universel. On perçoit un gris comme l’intermédiaire entre un blanc et un noir, une taille moyenne comme l’intermédiaire entre une petite et une grand taille, il n’en pas de même d’un bleu, d’un vert, d’un rouge qui, à valeur égale n’offrent pas d’ordre spontané.

Une variable est QUANTITATIVE (Q) lorsque la distance visuelle entre les catégories d’une composante ordonnée peut s’exprimer spontanément par un rapport numérique. On perçoit qu’une longueur est égale à trois fois une autre longueur, qu’une surface est le quart d’une autre surface. (Bertin 1967 : 48)

33It is a general agreement that expressions of proportions like "twice as large" only are possible for data at the ratio level. Hence, this indicates that Bertin’s term quantitative (Q) equalises the measurement level, ratio, since it is able to portray proportions. While MacEachren (1995) recognises and remarks within brackets, that

… he [Bertin] did not distinguish between interval and ratio levels (p : 270)

34I would, based on the characteristics of the term quantitative and the obviously fact that the term order (O) equalises the level ordinal, state that the level interval is missing in Bertin’s level of organisation. The next arising question is if the interval level constitutes a relevant level in-between ordinal and ratio. In order to find an answer let us first take a look at the conventional way of differentiating between interval and ratio scale. The distinction between them

refers to whether the variables’ units of measurement have an arbitrary or absolute zero (Walford 1995 : 14).

35If the variables’ unit have an absolute zero, statements of proportions are possible. According to Robinson et al, however, the distinction between interval and ratio is not relevant for the cartographic symbolisation.

In both instances a range is being displayed, and from the point of view of representation, it is immaterial whether or not the scale begins at an arbitrary zero (Robinson 1984 : 110).

36There is, however, another term that can be equalised with the interval level : the term range-grading.

Range-grading acts much like interval scale measurements. Often, for instance, we group features by attribute values into classes, such as family incomes of "less than $10,000", "$10,000 to $30,000", "$30,000 to $50,000", and "over $50,000." (Robinson et al 1995 : 273)

37While, according to Robinson et al, the distinction between interval and ratio level is without importance in cartographic symbolisation, the distinction between range-grading and ratio matters. Range-grading is the result of the classification activity executed on ratio data. Hence, after having decided ‘the choice of the number of classes’ and ‘the choice of the method applied for determining class limits’ the ‘choice of graphics representation’ remains. Since this process is often applied in choropleth and other statistical mapping, indicates that range-grading becomes a relevant level in-between order and ratio.

On the other hand, a distinction between ordinal, range-graded, or ratio portrayal does make sense cartographically (Robinson et al 1984 : 279).

38Having presented different systems of organising statistical data, I would state that a four-levelled measurement system is relevant for cartographic symbolisation. However, applying the graphic semiology is therefore not a straightforward task since the theory was constructed for a three-levelled organisation system. Before we get closer to the proposed solution to this inconsistency, let us first look at three schemata for symbolisation. We start with Bertin.

Symbolisation schema after Bertin

39Bertin’s theory is based on the graphic sign system in which eight visual variables are recognised : the two dimensions of the plane (x and y), size, value, texture, colour, orientation, and shape. Bertin considers colour to be without variation in value. When the colour intensity changes, it is due to change in the visual variable value. Morrison (1974) added a third dimension of colour : saturation. Bertin also mentioned saturation, but he merged it with hue into the single category he called colour. Although saturation and other visual variables have been recognised by some North American cartographers (Morrison 1974, MacEachren 1995), they will not be commented on in this paper.

40The visual variables according to Bertin have one or more of the following properties  :

  • quantitative (proportional) Q

  • ordered O

  • qualitative :

  • selective (differential) #

  • dissociative (variable visibility) Image 100000000000000E0000000E04021663E08409FB.png

  • associative (constant visibility) Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

41In cartographic symbolisation as in all graphics, the visual variables should be used according to their organisation level :









Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

Image 100000000000000D0000000C2BFAB94A4EE31B6B.png








Image 100000000000000E0000000E04021663E08409FB.png

Image 100000000000000E0000000E04021663E08409FB.png






















Table 3. Symbolising schema after Bertin (1981  : 231).

42Bertin stresses the importance that the level of organisation plays to graphics  :

La notion de niveau est ici fondamentale. C’est sans doute (avec l’imposition des diagrammes) la source du plus grand nombre d’erreurs graphiques (Bertin 1967 : 64).

43Hence, according to Bertin, a symbolisation that is not in accordance with the schema above will destroy the meaning of the data.

Symbolisation schema after Weibel and Buttenfield

44Another approach that follows the three-levelled organisation of Bertin, is the approach to Weibel and Buttenfield (1992). The purpose of their paper is to propose approaches to improve the quality of graphics produced in a GIS environment. As they consider the provision of cartographic training only as a partial solution, they recommend the development of an expert system for map design. All expert systems must be grounded on some knowledge base or rule base. Although the fundamentals of map design are largely explained by several textbooks, these

principles are rarely presented in the form of a set of strict rules which lend themselves to a straightforward implementation in automated cartography systems (Weibel and Buttenfield 1992 : 225).

45Weibel and Buttenfield (1992) use the three interrelated components : generalisation, symbolisation and production as a framework for identifying the major elements of map design (Buttenfield and Mark 1991). With respect to symbolisation, they find a relatively comprehensive and consistent theory for cartographic symbolisation in the work of Bertin (1983). They are referring to the same visual variables that Bertin identified and do not include other visual variables identified by others. They are also using the same organisation levels as Bertin. Each visual variable has particular properties to transcribe data belonging to a particular organisation level. These properties are summarised and reproduced below.









Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

Image 100000000000000D0000000C2BFAB94A4EE31B6B.png

Image 100000000000000D0000000C2BFAB94A4EE31B6B.png








Image 100000000000000E0000000E04021663E08409FB.png

Image 100000000000000E0000000E04021663E08409FB.png






















Table 4. Symbolising schema after Weibel and Buttenfield (1992, figure 3, page 229).

46In comparison with Bertin, there is a striking difference in how Weibel and Buttenfield denote the qualities of the visual variable value. While Bertin was strict and allowed only size to portray quantitative data, Weibel and Buttenfield gives this property also to value.

A four-levelled measurement system consistent with Bertin

47A suggestion have been proposed by Geels (1987) and taken up by Kraak and Ormeling (1996) about how to apply the conventional four level measurement system together with Bertin’s visual variables in a consistent system. They solve the problem of inconsistency by systemising four perceptual characteristics corresponding to the levels of measurements. These are as follows :

  • difference in quality - nominal scale

  • difference in order - ordinal scale

  • difference in distance - interval scale

  • difference in size (proportions) - ratio scale

Now all the building blocks would seem to be available to match the data components to the graphic variables that convey an idea of the kind of differences expressed by the measurement scales : qualitative differences by rendering nominal components, ordered differences for rendering ordered components, distance differences for rendering interval components and proportional differences for rendering ratio components (Kraak and Ormeling, 1996 : 150-151).

48Instead of separating interval and ratio scale with their point of zero, Kraak and Ormeling find it convenient to differentiate the scales according to their abilities to measure differences. At the nominal level one can differentiate between geographical phenomena. Variables at the ordinal level can be set in a succession and exhibit "greater than" relations between classes. As for the variable at ratio level one can decide proportions. Interval level

exhibit known distance or interval relations between map classes (Muehrcke 1976 : 131).

49Thus, at interval level, one can decide a succession, as well as distance - not between individual values, but between classes or groups of data. Kraak’s and Ormeling’s use of difference in distance might seem clearer if we exchange the term interval with the term range grading. By the manner, as already outlined, Robinson et al describe the range-graded level, it seems obvious that it constitute a natural belonging between ordinal and ratio level. It can easily be proved that at the range-graded level, differences in proportions cannot be derived. This is only possible at the ratio level. From the family example the families were classified into four brackets : "less than $10,000", "$10,000 to $30,000", "$30,000 to $50,000", and "over $50,000." Clearly, we cannot say that a family in the upper bracket is four times wealthier than a family in the lower bracket. We are, however, able to say that a family belonging to the lower bracket has a difference in income of $40,000 or more in comparison with a family in the upper bracket, hence a difference in distance. Such a difference in distance is not obtainable at the ordinal level, since this level does not indicate any specific numerical magnitude of difference.

50Following Kraak and Ormeling (1996 : 123) but with emphasis on range-graded rather than interval, this approach might serve as a solution to the inconsistency between Bertin and the conventional statistical literature. Thus, being confronted with the need to symbolise range-graded data, which often occurs after a classification, a symbolising schema also including this level finally becomes available. Bertin did not explicitly provide a solution for symbolising range-graded data, but by looking at his map showing water hardness in the USA (Bertin 1983 : 78), we can see that he uses value in order to portray range-graded data. It seems therefore as Kraak and Ormeling are in accordance with Bertin. Value as claimed by Bertin, is not able to connote difference in proportions (ratio) but according to his practice, they can provide differences of distance (range-graded). The table below summarises the use of the visual variable according to the level of measurement.




































Table 5. Symbolising schema after Kraak and Ormeling (1996  : 151). Capital ‘X’ indicates strong perception, small ‘x’ indicates weak perception.

51This schema shows striking differences with Bertin. While Bertin conceptualises texture as being a week ordinal variable, Kraak and Ormeling allow texture to be a strong ordinal variable and a week interval variable. Another difference is that Kraak and Ormeling do not subdivide the nominal qualities. Thus the special quality of texture being both ordered and associative (constant visibility) is not accentuated.

Towards a solution

Organisation systems adopted for two purposes of maps

52Most cartographic texts have emphasised a division into nominal, ordinal, interval, and ratio levels of measurement. These four categories are usually grouped into higher level categories in one of the following ways :

  1. A qualitative/quantitative dichotomy where qualitative includes nominal and quantitative includes ordinal, interval and ratio.

  2. A non-numeric/numeric dichotomy where the non-numeric information holds categorical information including nominal and ordinal and the numeric information includes interval and ratio level.


…cognitively, ordinal categories are, perhaps, more naturally grouped with the numerical categories of interval and ratio-level information (MacEachren 1995 : 253)

54the non-numeric/numeric distinction correspond with the objective of the present paper. That is, adopting cartographic symbolisation rules for the two purposes of maps  ; respectively communication device and visualisation tool. These two intentions of map use correspond to different levels of generalisation.

Image 10000200000001CF000000E4B63999F2DF4D649E.png

Figure 2 : The correspondence between level of measurement and mapping purposes.

55When the map is intended as a communication device, the aim is rapid perception and memorising overall information, and its imperative is simplicity. To obtain those goals, generalisation is needed. Thus if the purposes of maps is to be a communication device – one should generalise the data to these non-numerical levels of measurement.

the map is not merely data ordered in a particular way, but also a communication device. From time to time, we must sacrifice theoretical elegance for visual effect […] data collected at one level are often converted to a lower level before mapping so as to produce a clearer map (Unwin 1981 : 24).

56This is to follow Morrison (1974 : 123) who stated that marks on maps could only connote two levels of measurement, the nominal and the ordinal, thus excluding the numerical levels of measurement. However, Morrison does state that maps can connote higher scales ‘when imbued with meaning via a legend’. But as the aim of cartographic communication is rapid perception, maps as a communication device should demand a minimum of legend information.

57When the map is intended as a visualisation tool, a more detailed measurement level is needed. Therefore, for such purposes of maps, the numerical levels should be included. One of the main contributors to the adoption of the four-levelled measurement system in cartography is Muehrcke (1976). As Bertin (1967) and Robinson (1995) he also stresses the important link between the organisation level and the symbolisation for the geographical data.

Ideally the form or appearance of a map symbol should connote to the map user the scaling level of its underlying referent data in earth space, since this coordination of symbol dimensions to type of data would facilitate direct map reading, i.e., map use that demands the minimum of legend information (Muehrcke 1976 : 126).

58Thus a sensitive four-levelled measurement system would have a clarifying effect on the map user, but it complicates the map reading process, since it demands legend information.

In order for the mapmaker to employ interval scaling on a map it is necessary to resort to numerical legend annotation or to incorporate numerals into the map context itself, such as is the case with temperature maps, contour maps, and so forth (Muehrcke 1976 : 131).

59The map user in this context is a knowledgeable person applying the map, or other source of graphics, as a visualisation tool in order to explore relationships inherent in her data set. However, as described by Bertin, such maps might serve neither as a communication device nor as a visualisation device if they do not reveal relationships.

Isarithms precisely define a quantity at each point on the paper. But, except for drawings supplemented by a value variation [isopleth], the curves construct reading maps (Bertin 1981 : 151).

60Bertin’s distinction between reading and seeing maps is comprehended mainly in the context of representing several variables versus only one variable in the same map. In the above citation, however, Bertin applies the distinction between reading and seeing maps for cartography with one ordered characteristic.

Representing quantities and perception of quantities

61Visualisation of quantitative data raises questions that are not easily answered : how are quantities to be represented and how able are human beings in estimating these quantities ? In order to answer these questions we need to return to the statement : ‘Information is answer to question’. Revealing spatial relationships is to reveal geographical information. Thus geographical information is revealed if the following question is answered : "Given a characteristic, what is its geography (geographical distribution) ?" Geographical information consist of at least three information variables : two geographical variables (the dimensions of the plan) plus a third representing the "theme" it self (Baudouin and Anker 1984 : 5). It is this third information variable that Bertin denotes for z.

How to represent quantities in z ? … Its solution depends on an implantation, by point, line, or area, of the quantities in the plane, which implies a distinction between absolute quantities and ratios. It also depends on the properties of size and value variations (Bertin 1981 : 184).

62After a closer reading of Bertin, we see that he is differentiating between two types of quantities.

Graphics separates quantities into two types … totals per object and ratios between totals per object. The totals are also called "absolute quantities" (Bertin 1981 : 190).

63The term ratio used by Bertin corresponds with the term ‘derived ratio’ which is more common in the cartographic literature. One typical example of derived ratio is density, i.e. population density. To avoid confusion, the term density will replace Bertin’s term ratio. In what respect are the ordered visual variables size and value able to represent absolute quantities, densities, range-graded, and ordered ? The question should be examined for the three basic categories of symbols : point, line and area.

64According to Bertin, only size can portray proportions and thus absolute quantities. Its implementation is only possible for point symbols.

Only implantation by point can manifest absolute quantities in z (Bertin 1981 : 190).

65Can this serve as guidelines for representing absolute quantities ? Is human vision able to interpret quantities from the graphics ? Bertin provided the following test :

Lorsque la perception est quantitative, le rapport numérique entre deux signes est immédiat et ne nécessite aucun recours à la légende, il apparaît spontanément au lecteu  : ceci est double, est 8 fois cela. Le meilleur test sensible sera donc de demander au lecteur la valeur du signe supérieur lorsqu’il attribue la valeur 1 au signe inférieur (Bertin 1967 :69).

66Following the result from empirical research performed in the years after Bertin (1967), we have to admit that the characteristics describing the quantitative (absolute quantities) variable seldom is attainable for graduated circle maps due to underestimation. One of the most thoroughly studied research topics in cartography, at that period, was probably perceptual scaling of graduated circles.

The underlying assumption of this research was that map-readers could make numerical (interval or ratio) level judgement based on graduated symbol maps (if the symbols were scaled appropriately). The eventual result of this research effort seems to be that magnitude perception is too variable (both from individual to individual and within individual) for any scaling function to be adequate (MacEachren 1995 : 308, footnote 14)

67As noted by MacEachren, these research results support Morrison’s (1974) statements that marks on maps could only connote two levels of measurement : the nominal and the ordinal. Consequently, I would state that it is possible to portray absolute quantities by using the visual variable size. However, due to variation in individual magnitude judgements, one cannot expect (as Bertin does) that the map-reader is able to induce proportions (quantities) without legend inquiring. Without this anchor information, one should not expect an induction more precise than at the ordinal level, especially not in the case of graduated point symbols. The same statement could not be applied for the case of linear symbols, since map users do not have the same problem of estimating linear symbol dimension (Robinson et al 1995 :485-6).

68Then, what is the situation for area symbolisation techniques ? The only possibilities of utilising size and hence portray absolute quantities is by applying the anamorphose mapping technique where the enumeration units’ area are scaled proportional to the matching value (Cauvin and Reymond 1986). Value is the next visual variable in the ordered hierarchy, but most cartographers would exclude choropleth maps as possible representation for absolute quantities.

The data must be areally standardized for areal symbolisazation, or left in raw (unstandardized) form for point symbolisation : the symbols must be consistent with the level of data processing. (Weibel and Buttenfield 1992 : 227)

69This is in accordance with the guidelines provided by Bertin :

  • Consider quantities applied to areas, say communities, for example.

  • These quantities must be ratios.

  • To extend these quantities over the entire area, we must apply a grid of points on the map. The scale of this grid must be such that there is at least one point in each area (Bertin 1981 : 207).

70Following these guidelines, we can represent densities for areas either by using a grid of point symbols (size) or by using value. The latter is most common due to the choropleth mapping technique in which the visual variable value often is employed.












Surface area




Population density

Image 10000000000001FA000000F67DFEA4A766B75988.jpg

Figure 3 : Density represented by a grid of point symbols (size) and value (from Bertin 1981 : 191).

71When value is able to portray data at one level, it is also able to portray data at lower levels.

Image 1000000000000095000000503C330534BE226FD0.jpg

Figure 4 : A quantitative symbolisation can also transcribe an ordered, selective or assosiative information (from Bertin 1967 : 39)

72Based on the discussion above, I will provide the following schema as a point of departure for establishing a rule base for the symbolisation of statistical maps. I find it convenient to apply the conventional four-levelled measurement system as a ground. Bertin’s distinction between nominal data having representatively constant and variable visibility and/or selective qualities is applied. In addition, a division of the ratio level into absolute values and densities is as well used.

Image 10000000000002EC000001270C59ACFBB662BB90.jpg

Table 6 : The visual variables, the level of measurement and the purposes of maps.

73From the table above, it becomes obvious that the flexibility in symbolisation increases as the measurement level of precision decreases. However, the distinction between communication and visualisation might seem too severe. I apply a conservative point of view, and I am thus closely following an interpretation of Bertin involving only the non-numerical levels constituting the map-purpose of communication device :

Notons que la perception visuelle quantitative n’a pas la précision des mesures numériques (si elle avait cette précision, les nombres n’auraient sans doute pas été inventés). (Bertin 1967 : 48).

74Thus, if numerical information is in demand, the communicative map might not be the best device providing this information, because such information demands legend interpretation.

Towards the rule base

The democratising of cartography

75While cartographers 30 years ago represented a group of specialists, today, nearly anybody can be a cartographer due to the "Human Age" where hardware and software limitations no longer override the mapmaker’s requirements. We have seen an embedding of cartographic tools in the most outspread software for spreadsheets and in statistical packages. This embedding, however, does not grant that everybody will become skilled cartographer, mostly due to the lack of graphic education. Several authors voice this concern, among them Kennedy (1994) :

... currently available technology makes it possible for anyone with a computer and a laser printer to produce scientifically unsound but aesthetically attractive maps (p.16).

76Within this context, there were during the 1980s high expectations for knowledge-based systems guaranteeing the production of proper maps. In retrospective, one has to agree that these expectations were too high. Kraak and Ormeling (1996) outlines two reasons why :

First, it proves that a discipline such as cartography is far too large to fit a single knowledge-based system, and second, the discipline is difficult to lay down in a set of rules (p. 210).

77Although I agree with this statement, I believe that for a restricted domain of cartography, like a restricted domain of statistical mapping, it is possible to feed a knowledge base or rule base. This might seem as plain talk and a too optimistic attitude towards the covered and subsequent argumentation. Buttenfield and Mark (1991) have considered that expert systems for statistical map design are more difficult to realise than expert systems for topographical map design.

The design of statistical maps involves a more complex and intuitively guided decision process (Buttenfield and Mark 1991 : 130).

78On the contrary

the symbology of locational maps is more standardized and thus design constraints may be more readily identified. The specificity of map purpose for locational maps also places finite bounds on the range of appropriate symbology, … (Buttenfield and Mark 1991 : 130).

79What seems to be needed for the realisation of a rule base for statistical mapping is hence a standardised symbolisation capable of placing design constraints. If this is to be realised, The identification of the measurement level is paramount.

Identification of the measurement level

80Since knowing the level of measurement is a prerequisite for symbolisation, this information must in some way or other be provided. A common method to obtain necessary information required for a suitable graphic representation is by asking a set of questions. Following this method, Muller et al (1986) defined and implemented a rule-based system where they addressed the user of the system

a series of forty queries subdivided into nine categories which are considered necessary for the determination of a suitable graphic representation (Muller et al 1986 : 559).

81One of these nine categories was measurement level containing the sub-categories ratio, absolute, ordinal and categorical from which the user of the system has to make appropriate selections.

82Muehrcke (1976) called attention to the importance of knowing the measurement concepts, but he was aware that map-readers are unlikely to be disciplined with measurement concepts. He stresses this point because

… effective map reading requires familiarity with measurement concepts, … (Muehrcke 1976 : 125).

83Since nearly anyone now can produce statistical maps, the unfamiliarity might as well be present among the map producers. If, however, the map producer, due to unfamiliarity with measurement concepts, is unable to perform this input of requirement, it is essential that the identification of measurement level is feasible for the rule base. This is unfortunately not at all an easy task. There are few guidelines, if any, to identify the measurement level from the data themselves. A solution could be to establish an attribute for each variable structured as metadata. An example where metadata containing the level of measurement is already realised for demonstration purposes of atlas mapping (Forest, 1995). There are simple technical solutions available for storing additional information since most common spreadsheets or database files have options of labelling the variables. When the label denoting the measurement level is identified by an expert system for cartographic symbolisation, the software might then provide for the user reasonable default options.

84Because of the distribution and availability of statistical data, these data should be distributed holding standardised information about their measurement level and other useful information.

Standardised metadata for statistical data would help users understanding the contents of datasets they receive from others (Robinson et al 1995 : 262).

85Metadata holding information of measurement level has to be prepared by the supplier of statistical data. More focus on arrangement of standardised metadata for distributed statistical data would undoubtedly contribute to a more feasible realisation of expert system for statistical mapping. It is astonishing that in the presentation of elements of spatial metadata content, Robinson et al only represent those relevant for physical data (1995 : 264-266).


86The main objective of the present paper was to illuminate an inconsistency in the organising of data. Being the prerequisite for symbolisation, it is essential that we overcome this inconsistency between systems of measurement before we perform graphic information processing or graphic communication. Corresponding with these two aspects of graphics are the two aspects of map use : maps serving respectively as a communication device and as a visualisation tool. For these two mapping purposes, a symbolisation schema is proposed. This schema is hopefully a point of departure for a rule base for statistical mapping. However, this schema needs enlargement by including additional recognised visual variables. Thus, while this paper has discussed the "y-axis" in the schemata holding the levels of measurement, a discussion should also be performed for the "x-axis" holding the visual variables. The main issue then becomes how to perform an appropriate implementation of such schema serving as rule base for cartographic symbolisation. The question remains to be answered.

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Jan Ketil Rød, « The third choice », Cybergeo : European Journal of Geography [En ligne], Dossiers, document 154, mis en ligne le 17 novembre 2000, consulté le 06 juillet 2020. URL : ; DOI :

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Jan Ketil Rød

Department of Geography, Faculty of Social Sciences and Technology Management, Norwegian University of Science and Technology (NTNU) N-7055 Dragvoll, Norway, Tel : + 47 73 59 17 23 Fax  : + 47 73 59 18 78

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