Cartograms – Classification and Terminology

Polish Cartographical Review Vol. 51, 2019, no. 2, pp. 51–65 DOI: 10.2478/pcr-2019-0005 Received: 01.02.2019 ANNA MARKOWSKA Accepted: 20.05.2019 Institute of Geodesy and Cartography Geoinformation and Cartography Warsaw, Poland orcid.org/ 0000-0001-8446-6171 [email protected]

Cartograms – classification and terminology

Abstract. The author discusses new classifications of cartograms. Cartographic anamorphoses terminology and a multi-level classification, including not only cartograms but also anamorphical projections, have been proposed. The selected area cartograms’ classes were discussed in detail and compared. Keywords: cartograms, area cartograms, cartograms’ classification

1. Introduction rature considers cartograms mostly as maps which are the result from a purposeful modifi- A cartogram1 is a form of cartographic pre- cation of traditional maps, mainly choropleth sentation for which there is no unified termino- and diagram maps (R. Szura 1989), i.e. “maps logy and full classification. Earlier divisions of of visibly distorted geometry – in comparison cartograms, both Polish and foreign, were in- to Euclidean geometry – but better suited to complete. Anamorphical projections and ana- reader requirements connected to map’s func- morphic pseudomaps were often omitted. The tion” (A. Michalski, P. Tymków 2011, p. 19). author aims to discuss the terminology referring In the literature there appear various terms to cartograms, present a proposed multi-level referring to area cartograms. The most common classification of cartographic anamorphoses English terms are cartogram and value-by- and compare classes of area cartograms. -area map. The term cartogram was originally used for graphic presentation of statistical data, in cartographic meaning it was first used in 2. Cartograms – terminology 1851 to name a series of maps Cartogrammes a foyer diagraphiques by C.J. Minard (H. Friis A cartogram is a map, on which one feature 1974). Today the English term cartogram refers – distance (distance cartograms) or area (area to maps elaborated in a “scale other than a true cartograms) is distorted proportionately to the scale” (V.S. Tikunov 1988, S. Mayhew 2004, value of a given phenomenon (A. Faliszewska B.D. Henning 2011). 2011). According to J.C. Muller (1982, 1983), if In the sixth volume of The history of carto- cartograms are a particular type of projection, graphy… (M. Monmonier 2015) other terms for all traditional maps can be treated as carto- area cartograms are listed: anamorphosis, diagrams, and traditional equal-area maps can be gramic maps, map-like diagrams, varivalent pro- treated as area cartograms. However, the lite- jections, density equalized maps, isodensity maps, mass-distributing (pycnomirastic) map projections. Terms of cartograms often relate to 1 In Polish terminology the term “kartogram anamorficzny” the shape of basic units, e.g. rectangular carto- has been used for many years to describe area cartogram (R. Szura 1989). The Polish term “kartogram” means “cho- gram (E. Raisz 1934) or circular cartogram, more ropleth map”. often known as Dorling cartogram (D. Dorling 52 Anna Markowska

1996). Terms of many area cartograms contain 3. Cartograms – classification the name of the algorithm used in their preparation. When W. Tobler (2004) wrote about compu- For the purpose of map classification, seven ter elaboration of area cartograms, he mentioned classification criteria were set. The criteria are J. Dougenik, N. Chrisman and D. Niemeyer divided depending on which map class they (Continuous Area Algorithm, 1985) or Gastner- can be referenced: -Newman (Diffusion-based Method, 2004) algo- • cartographic anamorphoses: rithms. It is possible to find other names related A) mathematical basis, to algorithms used to generate cartograms B) transformed object, (A. Markowska, J. Korycka-Skorupa 2015), e.g.: C) transformed method; − algorithm of W. Tobler from 1973 (Rubber- • area cartograms: -map Method) and from 1986 (Pseudo-carto- D) graphic continuity; gram), E) graphic presentation; − algorithm of D. Dorling from 1990 (Cellular • anamorphical projections: Automation Algorithm), F) grid transformation; − algorithm of D. House and C. Kocmoud • distance cartograms: from 1998 (Continuous Area Cartogram Using G) the main distortion point’s location. the Constraint-based Method). Since the publication of W. Tobler’s (2004) A. Mathematical basis article there have appeared new algorithms, Division according to mathematical accuracy and therefore new terms referring to cartograms, of transformation was established as the main e.g. gridded cartogram (B.D. Henning 2011), criterion of classification of cartographic ana- circular-arc cartogram, rectilinear cartogram, morphosics (Z. Mudrych 1976). The term carto- table cartogram or mosaic cartogram (S. Nusrat, graphic anamorphosis is overriding in relation S. Koborov 2016). to cartograms. It includes all presentations

Fig. 1. Cartograms’ classification Cartograms – classification and terminology 53 having certain anamorphic features, though they or distances between selected points can be may not be drawn up according to strict ma- introduced (fig. 1, criterion C). Both transfor- thematical rules or may not maintain spatial mations are proportional to the value of a phe- relations (fig. 1, criterion A): nomenon (J. Olson 1976). Two classes of − anamorphic maps (app. 1, pt 1) – presen- cartograms can be proposed: tations which have been made according to − area cartograms – maps on which the area mathematical and spatial rules; of individual spatial units is changed depend − anamorphic pseudomaps (app.1, pt 2) – ing on the value of a phenomenon (app. 1, graphically close to anamorphic maps, but ela- pts 5a–10); borated in a more arbitrary way, often without − distance cartograms – maps on which the clearly determined mathematical rules or rules distance between selected points is changed connected to spatial relations. depending on the value of a phenomenon (app. 1, pts 4a–4b). B. Transformed object For distance cartograms a general division The second criterion of classification consi- is proposed according to the location of the ders if the anamorphic transformation is applied main points of distortion (fig. 1, criterion G): to the cartographical grid or to the thematic − monocentric distance cartograms – maps contents of the map (fig. 1, criterion B): on which the distance between the central − cartograms (app. 1, pts 4a–10) – anamor- point and given points changes with the value phic maps in which, depending on the value of of a phenomenon (e.g. distances between a se- the phenomenon, the area of individual areal lected metro station and next stations expressed units is changed (area cartogram) or the distance in the time of journey – app. 1, pt 4a); between selected points is changed (distance − polycentric distance cartograms – maps cartogram); which means that changes are in- on which the distance between a subsequent troduced depending on the thematic contents pair of points in a given network changes with of the map; the value of a phenomenon (e.g. distances − anamorphical projections (app. 1, pt 3) – between two subsequent stations of the War- anamorphic maps resulting from a transforma- saw metro expressed in the time of journey – tion of cartographical grid (J. Korycka-Skorupa, app. 1, pt 4b). et al. 2015). Anamorphical projections are a type of distortion projections in which cartographical grid D. Graphic continuity is distorted (S. Grabarczyk-Walus 2007). In The author of a map performs various ope- this group of projections a local scale change rations during the process of cartogram elabo- is achieved by transforming linear elements of ration (B.D. Dent 1999, B.D. Dent et al. 2009 the map by specially selected transformational − fig. 2). In an anamorphical transformation functions. Areas around the selected center the area of both contiguous and noncontiguous are enlarged. Thus it is possible to observe cartograms is changed. In both classes of area more details in the reader’s area of interest. cartograms the area of base units is modified According to the type of distortion of the car- proportionately to the value of a phenomenon. tographical grid of the base map three classes In contiguous and noncontiguous cartograms of anamorphical projections can be indicated mutual location (orientation) of base units is (fig. 1, criterion F): preserved. Therefore changes of area or orien- − one-direction anamorphoses (distortion along tation cannot differentiate classes of area car- one axis of the rectangular coordinate system), tograms and should not be the criteria of their − two-direction anamorphoses (distortion along classification. two axes), Spatial continuity of presentation differentiates − radial anamorphoses (distortion in radial contiguous and noncontiguous cartograms. directions from a given central point). Therefore this characteristic was chosen as a criterion of classification of area cartograms. C. Transformed method Another aspect differentiating contiguous and In the process of editing the contents of a car- noncontiguous equiform cartograms is the tograms, the changes in the area of base units shape of base units. According to the scheme 54 Anna Markowska

Fig. 2. The operations take to edit area cartograms (B.D. Dent 1999; B.D. Dent et al. 2009)

(fig. 2) contiguous cartograms always have fied terms diagram cartograms, rectangular a changed shape of base units, while in non- cartograms, square cartograms or circle carto- contiguous equiform cartograms the shape is grams are used; preserved. However, it is possible to construct − contiguous regular (mosaic) cartograms a noncontiguous cartogram, in which base (app. 1, pt 6) in which base units correspond in units are presented in the form of geometrical shape to spatial units, but their borders are figures, e.g. circles. For that reason the crite- geometrized, most often build of segments. rion of spatial continuity is overriding to the Further in the article a simplified term regular criterion of shape. Using spatial continuity as cartograms is used; a criterion of classification of area cartograms − contiguous irregular cartograms (shape-like the following classes can be proposed (fig. 1, cartograms) (app. 1, pt 7) in which base units criterion D): correspond to the shape of statistical units, − contiguous cartograms –area cartograms their borders are not geometrized, the shape on which spatial continuity is preserved, which of units depends on the distortion algorithm means that base units have not been separated applied. Further in the article a simplified term (app. 1, pts 5a–7); irregular cartograms is used. − noncontiguous cartograms – area carto- During the division of noncontiguous cartograms on which spatial continuity is broken, grams, firstly the cartograms which preserve which means that base units have been sepa- the shape of spatial units (noncontiguous rated (app. 1, pts 8–10). equiform cartograms – app. 1, pt 10) were diffe- rentiated from those which do not preserve the E. Graphic presentation shape (noncontiguous diagram cartograms – The last criterion of classification refers to app. 1, pt 8, and nocontiguous regular carto- the shape of base units. This aspect can be grams – app. 1, pt 9). Noncontiguous diagram considered in both contiguous and nonconti- cartograms have base units represented by geo- guous cartograms. Looking at the shape of metric figures (e.g. circles, squares, hexagons), base units contiguous cartograms could be whereas in noncontiguous regular (mosaic) car- divided into (fig. 1, criterion E): tograms the shape of base units corresponds − contiguous diagram cartograms (app. 1, to spatial units, but their borders are geometrized, pts 5a–5c) in which units of reference have often built of segments. been substituted by simple geometric figures, Figure 1 presents a classification of carto- e.g. squares, rectangles, circles (e.g. Dorling graphical anamorphoses. It includes the clas- cartogram, 5c). Further in the article a simpli- sification criteria mentioned earlier and terms Cartograms – classification and terminology 55 describing individual anamorphic forms. Look- value of a phenomenon, and the units have ing at area cartograms as forms of cartographic the shape of a rectangle (M. van Kreveld, presentation it can be noted that the proposed B. Speckmann 2004 – app. 1, pt 5a). criteria focus not only on the process of map In the process of elaboration of rectangular transformation (the presentation method), but cartograms several rules have to be followed: also the result of that process (the presenta- − rectangle area has to be proportional to tion form J. Korycka-Skorupa 2002). The first phenomenon value (e.g. size of population or group of criteria (method-related) includes: value of agricultural production); mathematical basis, transformed object and − in the case of adjoining units (e.g. voivod- transformed method. In the second group ships) their adjoining location has to be pre- (form-related) two criteria can be noticed: served on a diagram cartogram; graphic continuity of presentation and graphic − adjoining rectangles cannot border only method of presentation. with corners – sides always are connected. Some of the first rectangular cartograms 4. Classes of area cartograms were prepared by E. Raisz (1934). The legend made by E. Raisz (fig. 3) is very useful. A square Classes of area cartograms were differenti- shows a certain value of a phenomenon ated according to the last two criteria of the (e.g. 1 million dollars), which makes it possible classification of cartographic anamorphoses to estimate value of individual units. presented earlier (fig. 1): An important aspect of rectangular cartograms − graphic contiguous – information if units is that maps can be read not only at a general are connected or separated (contiguous carto- level (general view of the map), or indirect level gram / noncontiguous cartogram); (comparing relations between base units), but − graphic presentation – related to the also at the detail level of map reading – esti- shape of base units (geometrical figures, regu- mating approximate values of individual units, lar/irregular shape) and the location of centroids which is facilitated by a correctly elaborated of spatial units. legend. A rectangular cartogram can be computer generated (M. van Kreveld, B. Speckmann A. Contiguous cartograms 2004). Contiguous cartograms as a class of area cartograms preserve spatial continuity of presentation. Three subclasses of contiguous A1b. Square cartograms cartograms have been proposed: diagram carto- Square cartograms (Demers cartograms) grams, regular cartograms (mosaic cartograms), are a special kind of rectangular cartograms, irregular cartograms (shape-like cartograms). in which base units are presented as squares (M. van Kreveld, B. Speckmann 2004 − app. 1, pt 5b). A1. Diagram cartograms In diagram cartograms spatial continuity of presentation is preserved, and individual units A1c. Dorling cartograms are substituted by geometric figures. Diagram The name of the Dorling cartogram (circular cartogram subclasses have been distinguished cartogram – app. 1, pt 5c) came from the name by the shape of base unit, e.g. rectangular (app. 1, of its author – Daniel Dorling, the British geo pt 5a), square (app. 1, pt 5b). Diagram cartograms grapher who specializes in social-geographi- using circles are an exception. In the literature cal issues. they are known as Dorling cartograms, the term In the Dorling cartogram the value of a phe- Circular cartograms (app. 1, pt 5c) is rarely used. nomenon is proportional to the area of a circle These two terms can be used interchangeably (D. Dorling 1993). Initially D. Dorling did not (A. Faliszewska 2011). use circles but hexagons for data presentation (A. Faliszewska, J. Korycka-Skorupa 2010). In A1a. Rectangular cartograms Great Britain border on six other constituen- In this class of diagram cartograms spatial cies, and voting most constituencies were the continuity of presentation is proportional to the topic of many of D. Dorling’s works. 56 Anna Markowska

Fig. 3. Rectangular cartogram. The area of rectangles is proportional to the value of US production in 1929 (E. Raisz 1934)

A2. Regular cartograms (mosaic cartograms) (M.J. Alam, et al. 2013) or more sides (fig. 4) Regular cartograms attempt to maintain the called a mosaic cartogram (R.G. Cano et al. general shape of a given area while geometrizing 2015). Despite the availability of automated it. Individual base units are built of unit squares methods of elaboration of regular cartograms, (segments), e.g. one square represents 150 the popularity of this class of cartograms is still births (app. 1, pt 6). quite low. Algorithms are not available in popular By using a unit square it is possible to render GIS programs and there are no desktop or approximate shape of a whole country or con- web applications with an easy interface which tinent. The main problem of presentations of would enable quick preparation of regular car- this type is that they are time-consuming. Also, tograms. particular attention has to be paid to the data used. While presenting some statistical data it A3. Irregular cartograms (shape-like car- is difficult to maintain the shape of individual tograms) areas because of their high range. It may be The most important characteristic of irregu- necessary to omit the units with very low value. lar cartograms is that they maintain spatial Regular cartograms can be prepared auto- continuity of a phenomenon, meaning that there matically, applying algorithms which enable is no gap between units (B.D. Dent et al. 2009). elaboration of polygons of up to eight sides In order to generate a map of this type it is Cartograms – classification and terminology 57

Fig. 4. Regular cartogram (squares or hexagons). Number of votes cast for candidates in the presidential election in 2012 (R.G. Cano et al. 2015)

necessary to change the shape of individual of presentation is lost. Depending on the main- units while keeping their continuity. tenance of the shape of base units three classes In this class of contiguous cartogram the of noncontiguous cartograms have been pro- area of individual spatial units should relate to posed: noncontiguous equiform cartograms their geographical shape. For that reason they (maintain the shape), noncontiguous diagram are irregular as in the illustration, app. 1, pt 7. cartograms and noncontiguous regular carto- Elaboration of irregular cartograms is very grams (do not maintain the shape of base labor-consuming and almost impossible without units). The location of the centroids of noncon- using computer programs. Terms of irregular tiguous cartograms’ unites is an important issue cartograms often contain the name of the algo- (fig. 5, J. Olson 1976). It determines overlapping rithm used for their elaboration. A comparison of base units. of the characteristics of cartograms generated If geographical coordinates of unit centroids with different algorithms is included in appen- are maintained there is a high probability of dix 2. units’ overlapping (fig. 5A). This can be avoided if the location of central points is changed in the process of anamorphical transformation B. Noncontiguous cartograms (fig. 5B). The second class of area cartograms is noncontiguous cartogram in which spatial continuity B1. Noncontiguous diagram cartograms In noncontiguous diagram cartograms (app. 1, pt 8) each spatial unit is represented by a single geometric figure (e.g. app. 1, pt 8 – the area of hexagons is proportional to the size of population and area in individual voivodships in Poland in 1970, J. Ostrowski 1970). Presen- tations of this type are difficult to interpret, because: − they do not maintain spatial continuity, base units are separated, − base units are replaced by geometric figures, so their shape is lost, − the resulting cartogram is usually not drawn Fig. 5. Noncontiguous diagram cartogram – location into the borders of the presented area – only of units: A – overlapping, B – non-overlapping unconnected diagrams appear in the final form. (J. Olson 1976) 58 Anna Markowska

Fig. 6. Irregular cartogram (Gastner-Newman) – the distortions mainly concern hard-to-recognize countries. Number of tankers by country (www.worldmapper.org)

B2. Noncontiguous regular (mosaic) car- nomenon. Distances between units can show tograms how diversified the analyzed phenomenon is. Noncontiguous regular (mosaic) cartogram The bigger the spaces, the more diversified is a subclass of noncontiguous cartograms, in the intensity of a phenomenon. which base units are presented as geometrized units. These cartograms are built of seg- 5. Area cartograms – comparison ments which can be visible on a map (app. 1, pt 9), usually borders of individual segments may be Graphic contiguity of presentation is a basic not marked on a map. Noncontiguous regular criterion of division of area cartograms, there- cartograms are easier to interpret than non- fore the first to be compared are contiguous contiguous diagram cartograms, because they cartograms and noncontiguous equiform car- refer the shape of base units and the shape of tograms. Advantages and disadvantages of the whole area are partially preserved. contiguous cartograms and noncontiguous B. Noncontiguous equiform cartograms equiform cartograms were listed by B.D. Dent Noncontiguous equiform cartograms (app. 1, et al. (2009, tab. 1). pt 10) are cartograms in which the shape of A summary of selected area cartograms’ individual units is maintained, and only the area characteristics was prepared by S. Nusrat and is modified depending on the value of a pheno- S. Kobourov (2016, app. 2). Only automatedly menon. In order to maintain the shape of refer- generated area cartograms were considered, ence units it is impossible to maintain spatial starting with the irregular cartogram elaborated continuity on such maps. This class of carto- on the basis of the Rubber map method algo- gram is considered to be the most simple to rithm from 1973 (W. Tobler 1973). Some impor- elaborate, because the only transformations tant algorithms are missing from the discussed of the base map are those connected to the summary, e.g. Gridded Cartogram (B.D. Hen- change of the unit area and in some cases also ning 2011). involve moving the centroids of units (J. Olson The summary in appendix 2 can be useful 1976). when choosing the cartogram which we want An interesting aspect of a noncontiguous to elaborate. Thanks to that summary it can be equiform cartogram is that spaces between noticed that the characteristic of a cartogram base units are significant – they can be used which is maintained by most algorithms is con- for interpretation of the spatial location of a phe- tiguity. The authors (S. Nusrat, S. Kobourov Cartograms – classification and terminology 59

Table 1. Advantages and disadvantages of contiguous and noncontiguous area cartograms (B.D. Dent et al. 2009)

Area cartogram’s Advantages Disadvantages class

Contiguous ‒ in most cases it is possible to maintain ‒ Losing the general shape of presenta- cartogram the location and placement of spatial tion and relations between units makes units, so they can be identified with it difficult to read information on the related units on a traditional map, map (fig. 6), ‒ due to maintained spatial contiguity of ‒ sometimes very high distortion of the presentation the user does not have to shape of base units which leads to imagine connections between separa- problems with identification of individu- ted base units, as in the case of non- al units, contiguous equiform cartograms, ‒ there are still no proper GIS programs ‒ general view of the area, e.g. Poland or do elaborate all classes of contiguous the world is easier to maintain than in cartograms, so preparation of such noncontiguous equiform cartograms maps is very labor-consuming.

Noncontiguous ‒ simplicity of construction, ‒ spatial contiguity of presentation is equiform ‒ the true geographical shape of individual broken, cartogram units, ‒ separation of units makes it difficult to ‒ significance of gaps between base recreate the shape of the whole area units, which can be useful for the inter- (e.g. country or continent), pretation of a phenomenon. ‒ if some units are very small it may be necessary to move them to avoid overlapping.

2016) point out only several cases of not main- Shape preservation is important at the ele- taining the mutual location of units (Demer Car- mentary level of map reading. At the general togram or Dorling Cartogram). In the summary level of map reading topology preservation is only some classes of cartograms are marked more important. Estimation of a phenomenon as those in which values of phenomena are fully value basing on a cartogram should be done represented (Statistics – accurate). According only if the visual equalization is high. to the authors such situation is possible if geo- The mentioned characteristics were placed metry is not maintained or if the shape of base on coordinate axes, i.e. edges of the Carto- units is mostly lost. gram Cube. The ideal cartogram is located in The above discussion of the characteristic of the upper rear right corner of the cube, which area cartograms is summarized in the Cartogram means that it fully preserves the shape, loca- Cube (fig. 7, R.E. Roth et al. 2010). It is a gra- tion and placement of units and that the repre- phic method of presenting relations between sentation is one hundred percent (D.A. Keim, features affecting informational attributes of A. Herrmann 1998). Such a cartogram is im- maps (Z.F. Johnson 2008) possible to reach. From the perspective of the − topology preservation – if placement between user, shape preservation is more important at neighboring units is well maintained, the level of detailed map reading, while topology − shape preservation – to what extent the preservation is more important at the general original shapes of units are maintained (angles level of map reading. For the purpose of reading and proportions of side lengths can be analyzed phenomenon values for individual base units, here), the elaborated cartogram should have as low − visual equalization – to what extent the error level as possible – visual equalization area of a unit correctly represents the value of should be as high as possible (R.E. Roth et al. a phenomenon. 2010). These three aspects of cartograms are im- In the Cartogram Cube the classes of area portant depending on the issues which the cartograms discussed in that article (Dorling car- user reads from a map. In the process of iden- togram, regular cartogram, Gastner-Newman’s tification of cartogram’s base units their shape cartogram, noncontiguous equiform cartogram, is more important than topology preservation. noncontiguous diagram cartogram), as well as 60 Anna Markowska

Fig. 7. Cartogram Cube ‒ cube of preserving the characteristics of area cartograms classes. Contiguous cartograms are marked in blue, green – noncontiguous cartograms, violet – a traditional map, and red – perfect area cartogram. Map location in cube based on R.E. Roth et al. 2010 (R.E. Roth et al. 2010)

a example traditional map (fi g. 7) can be placed. togram within the Cartogram Cube to evaluate From fi gure 7 it can be concluded that regu- the quality of cartograms elaborated basing on lar and irregular (Gastner-Newman) cartograms the same database. are more useful than rest area cartograms’ The summary of features of various carto- classes. However, it should be noted that the grams performed by the authors mentioned placement of a cartogram in the scheme can above makes it possible to choose the most be described more as a certain space rather appropriate cartogram depending on map than a particular point. Each elaborated carto- purpose. Such elaborations are also helpful gram has different distortion, and there are no for a comparison of analyzed cartograms. indicators which would evaluate how correct a specifi c cartogram is. Such factors would be 6. Conclusions especially useful for irregular cartograms in which the shape of individual units and the The above classifi cation of cartograms is the whole area can be signifi cantly distorted. In the most complete of all presented in the literature case of highly distorted shapes of base fi elds it so far. The earlier summaries of cartograms would be better to use a diagram cartogram or excluded some cartogram classes, mainly a noncontiguous equiform cartogram. focused only on area cartograms, omitted Cartogram Cube can be useful if we make distance cartograms or anamorphical projec- different cartograms for statistical data referring tions. They can be described as incomplete to the same database and topic. Then we can typologies rather than multi-level classifi cations. set together cartograms depending on how they At the same time they did not include differen- maintain the shape, location and placement of tiation of classes of area cartograms. The termi- units and how they represent the value of a phe- nology proposed in the classifi cation, as well nomenon. It is worth considering how possible as the method of division of cartograms, are still it would be to use the space of a particular car- open to discussion and require further analysis. Cartograms – classification and terminology 61

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Appendix 1. Cartographic anamorphosics examples Cartograms – classification and terminology 63 64 Anna Markowska

Appendix 2. Automatically generated area cartograms (S. Nusrat, S. Kobourov 2016) Cartograms – classification and terminology 65