A cartogram is a form of map in which some spatially extensive variable such as population size or gross national product is substituted for land area. The geometry or space of the map is distorted (transformed) in order to convey the information of this alternate variable. The end result is a map or diagram in which the density of the variable under consideration is uniform throughout the map; variation in values are represented by variations in the size of mapped regions. Cartograms may be created use a variety of algorithms. One simple approach, due to Dorling (1995, CATMOG 59), is to associate the centroid of each mapped zone with the value of the variable to be mapped, and then to construct a circle around this point with area proportional to the magnitude of the variable. The size of the circles are typically scaled so that the largest is some fraction of a bounding rectangle encompassing all the original zones. Generally these circles will overlap, as illustrated for Zurich cantons in Figure 4‑48A. By systematically moving and rescaling the circles overlapping can be minimized whilst retaining approximate contiguity of the mapped regions (Figure 4‑48B).

Dorling (1995, p32) describes the algorithm, for which sample code is provided, as follows:

Each region ... is treated as an object in a gravity model which is repelled by other circles with which it overlaps, but is attracted to circles which were neighboring regions on the original map. Forces akin to forces of gravity are calculated, including velocity and acceleration, and the whole process is acted out as if it were occurring in treacle, to avoid any circles moving too quickly (as can occur when small objects orbit too close to large objects in the original gravity models).

This example illustrates a procedure that seeks to represent the variable ‘population’ by the areas of the circles, whilst retaining the approximate relative positions and contiguity of the communes whilst avoiding overlap. It results in a discontinuous map with no attempt to retain any of the original region shapes. This form of cartogram is supported in several GIS and mapping packages, including MAPresso (Java-based), MapViewer and GeoDa.

If the objective is to create a cartogram that retains full contiguity, then the original area-based map needs to be stretched, or ‘rubber-sheeted’, in some way. Many algorithms have been proposed for such cartograms, commencing with Tobler’s work on map transformations, Tobler (1963). More recently continuous transformation cartograms have been implemented using the rubber-sheeting algorithms of Dougenik et al. (1985) and the diffusion-based algorithm of Gastner and Newman (2004) − see Figure 4‑49B (Zurich cantons), Figure 4‑50 and Figure 4‑51B (which uses the North Carolina dataset) discussed in several sections of this Guide.

It is important to note that within-zone variation is not represented in this technique, hence representation of highly concentrated datasets (such as population) will provide a very particular view of the dataset. If zoned or gridded data is available within zones it may be useful to consider using such information when generating catrograms.

Figure 4‑48 Cartogram creation using basic Dorling algorithm

A. Zurich Canton, Switzerland. 171 communes, population mapped as proportional circles

B. Cartogram creation using Dorling algorithm

Created using MAPresso (Adrian Herzog)

 

The Gastner-Newman technique commences with the source map with its varying ‘population’ density and conceptually allows the individual members of this population to diffuse from higher density zones to lower density zones until such time as density is uniform. Zone boundaries are progressively moved during the diffusion process such that the net flow through them is zero, until they reach a stable position. It is important to note that here, as with many software implementations of geospatial algorithms, the final cartograms may differ depending upon the software package used to generate them and the parameters used or selected. For example, Dorling cartograms are produced very rapidly by GeoDa, with no option for parameter selection, and with some residual overlapping of the circular regions (Figure 4‑51C). If the MapViewer software is chosen with the same dataset the results are markedly different for almost all parameter selections (Figure 4‑51D). Of course, as another alternative, the shape of regions could be retained and their size altered to reflect the mapped variable if they are sufficiently well separated (i.e. a form of ‘explosion’ or multipolygon representation of the original mapped regions – see Figure 4‑51E).

A final cartogram procedure that warrants attention involves identifying a zoning of the original source data in which the population is approximately equal (e.g. census areas, electoral districts). These zones, which will generally be of varying sizes, are then replaced with equal-sized hexagons and re-arranged into a lattice that approximates the source map. This procedure provides a contiguous mapping of the source data, retaining some of the overall form of the conventional map, whilst eliminating most of the variations due to underlying population levels. The Atlas of mortality in Britain by Shaw et al. (2008) uses this approach for all of its maps (see for example, Figure 4‑52) – these show the cause of death for roughly 15 million cases over a 24 year period.

Figure 4‑51 Cartograms of births data, 1974

A. Source data

B. Glastner-Newman diffusion algorithm (ArcGIS 9 Cartogram Geoprocessing tool)

C. GeoDa’s Dorling cartogram

D. MapViewer’s Dorling cartogram

E. MapViewer’s non-contiguous ‘explosion’ cartogram

Figure 4‑52 Hexagonal cartogram showing UK mortality data, age group 20-24

Note: in this diagram each hexagon is divided into two halves which may have different principal causes of death within a given district, for this age group