[[File:Global population cartogram.png|thumb|upright=1.5|Mosaic cartogram showing the distribution of the global population as of 2018.
Each of the 15,266 pixels represents the home country of 500,000 people – cartogram by Max Roser for Our World in Data]]
A cartogram (also called a value-area map or an anamorphic map, the latter common among German speakers) is a thematic map of a set of features (countries, provinces, etc.), in which their geographic size is altered to be directly proportional to a selected variable, such as travel time, population, or gross national income. Geographic space itself is thus warped, sometimes extremely, in order to visualize the distribution of the variable. It is one of the most abstract types of map; in fact, som be called diagrams. They are primarily used to display emphasis and for analysis as nomographs.
Cartograms leverage the fact that size is the most intuitive visual variable for representing a total amount. In this, it is a strategy that is similar to proportional symbol maps, which scale point features, and many flow maps, which scale the weight of linear features. However, these two techniques only scale the map symbol, not space itself; a map that stretches the length of linear features is considered a linear cartogram (although additional flow map techniques may be added). Once constructed, cartograms are often used as a base for other thematic mapping techniques to visualize additional variables, such as choropleth mapping.
History
thumb|upright=1.15|One of Levasseur's 1876 cartograms of Europe, the earliest known published example of this technique.
The cartogram was developed later than other types of thematic maps, but followed the same tradition of innovation in France. The earliest known cartogram was published in 1876 by French statistician and geographer Pierre Émile Levasseur, who created a series of maps that represented the countries of Europe as squares, sized according to a variable and arranged in their general geographical position (with separate maps scaled by area, population, religious adherents, and national budget). Later reviewers have called his figures a statistical diagram rather than a map, but Levasseur referred to it as a carte figurative, the common term then in use for any thematic map. He produced them as teaching aids, immediately recognizing the intuitive power of size as a visual variable: "It is impossible that the child is not struck by the importance of the trade of Western Europe in relation to that of Eastern Europe, that he does not notice how much England, which has a small territory but outweighs other nations by its wealth and especially by its navy, how much on the contrary Russia which, by its area and its population occupies the first rank, is still left behind by other nations in the commerce and navigation."
Levasseur's technique does not appear to have been adopted by others, and relatively few similar maps appear for many years. The next notable development was a pair of maps by Hermann Haack and Hugo Weichel of the 1898 election results for the German Reichstag in preparation for the 1903 election, the earliest known contiguous cartogram. Both maps showed a similar outline of the German Empire, with one subdivided into constituencies to scale, and the other distorting the constituencies by area. The subsequent expansion of densely populated areas around Berlin, Hamburg, and Saxony was intended to visualize the controversial tendency of the mainly urban Social Democrats to win the popular vote, while the mainly rural Zentrum won more seats (thus presaging the modern popularity of cartograms for showing the same tendencies in recent elections in the United States).
The continuous cartogram emerged soon after in the United States, where a variety appeared in the popular media after 1911. Most were rather crudely drawn compared to Haack and Weichel, with the exception of the "rectangular statistical cartograms" by the American master cartographer Erwin Raisz, who claimed to have invented the technique.
When Haack and Weichel referred to their map as a kartogramm, this term was commonly being used to refer to all thematic maps, especially in Europe. It was not until Raisz and other academic cartographers stated their preference for a restricted use of the term in their textbooks (Raisz initially espousing value-area cartogram) that the current meaning was gradually adopted.
The primary challenge of cartograms has always been the drafting of the distorted shapes, making them a prime target for computer automation. Waldo R. Tobler developed one of the first algorithms in 1963, based on a strategy of warping space itself rather than the distinct districts. Since then, a wide variety of algorithms have been developed (see below), although it is still common to craft cartograms manually. It is likely impossible to preserve both of these, so some cartogram methods attempt to preserve one at the expense of the other, some attempt a compromise solution of balancing the distortion of both, and other methods do not attempt to preserve either one, sacrificing all recognizability to achieve another goal.
Area cartograms
thumb|Cartogram of [[Germany, with the states and districts resized according to population]]
The area cartogram is by far the most common form; it scales a set of region features, usually administrative districts such as counties or countries, such that the area of each district is directly proportional to a given variable. Usually this variable represents the total count or amount of something, such as total population, gross domestic product, or the number of retail outlets of a given brand or type. Other strictly positive ratio variables can also be used, such as GDP per capita or birth rate, but these can sometimes produce misleading results because of the natural tendency to interpret size as total amount. The various taxonomies tend to agree on the following general types of area cartograms.
Anamorphic Projection
This is a type of contiguous cartogram that uses a single parametric mathematical formula (such as a polynomial curved surface) to distort space itself to equalize the spatial distribution of the chosen variable, rather than distorting the individual features. Because of this distinction, some have preferred to call the result a pseudo-cartogram. Tobler's first computer cartogram algorithm was based on this strategy, for which he developed the general mathematical construct on which his and subsequent algorithms are based. Because they do not directly scale the districts, there is no guarantee that the area of each district is exactly equal to its value.
Shape-warping contiguous cartograms
thumb|upright=1.35|Contiguous cartogram (Gastner-Newman) of the world with each country rescaled in proportion to the hectares of certified [[organic farming]]
Also called irregular cartograms or deformation cartograms, but these are often more complex and slower algorithms than those that severely distort shape.
Non-contiguous isomorphic cartograms
thumb|upright=1.15|left|Non-contiguous isomorphic cartogram of the [[Czech Republic, in which the size of each district is proportional to the Catholic percentage and the color (choropleth) representing the proportion voting for the KDU-CSL party in 2010, showing a strong correlation.]]
This is perhaps the simplest method for constructing a cartogram, in which each district is simply reduced or enlarged in size according to the variable without altering its shape at all. these are actually the original form of cartogram, dating back to Levasseur (1876) Several examples of block cartograms were published during the 2016 U.S. presidential election season by The Washington Post, the FiveThirtyEight blog, and the Wall Street Journal, among others. This is a cartogram for the 2024 and 2028 elections, based on the 2020 Census apportionment:
[[File:Cartogram 2008 red blue.png|thumb|upright=1.6|Mosaic cartogram of United States Electoral College results (scaled by 2008 electors) of four past Presidential elections (1996, 2000, 2004, 2008)
|center]]
The major disadvantage of this type of cartogram has traditionally been that they had to be constructed manually, but recently algorithms have been developed to automatically generate both square and hexagonal mosaic cartograms.
Linear cartograms
thumb|A linear cartogram of the London Underground, with distance distorted to represent travel time from High Barnet station|right
While an area cartogram manipulates the area of a polygon feature, a linear cartogram manipulates linear distance on a line feature. The spatial distortion allows the map reader to easily visualize intangible concepts such as travel time and connectivity on a network. Distance cartograms are also useful for comparing such concepts among different geographic features. A distance cartogram may also be called a central-point cartogram.
A common use of distance cartograms is to show the relative travel times and directions from vertices in a network. For example, on a distance cartogram showing travel time between cities, the less time required to get from one city to another, the shorter the distance on the cartogram will be. When it takes a longer time to travel between two cities, they will be shown as further apart in the cartogram, even if they are physically close together.
Distance cartograms are also used to show connectivity. This is common on subway and metro maps, where stations and stops are shown as being the same distance apart on the map even though the true distance varies. Though the exact time and distance from one location to another is distorted, these cartograms are still useful for travel and analysis.
Multivariate cartograms
thumb|left|upright=1.15|Hexagonal mosaic cartogram of the results of the 2019 Canadian parliamentary election, colored with the party of each winner using a nominal choropleth technique.
Both area and linear cartograms adjust the base geometry of the map, but neither has any requirements for how each feature is symbolized. This means that symbology can be used to represent a second variable using a different type of thematic mapping technique. Cart, and the Cartogram Processing Tool (an ArcScript for ESRI's ArcGIS), which all use the Gastner-Newman algorithm. An alternative algorithm, Carto3F, is also implemented as an independent program for non-commercial use on Windows platforms. This program also provides an optimization to the original Dougenik rubber-sheet algorithm.
The CRAN package recmap provides an implementation of a rectangular cartogram algorithm.||area contiguous||with distortion|| Yes, algorithmically guaranteed
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| 2004||Gastner, Newman||Diffusion-based method||area contiguous||with distortion || Yes, algorithmically guaranteed
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| 2004||Sluga||Lastna tehnika za izdelavo anamorfoz||area contiguous||with distortion|| Unknown
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| 2004||van Kreveld, Speckmann||Rectangular Cartogram||area contiguous||no (rectangles)|| No
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| 2004||Heilmann, Keim et al.||RecMap||area noncontiguous||no (rectangles)|| No
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| 2005||Keim, North, Panse||Medial-axis-based cartograms||area contiguous||with distortion|| Yes, algorithmically guaranteed
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| 2009||Heriques, Bação, Lobo||Carto-SOM||area contiguous||with distortion|| Yes
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| 2013||Shipeng Sun ||Opti-DCN