Editing Cartogram (section)

== Area cartograms ==
[[File:Germany-population-cartogram.png|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 [[Proportionality (mathematics)|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 [[Level of measurement|ratio]] variables can also be used, such as [[List of countries by GDP (nominal) per capita|GDP per capita]] or [[Birth rate]], but these can sometimes produce misleading results because of the natural tendency to interpret size as total amount.<ref name="bertin" />  Of these, total population is probably the most common variable, sometimes called an ''isodemographic map''.

The various strategies and algorithms have been classified a number of ways, generally according to their strategies with respect to preserving shape and topology. Those that preserve shape are sometimes called ''equiform'', although ''isomorphic'' (same-shape) or ''homomorphic'' (similar-shape) may be better terms. Three broad categories are widely accepted: contiguous (preserve topology, distort shape), non-contiguous (preserve shape, distort topology), and diagrammatic (distort both). Recently, more thorough taxonomies by Nusrat and Kobourov, Markowska, and others have built on this basic framework in an attempt to capture the variety in approaches that have been proposed and in the appearances of the results.<ref name="nusrat2016">{{cite journal |last1=Nusrat |first1=Sabrina |last2=Kobourov |first2=Stephen |title=The State of the Art in Cartograms |journal=Computer Graphics Forum |date=2016 |volume=35 |issue=3 |pages=619–642 |doi=10.1111/cgf.12932|arxiv=1605.08485 |hdl=10150/621282 |s2cid=12180113 |hdl-access=free }} Special issue: 18th Eurographics Conference on Visualization (EuroVis), State of the Art Report</ref><ref name="markowska2019">{{cite journal |last1=Markowska |first1=Anna |title=Cartograms - classification and terminology |journal=Polish Cartographical Review |date=2019 |volume=51 |issue=2 |pages=51–65 |doi=10.2478/pcr-2019-0005|bibcode=2019PCRv...51...51M |doi-access=free }}</ref> The various taxonomies tend to agree on the following general types of area cartograms.

===Anamorphic Projection===
{{see also | Anamorphosis}}
This is a type of contiguous cartogram that uses a single parametric mathematical formula (such as a [[Polynomial | 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''.<ref name="bortins-demers">{{cite web |last1=Bortins |first1=Ian |last2=Demers |first2=Steve |title=Cartogram Types |url=http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html |website=Cartogram Central |publisher=National Center for Geographic Information Analysis, UC Santa Barbara |access-date=15 November 2020 |archive-date=29 January 2021 |archive-url=https://web.archive.org/web/20210129215218/http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html |url-status=dead }}</ref> [[Waldo R. Tobler|Tobler's]] first computer cartogram algorithm was based on this strategy,<ref name="tobler1963" /><ref name="tobler1973">{{cite journal |last1=Tobler |first1=Waldo R. |title=A Continuous Transformation Useful for Districting |journal=Annals of the New York Academy of Sciences |date=1973 |volume=219 |issue=1 |pages=215–220 |doi=10.1111/j.1749-6632.1973.tb41401.x|pmid=4518429 |bibcode=1973NYASA.219..215T |hdl=2027.42/71945 |s2cid=35585206 |hdl-access=free }}</ref> for which he developed the general mathematical construct on which his and subsequent algorithms are based.<ref name="tobler1963" /> This approach first models the distribution of the chosen variable as a continuous density function (usually using a [[Curve fitting|least squares fitting]]), then uses the inverse of that function to adjust the space such that the density is equalized. The Gastner-Newman algorithm, one of the most popular tools used today, is a more advanced version of this approach.<ref name="GSM-Fast-Flow-Based">{{cite journal |author = Michael T. Gastner |author2=Vivien Seguy |author3=Pratyush More |year = 2018 |title = Fast flow-based algorithm for creating density-equalizing map projections |journal = Proceedings of the National Academy of Sciences |volume = 115 |pages = E2156–E2164 |doi = 10.1073/pnas.1712674115 |pmid=29463721 |pmc=5877977 |issue = 10 |arxiv = 1802.07625  |bibcode=2018PNAS..115E2156G |doi-access=free }}</ref><ref name="gastner-newman">{{cite journal |last1=Gastner |first1=Michael T. |last2=Newman |first2=M.E.J. |title=Diffusion-based Method for Producing Density-Equalizing Maps |journal=Proceedings of the National Academy of Sciences of the United States of America |date=May 18, 2004 |volume=101 |issue=20 |pages=7499–7504 |doi=10.1073/pnas.0400280101 |jstor=3372222 |pmid=15136719 |pmc=419634 |url=|arxiv=physics/0401102 |s2cid=2487634 |doi-access=free }}</ref> 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===
[[File:PaullHennig2016WorldMap.OAha.CC-BY-4.0.jpg|thumb|upright=1.35|Contiguous cartogram (Gastner-Newman) of the world with each country rescaled in proportion to the hectares of certified [[organic farming]]<ref name=Atlas>Paull, John & Hennig, Benjamin (2016) [https://www.academia.edu/25648267/Atlas_of_Organics_Four_maps_of_the_world_of_organic_agriculture  Atlas of Organics: Four Maps of the World of Organic Agriculture]  Journal of Organics. 3(1): 25–32.</ref>]]
Also called ''irregular cartograms'' or ''deformation cartograms'',<ref name="markowska2019" /> This is a family of very different algorithms that scale and deform the shape of each district while maintaining adjacent edges. This approach has its roots in the early 20th Century cartograms of Haack and Weichel and others, although these were rarely as mathematically precise as current computerized versions. The variety of approaches that have been proposed include [[Cellular automaton|cellular automata]], [[Quadtree | quadtree partitions]], [[cartographic generalization]], [[Medial axis|medial axes]], spring-like forces, and simulations of inflation and deflation.<ref name="nusrat2016" /> Some attempt to preserve some semblance of the original shape (and may thus be termed ''homomorphic''),<ref name="kocmoud1998">{{cite book |last1=House |first1=Donald H. |last2=Kocmoud |first2=Christopher J. |title=Proceedings Visualization '98 (Cat. No.98CB36276) |chapter=Continuous cartogram construction |date=October 1998 |pages=197–204 |doi=10.1109/VISUAL.1998.745303 |isbn=0-8186-9176-X |s2cid=14023382 |chapter-url=https://www.researchgate.net/publication/3788051}}</ref> but these are often more complex and slower algorithms than those that severely distort shape.

===Non-contiguous isomorphic cartograms===
[[File:Cartogram projector cz.png|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.<ref name="torguson2009" /> In most cases, a second step adjusts the location of each shape to reduce gaps and overlaps between the shapes, but their boundaries are not actually adjacent. While the preservation of shape is a prime advantage of this approach, the results often have a haphazard appearance because the individual districts do not fit together well.

===Diagrammatic (Dorling) cartograms===
[[File:Wikipédia-Pays par liens.2011.07.svg|thumb|right|upright=1.35|Diagrammatic (Dorling) cartogram of the number of times each country is linked in the French-language Wikipedia.]]
In this approach, each district is replaced with a simple geometric shape of proportional size. Thus, the original shape is completely eliminated, and contiguity may be retained in a limited form or not at all. Although they are usually referred to as ''Dorling cartograms'' after Daniel Dorling's 1996 algorithm first facilitated their construction,<ref name="dorling1996">{{cite book |last1=Dorling |first1=Daniel |title=Area Cartograms: Their Use and Creation |date=1996 |volume=59|series=Concepts and Techniques in Modern Geography (CATMOG) |publisher=University of East Anglia}}</ref> these are actually the original form of cartogram, dating back to Levasseur (1876)<ref name="levasseur1876" /> and Raisz (1934).<ref name="raisz1934" /> Several options are available for the geometric shapes:
* '''Circles''' (Dorling), typically brought together to be touching and arranged to retain some semblance of the overall shape of the original space.<ref name="dorling1996"/> These often look like [[proportional symbol map]]s, and some consider them to be a hybrid between the two types of thematic map.
* '''Squares''' (Levasseur/Demers), treated in much the same way as the circles, although they do not generally fit together as simply.
* '''Rectangles''' (Raisz), in which the height and width of each rectangular district is adjusted to fit within an overall shape. The result looks much like a [[Treemapping|treemap diagram]], although the latter is generally sorted by size rather than geography. These are often contiguous, although the contiguity may be illusory because many of the districts that are adjacent in the map may not be the same as those that are adjacent in reality.

Because the districts are not at all recognizable, this approach is most useful and popular for situations in which the shapes would not be familiar to map readers anyway (e.g., [[United Kingdom Parliament constituencies|U.K. parliamentary constituencies]]) or where the districts are so familiar to map readers that their general distribution is sufficient information to recognize them (e.g., countries of the world). Typically, this method is used when it is more important for readers to ascertain the overall geographic pattern than to identify particular districts; if identification is needed, the individual geometric shapes are often labeled.

===Mosaic cartograms===
[[File:Germany_population_states_hexagonal.svg|thumb|Population cartogram of the states of Germany. Each hexagon represents 250&thinsp;000 people. The cartogram is topologically correct in that any states that touch on the cartogram also touch in reality.]]
In this approach (also called ''block'' or ''regular cartograms''), each shape is not just scaled or warped, but is reconstructed from a discrete [[tessellation]] of space, usually into squares or hexagons. Each cell of the tessellation represents a constant value of the variable (e.g., 5000 residents), so the number of whole cells to be occupied can be calculated (although rounding error often means that the final area is not exactly proportional to the variable). Then a shape is assembled from those cells, usually with some attempt to retain the original shape, including salient features such as panhandles that aid recognition (for example, [[Long Island]] and [[Cape Cod]] are often exaggerated). Thus, these cartograms are usually homomorphic and at least partially contiguous.

This method works best with variables that are already measured as a relatively low-valued integer, enabling a one-to-one match with the cells. This has made them very popular for visualizing the [[United States Electoral College]] that determines the election of the [[President of the United States|president]], appearing on television coverage and numerous vote-tracking websites.<ref>{{cite news |last1=Bliss |first1=Laura |last2=Patino |first2=Marie |title=How to Spot Misleading Election Maps |url=https://www.bloomberg.com/news/articles/2020-11-03/a-complete-guide-to-misleading-election-maps |newspaper=Bloomberg |date=3 November 2020 |access-date=15 November 2020}}</ref> Several examples of block cartograms were published during the 2016 U.S. presidential election season by ''The Washington Post'',<ref>{{cite news|title=Poll: Redrawing the Electoral Map|url=https://projects.fivethirtyeight.com/2016-election-forecast/?ex_cid=rrpromo#plus&electoral-map|archive-url=https://web.archive.org/web/20160710121436/http://projects.fivethirtyeight.com/2016-election-forecast/?ex_cid=rrpromo#plus&electoral-map|url-status=dead|archive-date=July 10, 2016|newspaper=Washington Post|access-date=4 February 2018}}</ref> the ''FiveThirtyEight'' blog,<ref>{{cite web|title=2016 Election Forecast|url=https://projects.fivethirtyeight.com/2016-election-forecast/?ex_cid=rrpromo#plus&electoral-map|archive-url=https://web.archive.org/web/20160710121436/http://projects.fivethirtyeight.com/2016-election-forecast/?ex_cid=rrpromo#plus&electoral-map|url-status=dead|archive-date=July 10, 2016|website=FiveThirtyEight blog| date=29 June 2016 |access-date=4 February 2018}}</ref> and the ''Wall Street Journal'',<ref>{{cite web|title=Draw the 2016 Electoral College Map|url=http://graphics.wsj.com/elections/2016/2016-electoral-college-map-predictions/|website=Wall Street Journal|access-date=4 February 2018}}</ref> 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) {{legend|#ff0001|States carried by the Republican in all four elections}} {{legend|#FF5E5E|States carried by the Republican in three of the four elections}} {{legend|#cccccc|States carried by each party twice in the four elections}}
{{legend|#6D5EFF|States carried by the Democrat in three of the four elections}}
{{legend|#2300ff|States carried by the Democrat in all four elections}}|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.<ref name="cano2015">{{cite journal |last1=Cano |first1=R.G. |last2=Buchin |first2=K. |last3=Castermans |first3=T. |last4=Pieterse |first4=A. |last5=Sonke |first5=W. |last6=Speckman |first6=B. |title=Mosaic Drawings and Cartograms |journal=Computer Graphics Forum |date=2015 |volume=34 |issue=3 |pages=361–370 |doi=10.1111/cgf.12648|s2cid=41253089 |url=https://pure.tue.nl/ws/files/90083986/MosaicMapsRevised.pdf }} Proceedings of 2015 Eurographics Conference on Visualization (EuroVis)</ref><ref name="tilegrams">{{cite web |last1=Florin |first1=Adam |last2=Hamel |first2=Jessica |title=Tilegrams |url=https://pitchinteractiveinc.github.io/tilegrams/ |publisher=Pitch Interactive |access-date=15 November 2020}}</ref>