The results of the popular vote in the 1996 U.S. presidential race are visualized above using traditional thematic mapping. Each state is colored either a shade of red or a shade of blue, denoting the majority winner of each state as Clinton or Dole, respectively, with color saturation indicating the magnitude of the winning percentage. There is a significant problem with this visualization. Without prior knowledge of population density across the country, the viewer has no clear indicator as to who actually won the election. While this map provides a medium of familiarity, it produces an intrinsic distortion of the very data we are trying to analyze. Since elections are not won on square miles, the results would be better visualized on a map more representative of population. These same election results are shown below on a 1996 equal population cartogram generated using the Constraint-Based Method.

802.11b Wireless Networks

One of the most interesting area of wireless network provision in the last couple years has been the emergence of community LANs based on sharing network access using the the 802.11b standard (commonly known as Wi-Fi or WLAN). Many local groups are forming in cities. This is largely an unplanned, activist lead movement, working to share access for free.

Here are example maps produced by different groups in London, New York City, and Seattle. These maps show the location of WLAN base station nodes that provide wireless access.

People are searching for active 802.11b nodes by so called 'war driving' - i.e. driving around sniffing for unsecured wireless networks. (This is named after the 80s idea of war dialing.)

This example map by Frank Keeney shows the results of his war driving in the Los Angeles area.

A useful review article, "802.11b Access Point Mapping", by Simon Byers and Dave Kormann, Communications of the ACM, May 2003.

Many other WLAN groups are organising in cities and towns in many countries. A comprehensive list is provided by the Personal Telco Project. Also worth checking out is Freenetworks.org.

Background information on the 802.11b free network movement:

"A LAN line", The Economist,11th January 2001.

"Motley Crew Beams No-Cost Broadband to New York High Speed Freed", Village Voice, by Peter Meyers, 15th August 2001.

Animation of the 1996 U.S. Population Cartogram construction process

Demonstration of "alternating relaxation" in the the Constraint-Based Method where the focus alternates between achieving correct region areas and restoring region shapes (Kocmoud and House, 1997).

Currently, there are no commercially available software packages to create a cartogram. The best alternative thus far is an ArcView avenue script, but it is not widely used. Each iteration of the script in this example took over 45 minutes. With five, ten or fifteen iterations, it could be quicker and much more aesthetically pleasing to produce a cartogram by hand.

Quoting Steve Demers:

" I definitely prefer for aesthetic purposes, [making a cartogram] by hand. The time you save automating, you end up using anyway fixing everything that the computer screwed up. You can see in several places in the final cartogram where the congressional districts lose topology. ...there is actually a hole in the cartogram. There are just too many things that can go wrong in automation. Making a cartogram takes a lot of cartographic licence, style and creativity- things a computer just cant do."

click for entire photoDisplayed above are the results of running the "contiguous cartogram" Arcscript by Andy Agena, from ESRI.com after 5, 10 and 15 iterations. For an animation, click the image to the right:

Compared to the hand-made image, the computer generated cartogram is not as easily readable.

Cartogram, map with shadings from black to white (distribution and intensity of illiteracy in France), the first (unclassed) choropleth map, and perhaps the first modern statistical map. (This cartogram dates from 1826 (Dupin 1827, Plate 1, vol.2) according to Robinson (p.232), rather than 1819 according to Funkhouser).

Dupin, C. (1826). Carte figurative de l'instruction populaire de la France . Jobard. BNF: Ge C 6588 (Funkhouser (1937, p.300) incorrectly dates this as 1819).

Dupin, C. (1827). Forces productives et commerciales de la France. Bachelier.

Robinson, A. H. (1982). Early Thematic Mapping in the History of Cartography . Chicago: University of Chicago Press. ISBN 0-226-72285-6.

Funkhouser, H. G. (Nov. 1937). Historical development of the graphical representation of statistical data. Osiris, 3(1):269-405. Reprinted Brugge, Belgium: St. Catherine Press, 1937.

There are many difficulties in showing rates of incidence or proportions in maps, when both the areas of geographic regions, and the populations in those regions vary, often inversely. In spatial epidemiology, for example, Standardized Mortality Ratios are often used, expressing the ratios of the number of deaths in each area to those expected on the basis of some externally specified (typically national) age-sex specific rates.

This figure uses a Chi-square metric to depict the distribution of number of cars, O, in each ward in Leicestershire, UK, expressed as a signed chi-square contribution, (Oi - Ei)/ Ö Ei, relative to the expected number, E, per capita. A diverging colour scheme applies hues of red and blue to those areas with higher and lower than expected values with colour saturation showing the magnitude of the variation. Thus whiter zones are close to the expected value and deeper blues and fuller reds show the extremes. This map still confounds area and population with visual impact, which the use of a cartogram base, with circle areas proportional to the population, helps avoid.

Figures from Maps of the Census: A Rough Guide, by Jason Dykes and David Unwin (http://www.agocg.ac.uk/reports/visual/casestud/dykes/abstra_1.htm).

Abstract:

This Case Study describes the considerations that are needed to produce maps of data from the Census of Population. The `area value' or choropleth map is the standard means of displaying such information on paper. It is a very imperfect visualisation device. First, it is necessary to be careful about the numbers that are mapped and, in particular, never to map absolute numbers. Second, choropleth maps are very sensitive to the mapping zones being used. To produce maps that do not distort the underlying distributions it is necessary to understand how the zones were defined and the effects of their varying sizes on the mapped pattern. Third, there are a series of strictly cartographic considerations related to how these maps are classed and the symbolism used. All of these issues are illustrated using data from the 1991 Population Census for Leicestershire, UK.

These problems lead to a consideration of the need to develop new mapping tools. Dynamic maps can take advantage of an interactive software environment to overcome some of the limitations of the static map. The possibilities which they provide for interactive engagement with data make them appropriate tools for exploratory analysis, or visual thinking. A mapping tool is introduced, which exemplifies this form of map use and examples of the techniques that might be used to visualize the UK Census of Population are provided.

Cartogram of child mortality.

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.

In the previous section we referred to the connectivity between objects, or topology. In a non-contiguous cartogram topology was sacrificed in order to preserve shape. In a contiguous cartogram, the reverse is true- topology is maintained (the objects remain connected with each other) but this causes great distortion in shape.

This leads to the single most difficult, but intriguing problem in creating cartograms. The cartographer must make the objects the appropriate size to represent the attribute value, but he or she must also maintain the shape of objects as best as possible, so that the cartogram can be easily interpreted. Here is an example of a contiguous cartogram of population in California's counties. Compare this to the previous non-contiguous cartogram.

This type of cartogram was named after its inventor, Danny Dorling of the University of Leeds. A Dorling cartogram maintains neither shape, topology nor object centroids, though it has proven to be a very effective cartogram method. To create a Dorling cartogram, instead of enlarging or shrinking the objects themselves, the cartographer will replace the objects with a uniform shape, usually a circle, of the appropriate size. Professor Dorling, for the reason described above in the non-contiguous cartogram section, suggests that the shapes not overlap but rather be moved so that the full area of each shape can be seen. Below is an example of a Dorling cartogram, using the same population of California counties example.

Another Dorling-like cartogram is the Demers Cartogram, which is different in two ways. It uses squares rather than circles; this leaves fewer gaps between the shapes. Secondly, the Dorling Cartogram attempts to move the figures the shortest distance away from their true locations; the Demers cartogram often sacrifices distance to maintain contiguity between figures, and it will also sacrifice distance to maintain certain visual cues (The gap between figures used to represent San Francisco Bay in the Demers Cartogram below is a good example of a visual cue.) The 25 Most Populated Counties in California are labeled in each of the two cartograms below for reference.

The circles are proportional to population size and sorted to 10 quantiles.

This a cartogram for the county-level election returns.

The blue areas are much magnified, and areas of blue and red are now nearly equal. However, there is in fact still more red than blue on this map, even after allowing for population sizes. Of course, we know that nationwide the percentages of voters voting for either candidate were almost identical, so what is going on here? The answer seems to be that the amount of red on the map is skewed because there are a lot of counties in which only a slim majority voted Republican.

This is a cartogram for the county-level election returns using a nonlinear color scale.

The (contiguous 48) states of the country are colored red or blue to indicate whether a majority of their voters voted for the Republican candidate (George W. Bush) or the Democratic candidate (John F. Kerry) respectively. The map gives the superficial impression that the "red states" dominate the country, since they cover far more area than the blue ones. However, as pointed out by many others, this is misleading because it fails to take into account the fact that most of the red states have small populations, whereas most of the blue states have large ones. The blue may be small in area, but they are large in terms of numbers of people, which is what matters in an election.

We can correct for this by making use of a cartogram, a map in which the sizes of states have been rescaled according to their population. That is, states are drawn with a size proportional not to their sheer topographic acreage -- which has little to do with politics -- but to the number of their inhabitants, states with more people appearing larger than states with fewer, regardless of their actual area on the ground. Thus, on such a map, the state of Rhode Island, with its 1.1 million inhabitants, would appear about twice the size of Wyoming, which has half a million, even though Wyoming has 60 times the acreage of Rhode Island.

Here are the 2004 presidential election results on a population cartogram. The cartogram was made using the diffusion method of Gastner and Newman. Population data were taken from the 2000 US Census.

The cartogram reveals what we know already from the news: that the country was actually very evenly divided by the vote, rather than being dominated by one side or the other.

The presidential election is not decided on the basis of the number of people who vote each way, however, but on the basis of the electoral college. Each state contributes a certain number of electors to the electoral college, who vote according to the majority in their state. The candidate receiving a majority of the votes in the electoral college wins the election. The electoral votes are apportioned roughly according to states' populations, as measured by the census, but with a small but deliberate bias in favor of smaller states.

To represent the effects of the electoral college, the sizes of states have been scaled to be proportional to their number of electoral votes.

This cartogram looks very similar to a popular vote cartogram, but it is not identical. Wyoming, for instance, has approximately doubled in size, precisely because of the bias in favor of small states.

The areas of red and blue on the cartogram are now proportional to the actual numbers of electoral votes won by each candidate. Thus this map shows at a glance both which states went to which candidate and which candidate won more votes -- something that you cannot tell easily from the normal election-night red and blue map.

Cartogram of energy consumption (including oil).

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.

Panregions: major regional world cities

This cartogram places cities in their approximate relative geographical positions.

There may or may not be hierarchical patterns within the spatial organisation of individual firms at the global scale (it depends on their particular strategies), but when aggregated the result is a world city network. This network is illustrated as a pattern of nodes in this figure. The cartogram includes all cities that have at least one fifth of the highest city connectivity (i.e London's) which creates a roster of 123 'world cities'.

Cartogram of greenhouse gas emissions.

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.

Cartograms are most often used to show population data, but there is no reason why they need be limited to population. They can in principle be used to show almost any quantity. Here is a cartogram of the world in which the sizes of countries are proportional to Gross Domestic Product, which is a measure of how much wealth a country's economy generates, and hence, to an extent, of the wealth of the country's inhabitants.

Notice how America and Europe dominate this map, along with Japan (yes – that huge dark-green island on the right really is Japan), while Africa dwindles almost to invisiblity.

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.

The map of Congressional Districts was edited beginning with Adobe Illustrator. To make sure each area is the right size, a standard sized reference box was created and moved over each area. Each district was compared to the box by eye, as they were drawn. The illustrator file was exported as a CAD file, then using ArcToolbox, the CAD file was converted to a GeoDatabase and then to a shapefile so it could be opened in ArcMap. ArcMap was then used to calculate the areas, and choropleth the districts by area. (districts that were too small were colored in a blue monochromatic scale, and the districts that were too large in a red monochromatic scale.)

The edited map was printed and the process started over again with illustrator, moving vertices around some more. (Anytime there was a blue district next to a red district one of the common-vertices could simply be moved into the blue district.) The process was performed three times for good measure.

Antonio Scarponi created an animated map of the world showing the growth of Internet users from 1993 and predicted to 2015. The three images above are single frames from the animation showing the state of the Internet world in 1996, 2001 and then projected for 2007.

The map uses a continuous cartogram representation where the size of the country is based on the number of Internet users rather than geographical area. Cartograms can be a very effective means of visualizing demographic data as they highlight areas based on where most people rather than simply territorial area.

A non-contiguous cartogram is the simplest and easiest type of cartogram to make. In a non-contiguous cartogram, the geographic objects do not have to maintain connectivity with their adjacent objects. This connectivity is called topology. By freeing the objects from their adjacent objects, they can grow or shrink in size and still maintain their shape. Here is an example of two non-contiguous cartograms of population in California's counties.

The difference between these two types of non-contiguous cartograms is a significant one. The cartogram on the left has maintained the object's centroid (a centroid is the weighted center point of an area object.) Because the object's center is staying in the same place, some of the objects will begin to overlap when the objects grow or shrink depending on the attribute (in this case population.) In the cartogram on the right, the objects not only shrink or grow, but they also will move one way or another to avoid overlapping with another object. Although this does cause some distortion in distance, most prefer this type of non-contiguous cartogram. By not allowing objects to overlap, the depicted sizes of the objects are better seen, and can more easily be interpreted as some attribute value.

Cartogram of people living with HIV/AIDS.

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.

In this map the sizes of countries are proportional not to their actual landmass but instead to the number of people living there; a country with 20 million people, for instance, appears twice as large as a country with 10 million.

Although the figures for populations of countries are well established and familiar to many, the cartogram provides a new way of looking at them and in particular makes clear the enormous disparity in the population of different regions. Note how large India and China have become: between them these two countries account for more than a third of the population of the world. On the other hand, notice the near-disappearance of Canada and Russia, the world's two largest countries by land area, which have relatively few people in them.

Notice also how the lines of latitude and longitude have become distorted by the growing and shrinking countries. This is an unavoidable consequence of the cartogram transformation: in order to give the countries the right sizes and still have them fit together you need to warp things a bit. The method used here, however, does a pretty good job of keeping the map recognizable.

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.

Areas are proportional to national populations.

The cartogram distorts geographical space to depict the data relationships in the attribute space more saliently. One can go even a step further, and use the map as a spatial metaphor entirely, for example to depict relationships in attribute space without the geography. The Self-Organizing Map (SOM) on the left is an example. A SOM (in essence a neural net) re-arranges a state's location in a hexagonal grid space according to its socio-demographic similarity with other states. States that resemble each other socio-demographically are placed closer to one another in the SOM than less similar states. States at the edges of the map are socio-demographically more different to other states than states towards the center of the map. Twenty two census variables were used to compute the SOM. The distribution of high and low values for each variable across the SOM map can also be depicted by a map.

Facial features indicate the social & economic characteristics of the constituencies; colour shows the proportions of the vote for the parties.

Cartogram of total spending on healthcare.

Technical details: These cartograms were created using a variant of the diffusion algorithm of Gastner and Newman. Data for the population cartogram were taken from the Gridded Population of the World compiled by the International Center for Earth Science at Columbia University; elevation and bathymetric data were taken from the NOAA 2-minute Gridded Global Relief data set. Data for the other cartograms came from the United Nations Statistics Division and from the databases of the World Health Organization. In all of the cartograms on this page, Antarctica has been treated the same as the sea, meaning its area is unchanged although its shape may be distorted slightly to make room for changes in the sizes of other parts of the world.