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Dorling, D. (1991) The Visualization of Spatial Structure, PhD Thesis, Department of Geography, University of Newcastle upon Tyne

Chapter 3: Artificial Reality

People ask me, "What's so good about artificial reality?" And I say, "What's so good about reality?"
— Myron Krueger, quoted in New York, August 6, 1990
[Haggerty M. 1991]

3.1 Imagining Reality

Reality requires resemblance to the original, artificial implies made by art — not natural. Artificial reality is a manufactured version of the original, created through imaginative skill to show more about reality than is directly visible. If we were to paint things just as we saw them, our purpose would be merely to store their likeness. Instead we wish to investigate their being. To bring out more than the mere surface details of reality we must create images which might not look directly like it — but tell us far more about it and what lies beneath it (Print XLVIII). Almost everything we study is either too small or too large to be studied as it really is. We create models of our world through which to understand it. When these models are expressed in picture form, an artificial reality is created.

Chemical models of molecules are a pioneering example. Not only do they represent great magnifications of reality to a size we can see; but, more importantly, they distort, simplify and elaborate the object, to enhance our understanding. The atoms making up the molecule are drawn as planets, brought close together, and linked with rods to imply connection. Unnecessary detail is omitted, and different elements are colour coded to aid interpretation. Highly complex molecular modelling is now part of the forefront of visualization on the scientific computing agenda.

Transforming reality is an ancient occupation. From charting the Heavens to depicting anatomy, we represent things not as they are, but as we think they should be understood. That we create artificial realities is partly of necessity — reality being too large, small, or chaotic — but it is mainly to expedite understanding36. In visualization we enhance reality. The most important decision in so doing concerns what view of space to adopt as the basis for our pictures (Print XLIX, Bunge W.W. 1966, Post J.B. 1973, Denes A. 1979, Krueger M.W. 1983).

Artificial reality is a way of understanding reality. We create it because we can, and our capability to do so is enormously expanded by visualization through the computer. The contents of our imaginations are made visible to feed back into our and others' minds, in great loops of creativity. The ability to share so vividly our thoughts will tax our abilities to accept and cope with each other's artificial realities37. Through seeing how others see, we all change our views.

3.2 Abstract Spaces

Viewed from a few hundred metres above the surface of the earth people appear like ants, milling around aimlessly. Another few hundred metres and we cannot see the people, only the buildings they constructed, the land they cleared and the roads and bridges which connect and separate these things (Print L). A few kilometers above the earth and all this evidence of human activity has disappeared; we are left with only the flares of oil wells and the line of some ancient Chinese wall to indicate that any people occupy the land below38.

The creation of artificial spatial realities is necessary to our sense of self importance. We think that what we do is central to this world, that the thin slices of concrete we have placed upon its soil is a great achievement, that our impact is of crucial importance to its future. When we map our world we are mapping it for ourselves; to navigate it, control it, and understand it as it relates to us. The objective view of the earth, the blue-green blob seen from the moon, tells us nothing of the world of people, who might as well not be there for all we can see. To see ourselves on this planet we had to create first the abstract spaces upon which our paths could be drawn39 (Print LI).

Early maps of the world were centred on the religious capitals, the land was magnified where most people were known to live and most detail could be drawn. There were maps of kingdoms and empires, spaces which contained land, the land which contained people. The world was flat, as you could not sail around it. Rivers were drawn as wide barriers, mountains enlarged as impassable obstacles. Toady roads are drawn far wider than their real width.

Map projections were first deliberately devised to aid navigation, straight lines on the map maintained their compass orientation40 (Figure 7). The shape of the world changed again, and suddenly it was full of oceans and seas where once the land had crowded to fit in the names of places and courses of rivers. The shape of the physical world had not changed but the world of man had, just as it has in the pictures drawn here (Print LII, Tobler W.R. 1966, 1977, Monmonier M.S. 1977a, Williams R.L. 1976, MacLane S. 1980, Stooke P.J. 1985). Trade and conquest by sea became paramount and the images changed to reflect this.

As all the lands were overrun by the former seafaring people, their area had to be subdivided and their actual size grew in importance. The shapes on the maps changed again as land area was maintained at the expense of compass direction. Oceans, now easily traversed, shrunk as they were cut out of the atlas, and the parcels of farm land to be settled, fought over and traded, were clearly depicted. The centres of the maps moved from Mecca and Jerusalem to Venice and London, and the names were of dominions rather than provinces41 .

This century is a time of air travel and world wars, of twenty four hour money markets and starvation on continental scales. The shape of the world has changed again, but to no single accepted projection. There are numerous one world projections, from those which claim the peoples of the world are best represented by their land areas, to those in which distance represents the shortest paths of ballistic missiles.

The shape of our world has always been an artificial reality of the times, whatever religious or scientific accuracy was claimed for it42. We shape our world for our own purposes, to see where we are ourselves and to see what each other has. We are still imagining the abstract spaces to draw ourselves upon. They have never been naturally given.

3.3 Area Cartograms

Cartograms are maps in which the particular distortion chosen is made explicit43. Area cartograms are drawn so that areas representing places on the paper are in proportion to a specific aspect of those places (Tobler W.R. 1961, 1963b, Forster F. 1972, Wilkie R.W. 1976). The aspect most commonly chosen has been total human population, another age might have chosen only adult men. Population cartograms give another shape to the world, reminiscent of ancient medieval projections, so obviously population centred. By choosing to draw the surface of the earth as a population cartogram, we give all its people equal representation in the image (Prints LIII, LIV & LV). In the process we lose much that is familiar, but then, we do not learn through familiarity44.

Population cartograms far outnumber any other kind. They were first drawn around a hundred years ago, and largely ignored until the last couple of decades. In the 1960s algorithms were developed to construct them by machine, as their manual creation has always been immensely tedious. Most used today are still generated manually, although interesting mechanical means were also developed for their production. Much effort went into this pioneering work, because much was hoped of the media. What can be done today is largely a realization of what people were trying to achieve in the past45.

The depiction of electoral geography is a frequent use of population cartograms. Here the population base is often the electorate. On any traditional map of an urbanized country, the majority of political constituencies are literally not visible to the naked eye (Prints LVI & LVII). The problem is particularly acute in countries such as Canada and Australia, but still fundamental in all other regions. The argument is not that the conventional map distorts the message, it is that it cannot even contain half of it. Numerous insets, and insets within insets, or dynamic zooming using a computer could be employed to try and see what is going on, but they cannot form what is required — a single gestalt image, a unique impression (Print LVIII).

Medical epidemiology is another large and rapidly expanding area which employs cartograms. What is the point of drawing diagrams of the incidence of disease using map projections which literally hide the cases you are trying to map? These cartograms are most useful in searching for structure in the incidence of a disease which is thought to strike the population at random. Where is the illness most prevalent, and how is it related to other features of the social landscape?

Population cartograms will gain in popularity as they become easier to employ and better understood in general46. For visualizing the spatial distributions of social structure there is no alternative, if we wish to see the detail of substance. A traditional map can take many projections, so too can a population cartogram. An infinite number of correct population cartograms can be constructed for any aspect of any set of places. This is both an asset, as it allows us to choose to what other properties we wish our abstract space to conform, and a hindrance, as the superficial appearance of the same population cartogram will vary from one author to another47.

3.4 The Nature of Space

The challenge is to construct a graphical representation of real space which portrays sections of the community in places as areas in proportion to their populations. This is usually achieved through an iterative transformation of the conventional Euclidean geometry and topology of the area, slowly stretching some parts, while squashing others, until the places' sizes are in proportion to their populations, instead of being in proportion to their land area.

This process can be tempered by deciding that the topology of the space should be preserved throughout the transformation. In other words, that these places which were neighbouring should remain so after transformation, and that those which were not so, should not become neighbours. Thus we are aiming to create a topologically and geometrically correct contiguous area cartogram48. Even so, it is still possible to create a multitude of these for any given area (Golay M.J.E. 1969, Sen A.K. 1975, 1976, Coombes M.G. 1978).

Further constraints can be added. The most common are that the outer boundary of the area be preserved and that the lengths of interior boundaries be minimised, so creating a cartogram, the shape of which looks familiar and whose interior is least convoluted. While it is possible to achieve both these aims simultaneously, happily now producing a unique solution, they are somewhat contradictory. Maintenance of the original perimeter dramatically restricts simplification of the internal boundaries. A population cartogram of Britain was produced which precisely preserved the original coastline, but a confused internal structure resulted (see Print XLIX). Here, I concentrate on creating the simplest population cartogram, only roughly following the physical outline of Britain, so that the patterns are depicted with the least visual distortion and the greatest interal detail49 (Print LIX).

The practicalities of the situation — designing an algorithm which could be successfully implemented with the equipment and knowledge that we have today — led to a further compromise in this work. The shapes of internal places were made circular (Figure 8), and hence as simple to gauge as possible. Strictly speaking the contiguity and topological constraints were now broken, but in practice the vast majority of places still bordered their former neighbours after transformation. Various methods could be employed to make the cartogram, once created, appear continuous again — building Thiessen polygons around the circle centres is simplest (see Print CLXXII).

Thus we can now create a space of places, the areas of which are in proportion to their populations, and which maintain, as far as possible, their original topology. Such a cartogram is particularly useful for visualization as it presents a much clearer image than one which would have to twist and wind to satisfy strictly all the conditions — all of the time.

3.5 Producing Illusions

What places should now be chosen, out of which to build these abstract spaces? How will the choice of which hierarchy and division of areas to use alter the image (Prints LX & LXI)? Cartograms from the same population-count of all the major administrative divisions of Britain have been constructed and many are shown here. The answer to the question of robustness is that the choice of areal unit does not substantially alter the final shape of these images — a reassuring outcome. In fact all thoughtfully constructed cartograms of Britain tend towards the same rough structure, which loosely implies that an ideal solution exists. There is a sense of aesthetic acceptance to be realised also. The following papers document various attempts to control or automate the process: Hunter J.M. & Young J.C. 1968, Hunter J.M. & Meade M.S. 1971, Skoda L. & Robertson J.C. 1972, Tobler W.R. 1973a, 1986c, Olson J.M. 1976b, Kadmon N. & Shlomi E. 1978, Eastman J.R., Nelson W. & Shields G. 1981, Dougenik J.A., Niemeyer D.R. & Chrisman N.R. 1983, Nelson B. & McGregor B. 1983, Cuff D.J., Pawling J.W. & Blair E.T. 1984, Selvin S., Merrill D., Sacks S., Wong L., Bedell L. & Schulman J. 1984, Dougenik J.A., Chrisman N.R. & Niemeyer D.R. 1985, Kelly J. 1987, Cauvin C., Schneider C. & Cherrier G. 1989, Torguson J.S. 1990.

This degree of autonomy in the shape of the cartogram, from the influences of the areal division which was used to create it, was only achieved by choosing a careful definition and measure of contiguity. Two places were said to be contiguous if they shared a common border or were linked by a major tunnel, road or rail bridge. The measure of contiguity was not absolute, but estimated as the proportion of the perimeter of an area made up by the border in question, or the length of estuary coastline which a bridge, for instance, rendered traversable (see Print L).

The algorithm for creating the cartograms began with each place at its Euclidean location, represented as a circle whose area was in proportion to its population. Overlapping circles repelled each other while circles were attracted to their neighbours in relation to the strength of their contiguity measure. Places which bordered the sea expressed a degree of inertia because part of their perimeter, being coastline, did not make up a common border, and this helped to maintain prominent peninsulas and other landmarks. Thus, although the exact shape of the coastline was sacrificed, many of its key locational features were retained.

The sustained combination of all these forces in parallel (Figure 9) created the new pictures of Britain used in this dissertation50. An algorithm was used where the solution evolved towards the desired goal (see Appendix A), releasing and tightening constraints to allow the conditions to be attained, and to ensure that the final pictures looked acceptable.

3.6 Population Space

The very shape and layout of the cartogram is of interest even before we begin to use it to depict other information. The population cartogram tells us a lot about the human geography of places — how they are related to each other in a new and intriguingly unfamiliar way51 (Prints LXII & LXIII, Warantz W. 1975, Tobler W.R. 1976a, Wilkie R.W. 1977, Finamore P.M. 1982, Härö E.S. 1989, Löytönen M. 1991).

The population of Britain is more drastically dominated by London than most human geographers would imagine. Greater London itself contains over an eighth of the population. Combined with those areas under London's immediate influence in population space, we can count nearly half the people of the island. This structure is repeated recursively — Glasgow making up more than half of Scotland and dramatically influencing the geography of the rest of that country. The areas of influence of the other great cities are clearly shown, as is how they compare and combine, are divided and divide space up amongst themselves (Print LXIV). The separation of Wales into North and South, and Scotland from England, highlights divisions which are well known, but missing from conventional depictions.

To make the reading of the cartogram simpler, and to learn more about population space, we can transform the major networks of infrastructure, which service the population and along which they move, to lie upon the space. The layout and purpose of the mainline railway network is clear on the cartogram (See Print XLV). It provides a series of arteries attempting to reach all areas equitably, in accordance to their populations.

The road network of Britain is much more complex, and only the motorways and designated main routes are shown in the illustrations drawn here. Again the even spread across the country can be noted (See Print XLVII). Intriguingly though, the network is most sparse in population space where it is most concentrated on the ordinary map — in London. It is no wonder that congestion is greatest where there are least roads per head of population. Here the physical practicalities and human desires combine, so that both versions of reality are useful in understanding the situation.

3.7 Stretching Spacetime

The difficulty of constructing these area cartograms is due to the fact that they are two-dimensional entities (Prints LXV & LXVI). One-dimensional cartograms are simplicity itself to produce. Imagine a one-dimensional, temporal cartogram of the population of the world from when the species began until the present day. Such a cartogram would consist of a single line, with dates marked along its length (Figure 10). The distance between any two dates would be in proportion to the number of people living between those times. Thus, the time line would be very compact at the beginning, having its years widely spread towards the end. More importantly, it is unambiguously the only solution to the problem. The number of dimensions of a cartogram can vary, limited in type only by the imagination. A half-way house can be envisaged of a one and a half-dimensional cartogram, where some information independent of time is depicted vertically up from the time-line cartogram, for example, the proportion of the population living in the various continents. Such a cartogram would be just as simple to construct and while appearing two-dimensional the information is of one dimension (place) within another (time).

The term linear has already been reserved in the literature on cartograms to mean something other than one-dimensional. A linear cartogram is one where the distances between places is deliberately altered for a given reason52, the most well known of these being to fit place names on, and so simplify, a map of a city's underground system. Another well known option is to make the distances between places proportional to the time or cost required to travel between them. This can only be achieved for the, say, shortest travel time distances between all places, when the two-dimensional space in which the linear cartogram resides, is itself warped in the third dimension.

What happens when we go beyond the two spatial dimensions and also attempt to incorporate time? At one level an analogue to the one and a half-dimensional cartogram can be made. The linear cartogram where distance is made proportional to travel time can be projected as a surface above an area cartogram. Thus a two and a half-dimensional linear area cartogram is created as a surface of travel time above population space (Angel S. & Hyman G.M. 1972, 1976).

Even simple two-dimensional population space changes in time (Print LXVII), dramatically so over long periods. A series of area cartograms has been constructed for this dissertation of the British electorate by parliamentary constituency from 1955 to 1987 (see Appendix B). The ten images show the gradual deformation of the space as the electorate grows nationally, the South East swelling in particular while the inner cities shrink. The fact that the definition and number of places changed also over this period, was easily incorporated.

A true three-dimensional volume cartogram of population spacetime is difficult to imagine. Such an image would have to be based upon the axiom of giving each life equal representation rather than each area. As lives have temporal extent they would have to be drawn as life lines. It is hard to imagine what further constraints would be employed in constructing such spaces. Obviously volume should be in proportion to individual lives, and contiguous places in space should touch each other, as should places connected with themselves, both forwards and backwards in time. If we then choose to minimise the area of internal boundaries, which are now planes rather than lines, we will warp time into space and vice versa53. A place which many people left will slip back in time, a place growing in size pushes forward. What are we creating and how can we understand it — let alone view it?

The computer algorithm employed here could be adapted to create all the variants mentioned above. The elusive formulae that people have searched for in the past, to achieve these transformations in a single step, are no longer required.

The nature, creation and use of spaces above two dimensions is the subject of the last part of this dissertation. For now we see how the unusual, but understandable, two dimensional population spaces can be gainfully employed in the visualization of spatial social structure.



XLVIII Experiments with area cartograms (Colour).
XLIX Continuous area cartograms of the British population (Colour).
L County boundaries showing bridges which maintain ward continuity.
LI The evolution of a cartogram of population by County.
LII The County population cartogram with arrows representing topology.
LIII Local authority districts on an equal land area projection indexed for identification.
LIV Local authority districts — indexed list in alphabetical order.
LV Local authority districts cartogram indexed for identification.
LVI Parliamentary Constituencies on an equal area projection indexed for identification.
LVII Parliamentary Constituencies — indexed, list in alphabetical order.
LVIII Parliamentary Constituency cartogram indexed for identification.
LIX Census wards—1981 resident population area cartogram.
LX The concentration of unemployment by ward.
LXI The distribution of unemployment by ward.
LXII Counties and Scottish regions — four colour map.
LXIII Counties and Scottish Regions on the enumeration district cartogram.
LXIV Enumeration district population cartogram.
LXV 1981 equal population grid squares.
LXVI 1981 population enumeration district cartogram showing the national grid.
LXVII The changing distribution of total population in Britain, 1971-1981.


The Mercator Projection maintains all compass directions as straight lines and was therefore extremely useful in an age of maritime navigation. It distorts areas considerably, for instance, Greenland is very much magnified while Africa is relatively compressed. The shape of the areas is also altered, but this is inevitable to some extent in flat representations of the surface of the Earth.
It is surpising that centuries after its inception this image, and other images like it, should still present the accepted view of the world. A series of lines, dividing land from water, which medieval explorers used to find their way across the oceans.

Figure 7: The Mercator Projection

The algorithm which was developed to create the area cartograms worked by repeatedly applying a series of forces to the circles representing the places. Circles attract those they are topologically adjacent to; the strength of this attraction being greater the larger the distance is between them and the longer their common boundary.

They repel those with which they overlap, with a strength proportional to the overlap. Friction is applied to prevent unsatisfactory local solutions being settled on too soon.

The repulsion factor must always be slightly greater than the attraction or else where, for example, each of four zones are all connected to the other three, an overlap will always remain.


Figure 8: The Algorithm at Work

The following set of equations show how the damping factor K of 0.25 was derived for the cartogram algorithm. x is position, v is velocity and An used as an ansatz:

To avoid oscillation the solutions to A must be real, therefore the discriminant must be positive: 1-4K >= 0
Thus the largest non-oscillating K is 0.25.

Figure 9: Deriving a Constant

Many different types of area cartogram can be imagined. Here are some world scale examples -

One dimensional: World population over time:

One-and-a-half dimensional: World Population over time by Continent:

Two-dimensional: World population over space by continent:

Two-and-a-half-dimensional: World Population with income as height:

Figure 10: Many-dimensional Cartograms


36 [a] To understand reality we must first transform it:
Graphics is a very simple language. Its laws become self-evident when we recognize that the image is transformable, that it must be reordered, and that its transformations represent a visual form of information-processing. [Bertin J. 1981 p.183]

[b] Transformation of space is the essence of geographic understanding:
Stated in simplest terms, a creative, productive person is one who can take the ordinary sensory data available to all of us and process them in new ways. It is this dual process of gathering data and transforming these data creatively the geography students should aim to master. The first step is to gain expertise with the techniques of the field. whereas writers need words, mathematicians need numbers, and artists need visual perceptions as primary raw materials of their respective fields, geographers need all three. It is in that uniquely geographical technique, the map, that words, numbers and graphic forms are moulded into a hybrid view of environmental relations.
But knowledge of geographical techniques (including maps) alone provides little more than the starting point for creative thought. The second and most important step in being creative involves transcending these basic raw materials of the field and intuitively seeing possibilities for transforming these ordinary geographical data into new creations. Here, again, the immense transformational power of cartographic abstraction provides the geographer with a special means toward environmental understanding. [Muehrcke P. 1981 p.6]

[c] The transformation can produce an artificial reality:
One tendency, which is not necessarily desirable, is the use of displays to reproduce the real world. The technical evolution of television suggests that other displays will undoubtedly progress toward greater and greater realism. But it is probably an error to try to automatically guide aesthetic displays in this direction. An abstract conceptual or symbolic space may be more effective than a completely faithful rendering of a real environment. What is important is that the displayed space appear sufficiently compelling so that the participant suspends belief and accepts the experience as real, even if the world it portrays is not. [Krueger M.W. 1983 p.79]

37 [a] How should we view the physical world?:

The map is not some inferior but more convenient substitute for a globe. Map projections are not simply choices of lesser evils among distorting possibilities. On the contrary, the map allows the geographer to twist space into the condition he wishes. For purposes of finding lines of constant compass direction, the Mercator projection is far superior to the actual surface of the earth. The earth itself lacks the spatial property of having such lines being straight lines. [Bunge W. 1966 p.238]

[b] Distortion can be used to great advantage:
Second, the distorting features are perceived not only negatively as an impurity, which interferes with the true form of the invariant object; they are also seen positively as the effect of a condition that overlays the true shape of the object. The effect is understood as the logical consequence of the object's position in space relative to the observer. [Arnheim R. 1970 p.51]

[c] We distort both to simplify and illuminate:
The map is our experimental tool. It allows us to twist space into desired shape. What projection will yield the "uniform surface" we need so that we can meaningfully test geographic theory? This geographic mapping is crucial before we can see the world clearly, so that "all the spatial shimmer is taken away and the underlying symmetry focuses for pertinent analysis." [Warntz W.W. 1973 p.55]

[d] Effective maps are not necessarily realistic:
One of the most often cited applications of graphic imagery is the use of maps. We found that the most effective maps may not be the most realistic, but are those which actually "distort" reality by eliminating information and by visually clarifying the topological and functional connections among geographical entities. (e.g., a pocket subway map). [Mills M.I. 1981 p.115]

[e] Here, we create and stress distortion:
The Distortion series by Agnes Dene (see fig. 26) is a series of projections for representing the earth. The series stresses distortion, rather than trying to minimize it. The projections were developed geometrically. Through her book, (Denes 1979) we know what Denes intended to point out. That is that distortion is an unavoidable part of the process in which concepts are transformed into substance. Denes description of her research method is comparable to descriptions of the scientific process. [Varanka D.E. 1987 pp.66-70]

38 [a] Recognition of the fundamental difficulty of showing the activities of people on the Earth has a long history:
Such being the points which it is desirable to compass in a pictorial representation of a given crop, say wheat, several modes of obtaining the relative power, in this respect, of different sections, States, or counties, suggest themselves to the mind. More than one of these might, perhaps, be advantageously adopted, were it practicable in the construction of such a map to take the township as the unit of treatment; but as this is wholly impracticable in dealing with so large a field as the United States, the county — embracing, as counties generally do with us, town and county, manufacturing, commercial, and agricultural populations alike — being the lowest unit of treatment that can be taken for the purpose, the reflections and the tentative computations of the Superintendent satisfied him that no one simple ratio could be found which would not, in many cases, grossly exaggerate, and in other cases as unjustly disparage, the importance of the crop to the county and the county to the crop. [Walker F.A. 1870 p.367]

[b] Thematic mapping brought about a conflict in cartographic history:
The second major revolution in cartography identified by Robinson (1976) was the emergence of "thematic" mapping. Again, technical advances facilitated expansion of this new revolution. Robinson included computers, along with further advances in printing, plastic drafting materials, and new photographic films, as the significant advances. This second revolution, which began more than a century prior to the invention of computers, also derived from a desire to view geographical reality from a different perspective. Whereas in the first case we progressed from an abstract theological basis of maps to a spatial planimetrically accurate one, with thematic maps we had almost the reverse. Emphasis was on communicating relationships and spatial problem solving, often at the expense of precise locational specification. [MacEachren A.M. 1987 p.100]

[c] The thematic / topographic debate continues today:
Other theoretical foundations and methodological approaches are essential here, based on studies of the specific geographical relationships between the phenomena being mapped. "The fetish of geometric accuracy", writes K.A. Salishchev, "which is becoming increasingly clear in respect to topographic maps and maps in general that are intended for cartometric work, turns out to be unwarranted and even senseless for maps intended for other purposes, particularly thematic maps" [21, pp.123-124]. [Suvorov A.K. 1987 p.259]

39 [a] Many thematic creations were extremely misleading:
This choropleth method of mapping numbers using the area of the collecting unit as part of the symbol is widespread, and as a method I have no disagreement with it. It is the misuse of the method that is bad. Choropleth maps are appropriate only for those numbers that include area. I expect that many of you are so accustomed to seeing this method used to map any kind of number that you will not agree with me that some uses produce error. Beware that your response isn't more emotional than reasoned. Arguing that this method of mapping is sanctioned by custom is not persuasive. Until quite recently it was accepted custom for a doctor to bleed, purge, and puke his patient. But, even though some survived, it is no longer held that this once customary treatment is laudable. I suggest that simple mechanical accuracy in maps is not enough. Map makers should provide psychologic, or call it aesthetic, accuracy as well. . . .
This map could not be much more deceiving than it is even if a conscious effort were made to make it so. [Williams R.L. 1976 p.216]

[b] One researcher developed a technique akin to the cartogram, but which did not transform space:
It should be stressed that use of GRP [a method developed by the author] to represent a population Pi of each region i is justified only if the reader is aware that no account is taken of the different territorial size of each region and that this way of graphical presentation is extremely variable, with changes in the division of the territory into alternative systems of regions: consider, for instance, three alternative maps of the territorial distribution of the population of the U.S.A., one by counties, one by states, and one by large geographic divisions. If the GRP scale is kept fixed on the three maps, their general aspect will change markedly: the number of symbols will decrease from the first to the third while the size of the symbols will increase. However, the total blackness of the GRP will remain almost constant. [Bachi R. 1968 p.198]

[c] The use of cartograms hides areas with small populations and thus low denominators:
It is the contention of this author that many, many of the census maps for small areas of Britain (such as EDs) are highly misleading because the ratio used (e.g. percentage values) are based upon small denominators and have been variably influenced by adjustment. Even without adjustment, percentages based on small populations often reach extreme values and, when mapped, these may dominate the map. [Rhind D. 1983 p.185]

[d] Basic geography may even be better learnt using cartograms:
Through classroom experimentation, Fuson "has proven that geographic locations are easier to learn when map clutter is eliminated." Fuson also suggested using cartograms as teaching aids for place-names before attempting to tackle atlases which tend to show too much detail. [Phillips D.J. 1977 p.5]

[e] The use of cartograms allows absolute measures to be simply shown:
Absolute numbers should never be mapped by this means when using a standard set of boundaries. To do so is grossly misleading since large areas will automatically tend to be black. Two solutions exist:
— to standardize the data, most commonly by converting the variable to a percentage or other ratio form;
— to transform the map base, such that the basic areas are enlarged or reduced in size so as to represent the total numbers of people therein, then to map, say, absolute numbers of retired people on this new base map (see figure 6.5); [Rhind D. 1983 pp.187]

40 [a] Projection is the most important decision in "how to map":
An expertly chosen map projection helps to focus and amplify the geographic message of a map (Figure 1). [Hsu M.L. 1981 p.152]

[b] Tobler was one of the first to argue that distortion could produce realism:
As map projections, the transformations used in this chapter do not conform to the traditional geographic emphasis on the preservation of spherical surface area. Maps prepared using these transformations, however, from many points of view are more realistic than the conventional maps used in geography. [Tobler W.R. 1961 pp.162-163]

[c] The map can be anything we make it:
We need to recognize unequivocally that the map is a socially constituted image and our definition of the artifact itself should reflect that recognition. This is entirely lacking in works such as Robinson and Petchenik's The Nature of Maps or Keates' Understanding Maps and in the voluminous literature on cartographic communication and cognition. They represent a still largely positivist way of cartographic thinking. [Harley J.B. 1990 p.6]

41 [a] Writing on the historical development of cartography, Harley complains:
On other maps, towns occupy spaces on the map — even allowing for cartographic convention — far in excess of their sizes on the ground. Castle signs, too, signifying feudal rank and military might, are sometimes larger than signs for villages, despite the lesser area they occupied on the ground. [Harley J.B. 1988 pp.292-294]

[b] A few pages on, however:
Maps as an impersonal type of knowledge tend to 'desocialise' the territory they represent. They foster the notion of a socially empty space. The abstract quality of the map, embodied as much in the lines of a fifteenth century Ptolemaic projection as in the contemporary images of computer cartography, lessens the burden of conscience about people in the landscape. Decisions about the exercise of power are removed from the realm of immediate face-to-face contacts. [Harley J.B. 1988 p.303]

42 [a] Ancient maps look very disorganised to us:
In ancient times and the middle ages, maps were highly subjective. No impersonal codes and conventions. No uniform scale, orientation or even distances. [Hagen C.B. 1982 p.326]

[b] But spatial accuracy is beginning to be disregarded today:
Thus I would like to suggest that the "new" view encouraged by the current interest in "GIS" is the recognition that the spatial relationships between the objects on your map matter as much or more than their actual coordinates, particularly when using computers. [Gold C.M. 1989 p.21]

[c] Cartograms often look very much like old maps:
As an aside, it is interesting to note that even if the earth were a disk (as some ancients believed) and not sphere-like, the suggested transformations still would be of value. The maps obtained here as transformations also are reminiscent of maps produced in the middle ages. Other equally unusual maps can be considered transformations, ... [Tobler W.R. 1961 pp.140-141]

[d] Cartograms can also have a propaganda value:
Chimerical cartography was effectively employed in the propagation of ideas by the Nazi geopoliticians. Dr. K. Frenzel, addressing the German Cartographic Society in Berlin, October 22, 1938, declared: "Every map has a suggestive force! Man is an ocular creature. He reacts to that which he sees and can take in at a glance." [Boggs S.W. 1947 p.433]

[e] Your point of view very much affects whether you see propaganda, distortion, or realism:
A second map in Der Krieg 1939/40 in Karten uses similar techniques in its aim to persuade the Americans that Britain was the real threat to the Monroe Doctrine. It shows the entire western hemisphere and once again, demarcates states with strong political ties to the British Empire in yellow, with the smaller Caribbean countries outlined in large solid circles. As a result, the mass of yellow accounts for an area larger than the USA and Mexico combined, which obviously distorts their true proportions. This impressionistic map clearly communicates the threat of British imperialism [Murray J.S. 1987 p.241]

43 [a] A simple definition of a cartogram is that:
A cartogram is a combination map and graph. [Wilkie R.W. 1976 p.1]

[b] Some well known examples include:
Perhaps the two most commonly quoted cartograms (or transformed maps) are Edgar Kant's logarithmic map, designed at Hagerstrand's behest, to depict relative migration distances from a specific origin, and Arthur Getis' map transformation of income space in Tacoma, to test whether shopping centres were in fact regularly spaced in relation to demand. A second, less sharply focussed strand in the use of cartograms can be identified. This strand is concerned mainly with conveying a clear visual impression of a static spatial distribution, with the cartogram providing a base on which related information can be depicted, generally by means of choropleth mapping. [Holmes J.H. 1974 p.218]

44 [a] Even for unfamiliar locations, cartograms simplify:
Long classroom experimentation by the author has proven (at least to him) that geographic locations are easier to learn when map "clutter" is eliminated. [Fuson R. 1970 p.xi]

[b] And in the new electronic information age:
Videotex is a special form of TV image, and behavioural research on cartography and videotex is very limited. In one of the few published studies, Mills (1981) has argued that the most effective maps on videotex may be those that distort spatial reality. He also argues that as cognitive capacities such as visual memory ability vary from individual to individual, some viewers may be much more able to learn from maps than others. [Taylor D.R.F. 1985 p.31]

45 [a] Research on cartograms has been lead by the work of Waldo Tobler:
Besides offering new types of maps and atlases it seems to be indispensably necessary to develop new methods of use and evaluation of cartographic representations. In pattern recognition not only cartography is interested, but also geosciences. The efforts of W. TOBLER from my point of view can be regarded as a promising beginning in elaborating suitable methods of analysing spatial pattern and transforming cartographic figures for new considerations. [Kretschmer I. 1978 p.36]

[b] A mystical uniform plain was being sought:
Geography, like other mathematized sciences before, is searching for the correct coordinate system and point of origin. Tobler is our Copernicus. The Geographic Projection, the one we still seek , the one much more important than the infinite projections mastered, is the Uniform Plain, which is the geographic equivalent of "other things equal" assumptions in other sciences. Once the space is properly projected, the patterns (our primitives are the dimensions) both probalistic and extremum (with the function to be minimized some concept of nearest) should emerge and be more testable. Somehow the patterns and the coordinate system should be related functionally. [Bunge W. 1968 p.31]

[c] And several computer algorithms were written to find it:
The original computer algorithm was constructed by Waldo Tobler, which generates pseudo-continuous cartograms (Figure 2.7) according to partial differential equations (Tobler, 1963). Tobler's algorithm fixes a planimetrically correct base map to an underlying continuous surface, which is then projected onto a distorted plane, which represents the variable transformation (Dougenik, et al., 1983).
The Tobler algorithm is regarded as imaginative but highly inaccurate, slow due to the number of iterations required by the algorithm, and guilty of producing an over generalized end product (Figure 2.8). This led Nicholas Chrisman to write a competing algorithm which uses a different distorted plane approach. In this scheme, each region or polygon has an amount of "force" applied to it based on the variable's value being mapped (Dougenik, et al., 1983). The implementation of the Chrisman algorithm (Figure 2.9) currently exists as part of the mainframe GIS package ODYSSEY (Corson-Rikert, 1983). [Torguson J.S. 1990 p.20]

46 [a] Cartograms blatantly proclaim the distribution of population:
First, the projective distortions not only permit the discovery of the prototype inherent within them; they call for it actively. [Arnheim R. 1970 p.51]

[b] They allow us to entre an artificial reality and gain new knowledge:
The distortions of geographic space are wilder than any other science. Einstein's simple spatial bending is child's play compared to the weird house of mirrors geographers must straighten. Point to your home. Now go "straight" home. Do you follow your arm through the wall? How crooked your "straight" path is as you wend and wind your way "straight" down the geodesic time path to your house. What sort of mapping is necessary to show all these "straight" paths from all points as straight? In chapter two it was emphasized that not even the order of space is maintained. Tobler proved to us that space can repeat. It fascinates the curious to notice how the refraction of light seems to bend a hand thrust into water. In geography, thrusting your hand into the proper space can cause it to reappear in 15 places simultaneously if the space is folded under itself five times. In such an Alice in Wonderland World it is a miracle that geographers have discovered any underlying order whatsoever. How much progress would the astronomers have made if they had had to make their observations through the wildly swirling lens through which we must peer? Nor does the mapping of the isotropic surface end with the transformation of such regions as Iowa. The entire globe itself has to be reprojected. What Tobler calls "the geometry of geography" meaning the curvature of the geodesic surfaces after Gauss, is hardly confined to sheets of paper. These projected globes will show little resemblance to current desk top models which seem to have won so many geographers to their hearts as "distortion free". Since the notion of reprojecting a "perfect" globe makes Tobler's point so clear, I suggest to the profession that we refer to them as Tobler Globes in his honor. [Bunge W. 1966 pp.242-243]

47 [a] Not all writers favour the use of cartograms:
Exhibit 193 represents a method of showing data for states by drawing the area of the state on the map proportional to the numbers represented. Such maps, however, are not superior to the dot or bar maps just described (Exhibits 183 to 189, inclusive) for showing distributions of size. In many cases the method illustrated in Exhibit 193 would result in the states being so distorted that little if any resemblance of their true shapes would remain, and even their relative positions would be inaccurate. It is much more difficult to compare the irregular areas on such maps than it is to compare either circles or bars. [Riggleman J.R. 1936 pp.179-180]

[b] Some dismiss them, but offer no alternative:
There are three basic misuses of area in mapping census-type numbers: 1. Failure to compensate for differences in the areas of the collecting units. 2. Including map area in the symbol when land area is not part of the number being mapped. 3. Using a base map that distorts area. There is an ever increasing number of canned computer mapping programs. These make it easy for anyone with a handful of numbers to produce a map, but seldom do the instructions for the program's use include warnings on their possible misuse. Choropleth-type maps of census-type numbers are one of the most dangerous traps for the unwary. [Williams R.L. 1976 p.213]

48 [a] Continuous area cartograms are not necessarily as challenging to create as some would claim:
The most valuable and constructionally challenging type of cartogram is the contiguous cartogram (Figure 2.6b), where proximal relationships and contiguity are maintained. Note that some shape properties have been sacrificed, but the topology still gives an appearance most like a real map. There is no generalization of the number of vertices which occur in the latter cartogram type. No boundary generalization theoretically takes place. The map like quality of the contiguous cartogram is cause for its high appeal and desireability. Its creation presents the highest challenge to cartographers using both conventional and computer-assisted techniques. Contiguous cartograms are the most difficult and time consuming to construct (Muehrke, 1978), though a well-designed cartogram of any type requires much forethought in its inception and execution. [Torguson J.S. 1990 pp.17-19]

[b] For instance, when producing cartograms:
The assumption of continuity of a distribution is often not warranted. The data are often in the form of discrete locations, as on a population dot map, or grouped into areal units, such as census tracts, or refer to areal units rather than infinitesimal locations, as land values which refer to specific parcels of land. In these cases an analytic solution is usually not feasible and rule of thumb approximations are useful. [Tobler W.R. 1961 p.155-156]

[c] Preserving contiguity completely often produces confusing twisted images:
One other striking feature of map (C) compared with map (A) is that many of the registration districts have become very long and thin. This is a common feature of isodemographic maps, especially near the margins, and arises from the need to pack the units properly while at the same time preserving the abuttments. [Cliff A.D. & Haggett P. 1988 p.60]

49 [a] The circular cartogram can be seen as a development of the proportional circle and dot maps:
In any urbanised country, however, it is quite impracticable to achieve a happy combination of map scale, dot value and dot dimension which will give a satisfactory representation of both country and town areas. The use of different sized dots can go a little way towards the solution of this problem, but recourse must always be made to the employment of proportional symbols, or overprinting of population totals for major urban areas. [Dixon O.M. 1972 p.20]

[b] It is interesting that proportional circles have to be rearranged in conventional mapping:
It is generally unsatisfactory to use small, fixed-size symbols for such mapping, so proportional symbols (possibly shaded) are often used. Here luck also plays a critical role — to get such a computer map satisfactory on the first run is to be extremely fortunate. Iteration must therefore be expected — figure 6.2 illustrates an example from a second iteration in which certain of the centre points of areas were edited to minimize symbol overlaps. Of course, the human cartographer is much better than the computer in planning the map production, but to draw such a map manually would take very much longer than the twenty seconds which it took to compute and even the twenty minutes it took to draw. [Rhind D. 1983 p.182]

[c] An analogue simulation was used to produce the cartograms shown here:
I feel that at least in our initial period of theoretical research, we will find analogue computers of great use. For identical reasons, I predict we will seek to solve our intractable formal theory by approximate methods and the use of computers, i.e., we will beat our problems to death with machines. Mathematicians will detest the lack of elegance, but notice that while the mathematics of such a strategy may be gauche and muscular, the theory itself can be completely lithe and uncluttered. It is not uncommon in the history of science that the scientific theory is simple while the mathematics required to operate the theory gracefully, remains behind and must catch up later. [Bunge W. 1966 pp.284-285]

[d] Many past hurdles have also been overcome towards the possibility of producing linear cartograms:
Unfortunately, cartographers have serious technical problems to resolve before this powerful technique becomes generally operational. And, whatever the form of the transformation, problems arise first in specifying meaningful transformations and secondly in preserving desired properties such as boundary, shape and continuity relations. The search for permissible and meaningful transformations, the degree of control attainable with each, and the reversibility of the transformation processes provide important topics for future research. [Muehrcke P. 1972 p.46]

50 [a] An example has been produced which satisfies the following criteria:
Though the example is very simple, there are still an infinite number of solutions, but some seem more appropriate than others (see Fig. 6.6). Preservation of the internal topology is one condition which might be applied; preservation of the shape of the boundary is another, etc. [Tobler W.R. 1961 pp.156-157]

[b] The cartograms used in this work are pseudo-continuous:
There are two types of contiguous cartograms. The first is the pseudo-continuous cartogram (Figure 2.6a). Pseudo-continuous cartograms depict regions like a continuous map, but are endowed with the "pseudo" label (after Muehrke, 1978) due to the generalization of the polygon's topological structure. This can be contrasted with the contiguous cartogram, where the topology has been retained (Figure 2.6b). [Torguson J.S. 1990 p.17]

[c] A chaotic environment was initiated to produce these cartograms:
It is impossible in principle to predict in the general case whether the forces tending towards chaos or the forces tending towards quiescence will ultimately dominate the dynamics of the system of whether, for that matter, neither one will ever dominate. Indeed, for many such systems, the conflicting pulls towards order and chaos seem to provide an essential tension which keeps the ongoing dynamics on an indefinitely extended transient, far from equilibrium. [Langton C.G. 1986 p.129]

[d] Here, the density of people is used to curve the space they live in:
So gravity can be explained by assuming that matter curves space. But why should matter do this? Why should matter curve space?
One explanation is that space curvature is what matter is. William K. Clifford first proposed this theory in an 1870 paper called "On the Space Theory of Matter": [Rucker R. 1984 p.82]

51 [a] Some people thought it would not be possible to produce such pictures as are shown here:
Extreme shape distortions associated with these sharp density gradients present such severe problems that it is doubtful whether isodemographic maps could be used for testing locational characteristics when both urban and rural populations are shown on the one map. One quails at the problems of presenting Australia's population in similar detail, with 63 percent of Australians being located in the ten largest cities. [Holmes J.H. 1974 p.218]

[b] Cartograms can be useful for many purposes:
The SMRs so calculated have been portrayed on a specially prepared demographic base map (Fig 7.17). The demographic base map is designed to relate the SMRs to the local populations at risk to death from cancer and to complement the geographical map which related SMRs to the areas within which such populations reside. The advantage of the demographic map is that main centres of population such as London, Birmingham, Glasgow or Liverpool assume increased proportions while geographically large areas with numerically small populations, such as Dyfed in Wales or Inverness in Scotland, are reduced in area relative to the UK as a whole. Disadvantages of the demographic map include its somewhat unfamiliar shapes, distortion of reality and loss of continuity. [Howe G.M. 1986 p.131]

[c] These cartograms are the first to successfully show the human geography of Britain at this fine scale:
This is attractive at first sight but has several disadvantages. Distances within census tracts in the DEMP are not necessarily population-adjusted; the resulting display bears no relationship to geographical space and cannot be used in conjunction with, for example, maps of land use and, finally, the algorithm has never been successfully applied to U.K. census tract boundaries that are highly irregular. [Alexander F.E., Ricketts T.J., Williams J. & Cartwright R.A. 1991 p.159]

52 [a] One definition of a linear cartogram is that:
A linear cartogram operates like an area cartogram but instead of varying areas with values it is the map distances which vary with values. Its construction is analogous to the azimuthal equidistant projection, but rather than physical distance varying with map distance — cost of effort of travel are used. [Lai P.C. 1983 p.33]

[b] The linear cartogram problem and the need for its creation has been appreciated for some time:
In transportation systems analysis, it is convenient to represent the transportation system mathematically by means of a network. For many purposes, such as the problem of estimating travel flows in the network, the mathematical properties of nodes and links are sufficient for the analysis. For other purposes the map image representation of the network may offer important visual clues. The field of interactive computer graphics, for example, explores the coupling of human intuition, aided by visual clues, with the computational power of computers. Transportation networks are abstracted from the locations of streets and intersections, and the graphic representation of the network resembles a road map. Such maps are traditionally produced using Euclidean geometry as the basis for the representation of the spatial relationships. The choice of Euclidean geometry has obvious advantages in the study of many physical phenomena, but it does not always represent accurately the properties of networks that are of interest in the study of transportation phenomena. For example, it is well known that travel time through a network is a more important determinant of behaviour than travel distance. Travel speeds vary markedly from link to link in transportation systems, and these variations result in a space that is non-Euclidean in experience. The problem is to represent these non-Euclidean properties graphically, in a way that will aid analysis processes that depend on visual information. [Clark J.W. 1977 p.195]

[c] Disease mapping could benefit from linear, as well as area, cartograms:
The analysis thus indicates the importance of viewing the relationships between places in terms of the metric (such as time or cost, for example) which is most appropriate for the problem being tackled. The plotting of places in terms of accessibility metrics like time and cost distances is particularly valuable when communicable diseases are being studied and may frequently provide a fresh perspective on the disease patterns occurring. [Cliff A.D. & Haggett P. 1988 p.267]

53 [a] Even more bizarre transformations are conceivable:
Turning to the map of the world, how far from London can you extend your military power before you will lose one hundred thousand men? This "circle" obviously moves much farther over sea than into Europe where military resistance, say from the French, would make such a move very expensive per geographic mile. A map of the earth with hundred-thousand-man-lost circles centred on England in 1850 establishes the most distant place, not as New Zealand, but somewhere around Moscow. The paths of least deaths at right angles to the circles draw another set of real longitudes and latitudes. Moscow is antipode, the opposite side of the earth, the "down under" from London. To conquer the world is to conquer Moscow, not Auckland, and this is why the British kept moving toward Moscow from the Crimea the walls of Peking, the Khyber Pass, from Vladivostock. All "paths" from London lead to Moscow, and thus the mysticism of Mackinder's geopolitic is explained. [Bunge W.W. 1973 p.286]

[b] Geo-political information often needs to be transformed:
It would seem that here, since many of the relationships which are displayed are not most significantly geographical, but rather operational in nature, and that a schematic map of some sort would have been much more effective. It is not miles or kilometers across the surface of the earth in which these geo-political factors are arranged, but rather an interesting — but complicated — series of topological relationships. [McCleary G.F. 1988 p.148]

[c] We are limited only by our imaginations:
In the opinion of this author, the value of such maps can be great for geographers and other behavioural scientists — a value which seems limited only by the imagination of the scholars whose tools they should be. [Lewis P.F. 1969 p.406]

[d] But we must remember that others have to understand what we produce:
As a rule, the novel, dramatic character of cartograms may deceive unwary map readers. Great care and skill must be exercised when dealing with this particular type of map. The advantages of cartograms are substantial enough, however, that geographers would do well to gain sufficient sophistication to handle these maps effectively. [Muehrcke P. 1981 p.27]