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

Chapter 9: Volume Visualization

It certainly feels like time is passing; I’d be foolish to argue otherwise. But I want to show you that this feeling is a sort of illusion. Change is unreal. Nothing is happening. The feeling that time is passing is just that: a feeling that goes with being a certain sort of spacetime pattern.

[Rucker R. 1984 p.140]

9.1 The Third Dimension

A dimension in this dissertation is something which can be both measured and moved around in, allowing the existence of a geometry — the relative arrangements of objects in space. Thus, real world time is not strictly a dimension for us, as we cannot move around in it. Although we live in a three-dimensional world our vision waters it down to two dimensions. We build our cities and homes along two-dimensional lines, and usually only think with two-dimensional concepts.

Time can be viewed as a third dimension119 in the social world when phenomena beyond the simple single lives of individuals are being considered. A social order of opportunities, jobs, customs and culture exists and moves in time and space. A disease is a spacetime entity, and its social repercussions can only be understood when it is seen as such, knowing when, as well as where, it strikes. Other economic or political relationships often include a third dimension when two are too limited to contain their full implications. Of course, we could imagine working with even higher dimensions. You may be surprised how difficult even trivial three-dimensional structures are to grasp. Four dimensions is well nigh impossible, and two is almost always preferable (Hardy R.L. 1988, Bjorklund A. & Gustavsson N. 1987, Thompson J.M. 1988, Ke Y. & Panduranga E.S. 1990, Hart J.C., Kauffman L.H. & Sandin D.J. 1990).

When does a variable become a dimension? That is essentially a question of the resolution of measurement. If place is just one of a dozen regions of this nation it is a variable which should be put in tables, not mapped. Once a variable has numerous possible values it can be considered a dimension. Movement and measurement along the variable must be possible, and the three-dimensional space created should be theoretically continuous. Time in the study of the last ten general elections is too discrete to consider approximately continuous, and interpolation of votes between the elections would be meaningless.

If we have a third dimension, how can we see it, let alone understand it? This is the problem which is responsible for relegating this chapter to the end of the dissertation. Basically, the answer is — not easily. Do not hope to understand the picture unless it is very simple. The traditional way to see in such blocks is to show some two-dimensional slices, as we might cut open a human brain in a medical scan. It is a small step from there to take many slices, allowing animation. To create more of an illusion of three dimensions, perspective and various lighting effects can be employed. These too can be animated.

All we are really showing with traditional three-dimensional graphics is a series of surfaces — two and a bit dimensional, but a long way from three — often containing almost one-dimensional information (Print CLIV). Recently, several innovations have been made in computer visualization which can create far better illusions of more complex three-dimensional worlds. The problem is then no longer deceiving the eyes, but teaching the mind120.

9.2 Spaces, Times and Places


Places exist temporarily as well as spatially (Morrill R.L. 1970, Parkes D.N. & Thrift N. 1975, Carlstein T., Parkes D. & Thrift N. 1978, Holly B.P. 1978, Cebrian de Miguel J.A. 1983, Rucker R. 1984, Monmonier M.S. 1990). Over the years people move home; over the decades new homes are built and old ones decay; over the centuries towns are formed and decline (Print CLV). An animation of the national boundaries of the European continent over the last four hundred years would show near continuous turmoil. Nations exist only as pockets in space and time, as also, in the long run, does the world system of nation states. Regions coalesce, fragment and disappear. The plate tectonics of human geography is a violent spectacle. Even the patterns of spatial inequality can alter in the space of a few dozen years. We can never understand why something is, if we do not look at how it came to be, and what it is becoming121.

The theme of unemployment has run right through this work. Unemployment as a national phenomenon has a well defined spacetime geography. Monthly records have been kept by the eight hundred and fifty-two amalgamated office areas for every month since 1978 (and for every ward since 1983). Over one hundred and fifty temporal observations constitute a dimension in the above sense. How then to view this information?

Many attempts were made to show the structure, some have been mentioned in the previous chapter. The structure was just too complicated for a few views into a spacetime block to uncover (Prints CLVI & CLVII), so a series of time-slices were drawn, one for each year since the series began. To highlight the changes, deviations from the expected value were drawn, knowing the average for the year and the place. If this had not been done, the changes over these twelve years would not have been visually apparent. Similar problems were encountered when trying to show house price change in a single image (Print CLVIII).

The twelve cartograms were created using both counties and amalgamated office areas to show how spatial resolution changed the image (see Prints XCI & XC). One picture was drawn a year, partly because that is all that would fit on the paper, and partly because unemployment is known to vary seasonally. The images show dramatic changes in the social structure of Britain. Initially there was high unemployment only in an expanded celtic fringe, but gradually the picture changed until by 1990 unemployment was highest in the north and inner cities, leaving a ring of almost full employment around Outer London. Between those dates, at the height of the early 1980s recession, places like Liverpool were seen to do relatively well as their position improved in relation to other areas, though it became worse in absolute terms.

Work such as this requires very large matrices of data, over a million rates for the ward time series (which, as a consequence, is not visualized here). It would perhaps be better if the fate of individuals were better known, rather than these giant matrices of aggregate counts of people’s fortunes, but while that is all we have, we must use it. A second problem concerns the use of deviation from the expected, to highlight change. This makes the images seen dependent on the first and last years chosen to bound the study period. It might be better practice to show the changes between individual years, which would be insensitive to which time span were chosen. Finally the twelve slices only begin to capture the inside-out goblet shaped structure of spacetime regional inequality in Britain. To see it as it is, we need software which can sophisticatedly render a projection of a great many observations in space — or, a lot of imagination122.

9.3 Spacetime Continuum


In some cases, usually when the information is sparse, we do have a near complete record of individual cases. For instance a list of firms opening and closing, and the number of their employees, could be created to try and understand unemployment change. The situation where such information is most plentiful is with the medical records of rare diseases. The incidences of when these are detected are available to the level of the address of the sufferer and the day of diagnosis. Such records are relatively few, allowing more sophisticated techniques to be attempted than could be used with the voluminous unemployment data (Print CLIX).

Childhood leukaemia cases in the North of England over the last twenty years number several hundred incidents, with a high level of accuracy involved in their recording123. We can think of these cases as points, sitting geometrically in a block of two decades of Northern English spacetime (Print CLX). If we were to render these cases as simple points, then, because of their sparsity, we may well not pick up any slight increase or decrease in the density of cases or some more subtle spatial and temporal arrangement (Levison M.E. & Haddon W. 1965, Hunter J.M. & Young J.C. 1971, Dean A.G. 1976, Howe G.M. 1979a, 1979b, Slocum T.A. 1983, Selvin S., Shaw G., Schulman J. & Merrill D.W. 1987, Cliff A.D. & Haggett P. 1988, Schulman J., Selvin S. & Merrill D.W. 1988, Selvin S., Merrill D.W., Schulman J., Sacks S., Bedell L. & Wong L. 1988). To aid visual interpretations the points are represented by spheres (Print CLXI), their influence decaying with distance (Figure 26) from their incidence in both space and time124. This process can also create a truly three-dimensional surface, a single value existing at every point in time and space being related to the general prevalence of the disease about that time and around that place.

To see this space we could again resort to time-slices, in this case taken from an animation showing the development with one frame for every month in the period. But, with only a few hundred cases more elaborate software can be employed where the spheres are actually placed in an abstract three-dimensional space and a camera flown around it, recording the views of specific interest (Prints CLXII to CLXVI). Again these are shown here as individual frames, but no longer time-slices, as they were taken at an angle through the block, and show an image looking into it, obscured only when cases are eclipsed by one another. As can be seen in the illustrations, the cases are very evenly spread in population space (Knox G. 1964, White R.R. 1972, Chinese Academy of Medical Sciences 1981, Gardner M.J., Winter P.D., Taylor C.P. & Acheson E.D. 1983, World Health Organisation 1985, Howe G.M. 1986b, Williams-Pickle L., Mason T.J., Howard N., Hoover R. & Faumeni J.F. 1987, Glaser S.L. 1990).

The actual space in which we place the cases is a very important consideration. A simple Euclidean space has only been used to show how that image differs from one obtained when a more appropriate two dimensional population cartogram is used. The relationship between time and space is not simple125. Physicists use the speed of light as a common unit in both metrics, but we are not physicists. Here it was arbitrarily chosen to make one year equal to twenty five kilometres or the root of three hundred thousand people. The distribution of the childhood population at risk from leukaemia hardly altered over the period, in relation to the slight oscillations of the disease. But a more thorough study would have to consider carefully the construction of a volume cartogram, in which every life was equal, and placed as close as possible to those with which it shared life, as well their immediate ancestors and offspring. The relationship between time and space would depend on how far and how frequently people tended to move in their lives.

9.4 Three Dimensional Graphs


Things other than space and time can be projected to occupy three dimensions. A three dimensional graph is created by merely raising a third axis at right angles to the conventional two, and plotting points inside that space. It is not therefore what the term is commonly used to describe, a one dimensional bar chart, with each bar drawn as a pillar. Three-dimensional graphs are often used in statistical projection pursuit, where you need all the dimensions you can get to explore aspects of multi-dimensional spaces. They are also becoming common on microcomputers, where the ability to rotate, or at least rock them is essential to maintain the three-dimensional illusion. Some claim that stereoscopic displays are also an asset.

Here I have developed the peculiar electoral triangle into the logical three-dimensional analogue of a tetrahedron which attempts to show how the vote is shared between as many as four parties in a large number of areas (Figure 27). We need to be able to do this if we are to include Scotland in our analysis of electoral composition in Britain126 (Print CLXVII). In Scotland, in recent years, the nationalist party has consistently come in third or second place, but the third major British party is still in the reckoning and has quite a different pattern of support (Bochel J.M. & Denver D.T. 1984, 1986, 1988, 1990). The abstract creation of an electoral tetrahedron is quite simple. The points, representing the electoral divisions in which the vote is counted, are placed so that their distance from the four apexes is in exact proportion to the share of the four party vote represented by those apexes. To stick to convention, looking from above, the Conservatives have the right hand apex, Labour the left, the Centre party the top (still on the plane), and the Nationalists the apex in the centre (now above the other three).

The problems are familiar when we attempt to project the three-dimensional structure of voting onto a two-dimensional plane for drawing. The methods mentioned above have been used and are shown here, taking slides from an animation, and showing slices from particular angles. But another method has also been developed in this case. That is to unfold a net of the tetrahedron as four equilateral triangles (Print CLXVIII). A point is drawn on the side of the tetrahedron it was originally closest to. As the centre was relatively empty (due perhaps to tactical voting) this technique does not create results that are too ambiguous. In fact each triangle contains only those divisions where a particular one of the four parties came last. The net can actually be further subdivided into areas in which the exact order of the parties are known. Such an arrangement makes interpretation of a complex three-dimensional situation considerably simpler (Print CLXIX), although in explicitly using two dimensions something has to be lost — in this case the exact fortunes of the party coming fourth.

What do we do, though, if we wish to see how the four-way situation changed over time? A slight change in the number of votes could send a division spinning across the net, which in reality would hardly move in the tetrahedral space. This would be unfortunate, unless only changes of party position were of interest. What if there were a fifth party also of some importance — say the green vote rose up in the future. Could we show the two-dimensional net of the three dimensional shadow of a four-dimensional hyper-tetrahedron? Or would it be better to observe the rotating three-dimensional net of the four dimensional point cloud (Prints CLXX & CLXXI)127? These situations are avoided for the while, but remain to be addressed.

9.5 Flows Through Time


If the possibilities at the end of the last section appeared a little daunting, consider, for a moment, the problem of showing how a pattern of flows has changed over a number of years, not a single change but a complete succession. Just such a truly three dimensional matrix has been constructed from the National Health Service central register, for flows (in both directions) between every pair of ninety-seven mainland family practitioner committee areas for each of the last fourteen years128. Even this low level of spatial and temporal resolution produces over one hundred thousand separate counts of migration streams. How can we begin to see what is happening to the flows of people which create the spacetime pockets of existence we call places in Britain?

The basic two-dimensional flow maps showed numerous overlapping arrows. When the change between two years was sought, even depicting the single differences could require two separate images. Depiction of spacetime flows of people would have to be constructed in three dimensions. Theoretically there would be a plane to represent every year, which would contain the changing population cartogram. Places would be linked by tubes, the width of which, say, was in proportion to the number of migrants. To prevent the image becoming too tangled in practice, a measure of significant change would have to be found, similar to that used in the two-dimensional case. Otherwise almost ten thousand tubes would have to connect every pair of planes. At least the origin and destinations of migration would be obvious, even if the paths between them were, to say the least, a little confused (Figure 28).

The structure just described has not been created here, as it would just produce a perplexing mess129 (Bryanston-Cross P.J. 1988, Carr D.B. & Nicholson W.L. 1988, Hesselink L. 1988, Congdon P. & Champion A. 1989, Dickinson R.R. 1989, Helman J.L. & Hesselink L. 1990a, 1990b, Stillwell J., Boden P. & Rees P. 1990). To understand such a complex structure, even after generalization, requires the development of new techniques to look into three dimensions. A maze of tubes crisscrossing in spacetime will not reveal its structure through the illumination of its outside surfaces. Cross sections through the connections would be confusing, and it is also not possible to simplify such a complex organization to a plane and retain its essential form.

9.6 Volume Rendering


We have only really been talking about volume visualization in this chapter, not looking at it. That is because all the facilities we, in social science, have to date, and for a little time to come, can only show surface views of three-dimensional structures. That is why only one chapter of this dissertation has been devoted to the subject which is currently the subject of most interest in computer science visualization130 (Drebin R.A., Carpenter L. & Hanrahan P. 1988, IEE sponsored 1988, Papathomas T.V., Schiavone J.A. & Julesz B. 1988, Bak P.R.G. & Mill A.J.B. 1989, Herr L. 1990, Hibbard W. & Santek D. 1990a, 1990b, Laszlo M.J. 1990). This is conceivably because it is still being developed. Software is being written to look inside the surfaces, to create images on the screen which we could not see in a picture on paper. Volume rendering defines what can be done with this kind of software, which can only be described in these pages.

The key theme is translucence. Surfaces can be peeled off a volume, like two-dimensional contours, but really to see the structure you must be able to see all the contours at once. To do that in space, objects must emit and transmit light, so that they can be seen through, but also not be transparent, to still be seen themselves. There is obviously a limit to how many layers can be pierced; each obscures a little more than the last. The combination of translucence with perspective, animation and lighting allows us almost to see inside the volume as we move around and through it (Dutton G. 1978, Doctor L.J. & Torborg J.G. 1981, Papathomas T.V. & Julesz B. 1988, Sabella P. 1988, Meyers R.J. & Stephenson M.B. 1990).

Imagine the economic spacetime of Britain with the unemployed areas shown like dark storm clouds through which it is possible to see better times ahead (in time), or to the side (in space). The whole structure would be held in the holographic image where no pocket of prosperity or despair could remain hidden. What would the spacetime continuum of childhood leukaemia incidence look like seen through translucent space? Well, for one thing no case could eclipse another. More importantly, when two or more cases fall in almost exactly the same time and place they will appear much darker than is usual, rather than as a single occurrence. The three-dimensional electoral graphs would appear more like a cloud of dust particles, or a galaxy. The true density and sparsity of spatial divisions would be apparent, where, before, they had quickly obscured each other as a dark mass. Lastly there are the flows through time. It might be possible to trace the path of each migration stream through the myriad structure of pipes and columns.

Translucence is not true three-dimensional imaging. To argue that is rather like telling a two-dimensional being which can only see a one-dimensional strip that if objects were made translucent they would see two dimensional structure. They would not. They would merely see what was previously completely hidden from their view, and may, through rotating the angle and position from which they viewed the two dimensional space, come to guess some of its structure. However, they would never have the full luxury of being able to see simultaneously all that it contains and how it is arranged from above, because they are part of that two-dimensional space; just as, in the real world, we will never have that visual ability in three dimensions.

9.7 Interactive Visualization


To bring the discussion up to date, it is increasingly being claimed that there are two types of visualization — the mundane variety which would include this work, and the interactive kind, the most extreme example of which is found in the artificial realities of computer graphics. Interactive visualization, like interactive graphics, allows the viewers instantly to choose the direction and position from which they are viewing, what they are seeing and how it is depicted, lit, animated and so on. What you see moves, and so can you. This freedom allows any aspect of a structure to be examined at will. It is almost as if you could pick it up and turn it around in your hands. In some systems you see the object stereoscopically through two images in a pair of goggles — better still, etched on contact lenses131 (Becker R.A. & Cleveland W.S. 1988, Becker R.A., Cleveland W.S. & Wilks A.R. 1988, Becker R.A., Cleveland W.S. & Weil G. 1988, Friedman J.H., McDonald J.A. & Stuetzle W. 1988, Stuetzle W. 1988, Kerlick G.D. 1990, Wills G., Bradley R., Haslett J. & Unwin A. 1990, Haggerty M. 1991). Your wishes are executed through the movements of your head, hands and even entire body. For the majority of us, however, interactive visualization will not arrive for several years yet.

The basic questions of what it is we wish to see and how that should be drawn remain as important as ever. Interactive computer graphics will allow you to pick up the earth in your hand and view it just as if it were a real globe — but we can already do that in the classroom with the plastic model. What is exciting about visualization is the facility it offers for us to transform what we wish to observe to a form most amenable to our understanding and then change that, if it does not suit us, at a whim.

Interactive visualization will reach the micro computer screen by first offering the user the ability to link several displays of the same data to gain greater insight; say a rotating tetrahedron of the Scottish voting composition in one window coupled with an animated cartogram in another — showing how the distribution of divisions changed geographically over time. An area of Glasgow in the 1970s could be selected and the points representing those divisions would light up in the tetrahedron simultaneously. As you moved a pointer over the changing cartogram of Scottish divisions, other points would become lit and you could trace patterns between geographical, historical and political spaces.

Artificial reality allows us to go one step further; to be actually inside the tetrahedron; to look in a spacetime cartogram down at the 1986 regional election results, and up at the 1990 contest; to see the dark clouds of unemployment rising above Glasgow to meet the fine detail of the 1991 census in the distant sky. Below us would lie the remains of decades of industrial structure and behind us the same for England and Wales. All this will require considerable research and development, but if cannot specify our aims at this stage, how can we plan for the future? Whether these technical innovations will result in useful still pictures for printed work (such as this) also remains to be seen.

 

Prints

CLIV The distribution of population by county in 1981 (Colour).
CLV Population change in Britain by district, 1961-1991 (Colour).
CLVI The space/time trend of unemployment in Britain by cubes, 1978-1990.
CLVII The space/time trend of unemployment in Britain by rings, 1978-1990.
CLVIII The distribution of years of highest house price inflation in Britain, 1983 to 1989.
CLIX The distribution of childhood leukaemia in Britain, 1966-1983 (Colour).
CLX Six views of the childhood leukaemia spacetime distribution, 1966-1986.
CLXI Key to cancer types shown by spheres in the spacetime diagrams (Colour).
CLXII The distribution of childhood cancers in Euclidean spacetime (Colour).
CLXIII The distribution of childhood cancers in spacetime 1968-1979 (Colour).
CLXIV The distribution of childhood cancers in spacetime 1980-1991 (Colour).
CLXV The distribution of childhood cancers in Teesside spacetime, from the east (Colour).
CLXVI The distribution of childhood cancers in Teesside spacetime, from the west (Colour),
CLXVII The 1988 district election results: Scottish voting composition tetrahedron.
CLXVIII A schematic representation of four party voting compositions.
CLXIX The 1988 district election results: Scottish voting composition unfolded.
CLXX Four perspective views of the 1988 Scottish district elections composition (Colour).
CLXXI A ray-traced image of the 1988 Scottish district elections composition (Colour).

 

The one- and two-dimensional binomial smoothing discussed earlier can easily be extended to work in three dimensions.
The three matrices shown here give the smoothing factors around a single voxel in three dimensions. After a few passes application of this filter approximates to the trivariate normal distribution.

Below is shown a perspective view of the bivariate normal kernel which could be used to smooth a point distribution. A trivariate function was used to smooth the cancer cases in this dissertation.

The width and sharpness of the kernel is more important than the actual shape of the function chosen, in its effect on the final image. These parameters correspond to the number of passes of binomial smoothing undertaken. Often it is useful to explore the effects of a range of parameters, if possible, as the image changes before your eyes.

Figure 26: Three-dimensional Smoothing

The idea of the equilateral triangle can be extended into three dimensions in a tetrahedron to show the composition of the votes of four parties, amongst a number of constituencies. Position (x,y,z) on the triangle is calculated from the Conservative (C), Labour (L), Liberal/Alliance (A) and Nationalist (N) proportions of the vote as follows:

Position in the equilateral tetrahedron formed then gives the share of the votes in any one constituency, and the distribution of all constituencies, simultaneously:

To understand the distribution within the three-dimensional space it must be rotatable by the viewer. A two-dimensional net of the space can be opened out to expose some of the pattern on flat paper, but a lot of the dynamism of the graphic is lost. It is hard to imagine how this device could be profitably extended to show the composition of the vote amongst five parties. Three dimensions are hard enough to grasp.

Figure 27: The Electoral Tetrahedron

A complex three dimensional structure is sure to appear extremely confusing when forced to fit onto flat paper. These two graphics show some experiments to project the spacetime distribution of unemployment, and the use of tubes to show migration flows across space and time.

Eventually being able to rotate these images is not enough. We need to be able to get inside them to explore and discover what the structure to the patterns may be. For now we can only paint pictures on the outside of what we wish to be able to see from within.

Figure 28: Three-dimensional Structure


 

119 [a] The introduction of time as a third dimension renders many of our conventional techniques obsolete:

Once time becomes a dimension within which activities can be viewed, the map, because of its static cross-sectional view of phenomena, loses its usefulness. [Holly B.P. 1978 p.12]

[b] And we can only just begin to grasp the complexity of four dimensions:

If a fourth spatial dimension cannot be visualized, it is probably because geometry is concerned with relations that can use perceptual and physical space as a convenient image up to the third dimension, but no further. Beyond that limit, geometrical calculations — just as any other multidimensional calculations, such as factor analysis in psychology — must be content with fragmentary visualization, if any. This also means probably putting up with pieces of understanding rather than obtaining a true grasp of the whole. [Arnheim R. 1970 p.292]

[c] How can we begin to take our thinking beyond two dimensions?:
Since it has been proven that the traditional geographic map cannot hold the solution to our space straightening problem, what will? It seems to me that the mapping will have to be on the surface of some object in hyper-space.25 [footnote]25 I have been struck with this notion, unable to advance for four or five years. Also Tobler does not warm to it so I do not trust it, but can offer no alternative. [Bunge W. 1966 p.272]

[d] Too much concentration on temporal change may lead us to forget the underlying two-dimensional structure:

Practically useful though this selective attention to change is, it also has its drawbacks. It makes it difficult to become aware of the constant factors operative in life. This weakness shows up when the thinker or scientist needs to consider agents lying beyond those that display observable change. In physical as well as in psychological or social matters, the constant aspects of a situation are most easily overlooked, hardest to be understood. The characteristics of perception not only help wisdom, they also restrict it. [Arnheim R. 1970 p.21]

120 [a] We are well equipped for visualization, but still often find it difficult:
It is estimated that fifty percent of the brain’s neurons are involved in vision. 3D displays light up more neurons and thus involve a larger portion of our brains in solving a problem. [Dreil van J.N. 1989 p.2]

[b] Animation is almost always required to gauge depth correctly:
Although it is not obvious why it should be, small, rapidly repeated, changes in the viewing transformation are seen as continuous motion of a rigid object — the point cloud. We automatically see the three-dimensional shape of the point cloud, using the unconscious human ability to perceive shape from motion [Marr, 1982]. [McDonald J.A. 1988 p.184]

[c] Other techniques are less helpful:

Stereo pairs are not very convenient for showing results to large groups of people at conferences. So we made movies with the wells and points rotating about a vertical axis. The human visual system can easily see three dimensions with rotating points in space. In fact, the movie was just as effective as stereo pairs in showing the 3D patterns among the points. [Prueitt M.L. 1987 p.5]

[d] Showing a third dimension as depth through motion might also be more effective than the alternatives of using colour or glyphs:
We now recognize the great value of the dynamic aspects of the display, especially easily recognizable rotation. Two aspects, horizontal and vertical, are always before us. We now have a strong feeling that the third aspect which supports these two best is this dynamic aspect of rotation, more useful than stereoscopy, color, flicker, or distinctive characters. [Fisherkeller M.A., Friedman J.H. & Tukey J.W. 1988 p.108]

[e] Interactive graphics allow us to retain and greatly expand upon the advantages of physical models:

A commonly overlooked but important advantage of physical models is that no vantage point is assumed by the mapmaker. The viewer has the option to determine the vantage point thought to be best suited for the purpose at hand. More importantly, perhaps, the physical model can be viewed from successive vantage points to gain some notion of the extent to which the landscape configuration is distorted in any single view. This vantage point flexibility eliminates most problems of a geometrical nature that are normally associated with reading fixed-view maps. [Muehrcke P. 1981 pp.21-22]

121 [a] It is only now technically possible to draw easily the third dimension (by computer):

The limits on what can be done are, as usual, the vision of the user. With continuing improvements in processor speed, display quality and software techniques, the presentation of information in visually arresting forms will become faster, easier and cheaper. To take full advantage of these capabilities 3D representations are essential. [Kluijtmans P. & Collin C. 1991 p.550]

[b] Often we do not have enough information to move out of the plane:
Visualization techniques have released the world from its traditional two dimensional approaches to display and in so doing, have highlighted the three dimensional deficiencies in our sources of data in terms of availability and accuracy. Indeed it is the lack of data that is currently inhibiting the wider application of many of these techniques. [McLaren R.A. 1989 p.13]

[c] Recently the value of a three-dimensional perspective has been realised in other areas of geography:

It is by positioning our geography between space and time, and by seeing ourselves as active participants in the historical geography of space and time, that we can, I believe, recover some clear sense of purpose for ourselves, define an arena of serious intellectual debate and inquiry and thereby make major contributions, intellectually and politically, in a deeply troubled world. [Harvey D. 1990 p.433]
[d] We simplify the study of complex societies by placing them within their dimensions of geography and history:

Added to this is the interplay between space and time. Consider that space too is multidimensional and the tapestry we have the privilege to study unfolds in front of us, a tapestry in constant flux as society packs and repacks time-space and is itself influenced by such changes. But the complexity is sufficiently awesome that we must of necessity start with well ordered deductive statements about the physical environment and build up from the basics rather than, like much of economics or sociology, plunging into the middle of the n-dimensional pool and trying to swim back to the edge. We have to realize that man lives in many-dimensional time and space and is himself multidimensional — until we realize this we will continue to be trapped in the x, y, z and t. And yet paradoxically we may not escape the trap until we are fully aware of the constraints and limits the x, y, z and t impose on action in time-space. [Carlstein T., Parkes D. & Thrift N. 1978 p.4]

[e] Appreciation of spacetime requires us to take an unfamiliar vantage point:

My world is, in the last analysis, the sum total of my sensations. Sensations can be most naturally arranged as a pattern in four-dimensional spacetime. My life is a sort of four-dimensional worm embedded in a block universe. To complain that my lifeworm is only (let us say) seventy-two years long is perhaps foolish as it would be to complain that my body is only six feet long. Eternity is right outside of spacetime. Eternity is right now. [Rucker R. 1984 p.136]
122 [a] Animation provides a qualitative change of dimension:
If, while maintaining the reference axis invariable, we film a collection of minimally dissimilar graphs, each one of which represents a moment in time of the distribution of a characteristic in the spatial sample, and if we present them for viewing at the rate of 24 images per second, the result will be the continuous movement of volume if, as in the present case, the type of representation selected is one of visualized block diagrams in isometric perspective. The successive configurations of this volume manifest, with a qualitatively different expressive force, the basic outline of a particular process of space-time evolution [Cebrian de Miguel J.A. 1983 p.478]

[b] We can create a spacetime block from the frames of an animation;
Try to imagine a picture like figure 137 that encompasses the entire space and time of Flatland. This vast tangle of worms and threads would make up what we call the Flatland block universe. You could think of making a model of the Flatland block universe by standing above Flatland and filming the action as the polygons move around. If you then cut apart the film’s frames and stacked them up in temporal order, you’d have a good model of part of the Flatland block universe. [Rucker R. 1984 p.137-138]

123 [a] That cancer incidence location may be important has even been noted by professors of the theory of art:

Also any sensible enquiry limits beforehand the sort of property to look for. The cancer specialist may not spend time on finding out with which letter of the alphabet the names of his subjects start but he may conceivably be interested in where they were born. [Arnheim R. 1970 p.163]
[b] Childhood leukaemia is one of the most evenly spread of all diseases:

The most notable feature of these maps are the complete lack of any discernible geographic pattern and the similarity of the rates in each of the four cities. While Aberdeen is known to have a background level of naturally occurring radiation, this is not reflected in an increased leukaemia risk. [World Health Organisation 1985 p.186]
[c] Cancers are, in general, not particularly obviously clustered:

The pattern differs somewhat according to cause of death. For some specific causes, the north west — south east divide is clear (heart diseases, stomach cancer, bronchitis) but other causes (e.g. neoplasms) are more variable in pattern. [Curtis S. & Mohan J. 1989 p.177]

[d] The reason for visualizing disease is to lead others to investigate unusual patterns:
While the geographical representation of cancer on maps has been recognised as useful in describing the ‘cancer scenery’ of a particular country (Frentzel-Beyme et al., 1979), the real purpose lies in identifying geographical areas or hypotheses that require more detailed epidemiological study. [World Health Organisation 1985 p.41]

[e] Before we begin looking at leukaemia, we can assume, from past research, that we may not find any discernible patterns:

The limited amount of geographic variation for certain cancers may also provide insights into etiology. Leukaemia rates were nearly constant across the country, similar to the minor international differences that have been reported. This suggests that the role of environmental exposures may be less important or conspicuous than for other cancers. [Melvyn Howe in Blot W.J. & Fraumeni J.F. 1982 p.190]
124 [a] The need to smooth the distribution of incidents, to see structure in this particular data set, was identified many years ago:

If space and time are considered jointly as a three-dimensional block of space-time with co-ordinates of time, latitude, and longitude, and if incidence (occurrences related to the population at risk) is represented within the volume of the block, it follows that there must be some unevenness. [Knox G. 1964 pp.20-21]

[b] Other such work has also been carried out, on a crude scale, in the past:
Poisson probability maps were constructed for leukaemia mortalities in the administrative counties of England and Wales for each year from 1950 to 1966 inclusive. This was done in the belief that much of the variation of these data on the time-space scale of the ‘county-year’ might be ascribed to random variations. Thus the probability maps would appear as filtered data through which only the non-random recordings would appear. [White R.R. 1972 pp.177-178]

[c] A geographical rather than statistical approach is warranted as people are not uniformly distributed:

However, the separation of space and time made in these two chapters produces but a partial understanding of the dynamics of disease processes. Regions do not operate as isolated units and the incidence of disease varies simultaneously in both the spatial and the temporal domains. Indeed, it is the interdependence of time and space which, as Gould (1970, p.44) has noted, ‘allows us to substitute pattern, and therefore predictability and order, for chaos and apparent lack of independence — of things in time and space.’
The linking of time and space units is readily illustrated by the many infectious diseases such as influenza which arrive in one part of the globe from another and are then passed on after an interval of time to further areas, supporting the remark of statistician, Stephan (1934), that ‘data of geographic units are tied together like bunches of grapes, not separate like balls in an urn.’ [Cliff A.D. & Haggett P. 1988 p.169]

[d] The idea of representing incidents by a cone has been previously suggested in two dimensional geography:
It may be useful to think of total population influence in yet another way. Each person’s influence may be represented by a pile of sand, with the height of the pile at the place the person occupies and decreasing away from him. Suppose there is a similar sandpile around the place of residence of every individual. Now let all this sand be superimposed. At any point the total height of the sand will be the sum of the heights of all the individual sandpiles. The total height is a measure of total population influence at that point, and a contour map or a physical model may be made of the entire surface. [Warntz W. 1975 p.77]

[e] In general there is often too little reliable information to allow this approach to be taken:

Thus, if we think of space as the weft and time as the warp of a space-time fabric, then it is evident that the threads are broken in many places. The many countries which have never made returns to WHO sever the weft and the missing observations in each country over time break the warp. Together, the resulting holes in the data matrix make the inter-regional and time-series comparison of morbidity and mortality data extremely complex. [Cliff A.D. & Haggett P. 1988 p.72]
125 [a] The relationship between time and space can be relatively simple in pure physics:

Usually in drawing Minkowski diagrams, one adopts a system of units so that the path of a light ray is represented by a 45 degree line. Light moves at about one billion miles per hour, so the idea is to mark off the space axis in units of one billion miles and mark off the time axis in units of one hour. [Rucker R. 1984 p.151]

[b] But, even in basic physical geography the aspect of the third dimension produces problems:

In the special case of meteorology there are some particular issues and concerns. The small thickness of the atmosphere (relative to its horizontal extent) necessitates a “stretched” z-axis for visualizing weather phenomena. Also, the desire to view the distribution of several variables simultaneously has given rise to a few interesting solutions: one is to portray each variable by a different attribute (color for variable A, height for variable B, iso-valued contours for variable C, etc.); another is to assign different transparency indices to the various surfaces that represent the variables. Both of the above methods result in images that are highly “unrealistic”, illustrating that there may be instances in scientific computing in which the visualization technique may have to transcend “realism”. Finally, the need to associate the atmospheric phenomena to the underlying map and terrain imposes additional display constraints that must be addressed. [Papathomas T.V., Schiavone J.A. & Julesz B. 1988 p.329]

126 [a] Scottish election results have recently become a truly four-way affair:

Two-way contests, which were far and away the most common in 1974, have declined pretty steadily and significantly. In particular, straight fights between Labour and the Conservatives, which were again the most common, are now relatively rare. The increase in Conservative V Labour V SNP contests is a direct function of the larger number of SNP candidates. This also explains why, despite the fall in the number of SLD candidates, the proportion of four-way contests reached a high point of twenty-three per cent of contests in 1988. [Bochel J.M. & Denver D.T. 1988 p.v]

[b] The old two party system in Britain has become three, it could easily split further:
British electoral politics seem to be on the threshold of moving into a new era of permanent three-party competition, which may or may not be accompanied by a radical change in the voting system. For this sort of shift there are no real parallels, and even to sketch a future scenario still seems precipitate. [Dunleavy P. 1983 p.58]

[c] Movement to the apexes of the electoral tetrahedron would indicate that the following had occurred:

This apparent consolidation of strength in the parties’ own territories is an interesting phenomenon; it is unclear on the available evidence whether incumbency of itself gives an advantage or whether parties successfully targeted their campaign effort to exploit existing support. [Bochel J.M. & Denver D.T. 1990 p.vi]

127 [a] It is easy to get lost in all these dimensions:

We need to be able to tell which three-dimensional subspace of the euclidean data space we are looking at. We also need to see how the point cloud is oriented in that space. To satisfy these needs we draw, in a corner of the screen, an object called the coordinate axes. This object was called the dreibein (German for tripod) in previous PRIM systems [Fisherkeller, Friedman, and Tukey, 1975] and is sometimes referred to as the gnomon in the computer graphics literature [Foley and van Dam, 1982]. [McDonald J.A. 1988 p.185]

[b] How reliable are our visual and mental abilities when dealing with this complexity?:

The use of such a system poses interesting theoretical questions: Is exploring data by looking at projections “safe” — if you look at enough different projections of structureless data, will you find structure by chance? If it is safe, is it “effective”? — in what sense can the information in a d-dimensional point cloud be extracted from a few of its 3-dimensional projections? The method, properly applied, appears to be both safe and effective, even allowing for the fact that we do not know the statistical properties of the eye as pattern detector. [Donoho D.L., Huber P.J., Ramos E. & Thoma H.M. 1988 p.119]
[c] Our vision is an effective means of analysis and can be conditioned to become even more so:
Powerful viewing capabilities present a problem of over-exploring data and finding spurious structure. In our own experience, this has rarely been a serious issue, perhaps because human vision is a far better instrument for distinguishing between the real and spurious in scatter plots than commonly believed. On the other hand, we tend to use graphical methods for screening data and obtaining rough qualitative insights, while in-depth analysis and subtle quantitative judgements are left to more formal methods, once their applicability is established by the screening process. Whatever the reason for the relative reliability of visual judgements may be, there still arise occasions when one wishes to have tools for sharpening one’s perception of random fluctuations in data. It has been suggested that data analysts should gauge their eyes every once in a while on some artificially created pseudo-random data, like multivariate normal point clouds [Diaconis, 1983]. We have followed this advice on occasion, and found it helpful in establishing structure as real, and in realizing that the most frequent types of random structure, such as local clottedness and moderate outliers, are usually not of interest to the data analyst. [Buja A., Asimov D., Hurley C. & McDonald J.A. 1988 p.292]

128 [a] Migration patterns have been fairly consistent over time, but do fluctuate:

In 1989, the total number of moves between FPC areas within England and Wales, at 1.76 million, was 6 per cent less than the 1.88 million in 1988 (Table 1). There was little variation in the total number of moves during the years 1979 to 1985, which ranged from 1.50 million (in 1981) to 1.60 million (in 1985). However, in 1986 the number of moves increased to 1.83 million (a 14 per cent increase over 1985), with further increases to 1.87 million in 1987 and 1.88 million in 1988 (Figure 1). During this period, expansion of financial services, resulting in easier access to mortgages, and relatively low interest rates may have contributed to the increased number of moves. Similarly, the fall in the number of moves in 1989 could have been partly due to the rise in interest rates. The total number of moves in 1989 was still 13 per cent above the average for the seven years before 1986. [Bulusu L. 1990 p.33]
[b] The geography of migration alters along with the history:

For a while during the 1970s these counterurban tendencies were operating so powerfully that they replaced the North-South drift as a primary dimension of regional population change in Britain (Champion, 1983). Particularly impressive was the way in which the South East’s population began to decline in the late 1960s, following its rapid growth in the 1950s and the early 1960s. [Champion A. G. 1989 p.122]

[c] Two dimensional thinking often limits our descriptions of three dimensional processes to ripples or waves:
Analysis of flows suggests that population is moving further and further from conurbation centres in the form of a ‘wave’ or ‘ripple’ process. [Spence N., Gillespie A., Goddard J., Kennett S., Pinch S. & Williams A. 1982 p.281]
129 [a] With such complexity it may be better to show only some of the structure:

Rather than trying to simply display the data the idea is to extract certain topological information and to display this. As the authors point out, a jillion little arrows displayed in a cube would not reveal much about a three dimensional flow. [Nielson G.M., Shriver B. & Rosenblum L.J. (eds) 1990 p.261]

[b] Ways of reducing the visual complexity are currently under development:
To visualize complicated three-dimensional flow structures, one requires displays with strong three-dimensional depth cues. From experience, we have found that rendering of opaque or semi-transparent surfaces (such as the interface between two fluids or a contour surface) provides the best results. In particular, when combined with an interactive surface-peeling capability for examination of interior flow detail, surface rendering is preferable over displays of stacked contours or dot patterns. Alternatively the source-attenuation method provides transparency and is relatively easily implemented, but at the expense of strong depth cues; fluid flows tend to look like clouds unless interfaces or other surfaces are accentuated. ... [Hesselink L. 1988 p.474]

130 [a] The old approach was to show three-dimensional structures through two dimensional surfaces:

The second, newer approach to volume visualization is called direct volume rendering, volume imaging, direct voxel rendering, or just volume rendering. This approach maintains an explicit connection between the volume data set and the volume visualization. The algorithms use no intermediate geometric representation. The resulting voxel clouds, perhaps more visually ambiguous, permit users to explore directly the contents of their data. The scientist can slice-and-dice the visualization to explore arbitrary cross-sections of the original volume data set. Viewing is not limited to surfaces, although surfaces are sometimes portrayed. [Herr L. 1990 pp.201-202]

[b] Great claims are made for the future of interactive computer graphics:
Interactive computer graphics is the most important means of producing pictures since the invention of photography and television; it has the added advantage that, with the computer, we can make pictures not only of concrete, “real world” objects but also of abstract, synthetic objects, such as mathematical surfaces in 4D (see Color Plates 1.3 and 1.4), and of data that have no inherent geometry, such as survey results. Furthermore, we are not confined to static images. Although static pictures are a good means of communicating information, dynamically varying pictures are even better — to coin a phrase, a moving picture is worth ten thousand static ones. This is especially true for time-varying phenomena, both real (e.g., the deflection of an aircraft wing in supersonic flight, or the development of a human face from childhood through old age) and abstract (e.g., growth trends, such as nuclear energy use in the United States or population movement from cities to suburbs and back to cities). [Foley J.D., Dam A. van, Feiner S.K. & Hughes J.F. 1990 p.3]

[c] It be advantageous to see how the information looks from the data’s point of view:
It is certainly feasible, and may prove useful, to offer a “biod’s eye” view of the dataset as viewed by one of the biods, using stereoscopic viewing and other “virtual reality” techniques as they develop. [Kerlick G.D. 1990 p.127; "biod" is made up of the words "bird" and "icon"]
131 [a] Rather than wear goggles containing visual displays:
An alternative design would be to fabricate a display on a contact lens and a sensor would detect eye movements as well as head and body movements. This display must then generate the image that the eye would see. Since it would only need to illustrate the small area that the fovea would see, the resolution of the image could be very modest. [Krueger M.W. 1983 p.100]
[b] Statistical graphics is one pioneering area in which artificial reality is being applied:

The amazing computing power now available to data analysts carries with it the potential for new graphical methods — dynamic graphics — that utilize visual input and achieve virtually instantaneous graphical change. High interaction methods represent a new frontier in data analysis and are an important adjunct to conventional static graphics. [Becker R.A., Cleveland W.S. & Wilks A.R. 1988 p.47]

[c] Interactive visualization is very different from animation:
At the extreme end, we find ourselves manipulating plots which change so fast that they appear in motion for all practical purposes. this is the domain of real-time graphics: plots are recomputed and redrawn so rapidly that the visual effect of smooth motion is achieved, and at the same time the user is given the possibility of controlling the process at any point in time. This contrasts with animation, where sequences of views are precomputed, stored away, and retrieved at the time of viewing. Motion graphics can be generated either way, but non-trivial user control is possible only with real-time graphics. The price we pay is that currently affordable off-the-shelf equipment can handle a real-time approach only on fairly sparse pictures, such as plots of point scatters. [Buja A., Asimov D., Hurley C. & McDonald J.A. 1988 p.278]

[d] Yet another revolution is being heralded:

Just as ‘visualization’ has been invented to describe the process of providing more immediate access to very large amounts of data, ‘interactive visualization’ will be ‘invented’ to describe the process of providing more immediate access to the particular features that are of interest to the analyst at particular points in both the spatial and time domains of a given field. [Dickinson R.R. 1989 p.10]