Analytic Table of Contents

Chapter 6: Maps of Abstractions

With maps of abstractions, we reach the limit of representation by resemblance, anchoring resemblance as we have to visual form. A display of concepts as words in boxes or circles connected with arrows and lines to other boxes and circles does not resemble anything I see in my mind's eye or in the world. How then do we understand and assess diagrams and maps of file directories and hypertext structures, or of the meanings of words? What sort of space do concepts (or concept-blobs) float in? What is the meaning of up and down, right and left?

Representing abstract things and relations visually inevitably involves metaphor in that things which have no shape (because no body) or spatial relation to other abstract things are displayed as shapes with locations in a two-dimensional plane. To be sure, the commitment to visible form can be quite slight, as in the case of Magritte's concept blobs or other simple kinds of semantic mapping, but some form of "thing" and some way of indication relation of things, especially "connection," seems necessary even to draw a simple diagram. Simple blobs or dots connected by lines ("edges") or arrows do not make much use of the vast repertoire of visual signifiers, and so we develop "icons" and other visual devices to assist in conveying the abstract. If what we want to focus on is structure, however, insipid, visually spare representations prove especially useful, even if they are only weakly spatial, and we will unfold the attractions of simple graph theory in the chapter as an almost neutral way to represent connections of abstract things.


thumbnail of applet in operation

Figure 6.1
Thumbnail of Java Graph applet

As for the abstract things themselves, there is a very strong tendency to represent them with words. Words are, after all, the main means we have for wielding abstractions. And so we find ourselves in another sector of flourishing imagetext. The Java programming language is an especially handy way to treat words spatially. At the left, for example, is a little applet supplied as a demo with the Java Software Development Kit for several years (modified to illustrate the grammar of BODY elements in HTML). If you are viewing this on line, you will see the various HTML elements wiggling about and you can drag them and pin them if you want. Left-to-right and up-down ordering is not significant (that is to say any of the five "daughters" of BODY can occur in any order). What is significant is containment: BLOCKQUOTE and the lists (OL and UL) cannot occur in P(aragraphs), but P can occur in them. ("CDATA" means "character data"--words and numbers). The jittering does help cancel any notion that ordering signifies. The applet also makes a squeeping sound, which does not signify anything profound.

The terms map and territory are often used in semiotics to describe the relation between a representation and that which is represented. When we have other ways to experience and examine the territory--say, by driving in it--we can assess the accuracy of the map. One would not want to call the BODY applet a map because it is not oriented and won't hold still, which is to say it is only weakly spatial. When a "map" is not oriented and the things on it are not in order left-to-right etc., it is hard to call it a map. Without this basis, some scholars studying information and cyberspace "mapping" say that the map/territory distinction breaks down and the map becomes the territory. Martin Dodge and Rob Kitchin, for example, say in their Atlas of Cyberspace that in the case of web sites (among others), "the site becomes the map; territory and representation become one and the same." (p. 3) and they make this claim about a specific site and map later in the book (see below). Questions of accuracy or goodness of fit therefore do not apply. But that cannot be right. If a map of a web site omits some of its pages or connections between the pages, it is an inaccurate or incomplete or bad site map. And so on for other diagrams of abstractions (or events): these are visual/spatial representations of non-visual/spatial things. The "territory" is still territory, just non-spatial territory, and it is reasonable to broaden the sense of map to "visual representation of things." That admittedly leaves very little distinguishing maps from diagrams (broadly used). Perhaps we should say: diagrams become more map-like as they represent objects in a space and the spatial relation of the objects signifies some relation between the objects.

Maps of abstract structures can themselves be very abstract—just points (vertices, nodes), say, and lines connecting them—or they may be based on a visual figure, say a tree or a star burst or even an image of something. These latter maps can be called metaphors for the abstract structure. For example, in a fairly common image of the Indoeuropean Language Family, languages are leaves and language families are branches growing out of the root of a Proto-Language. It is useful to be able to refer to the abstract structure of a "descent" diagram in non-imaged terms (a "acyclic directed graph") rather than to call it a "tree" so that we do not confuse the metaphor with the structure. These abstract analytic terms are usually drawn from the mathematics of graphs (graph theory); we will then return to the rather contrasting notion of maps and begin to grasp what is complex about images of data structures. Then, in the main body of the chapter, we use this analytic framework to look more closely at three particular areas where abstract structures and spaces have been studied and various visual metaphors explored, namely semantic structures, file directories, hypertext generally, and web sites. In the hypertext section, we examine metaphors of rhizome, collage, and cinematic montage developed by literary theorists. The other topics are all major areas in the modeling and especially the visualization of information. We will also touch on the work of Martin Wattenberg, Ben Fry, and Lisa Jevbratt, where the boundary between IT and ART more or less disappears.

Data structures

We accept graph theory's use of the term graph 1 as a special term of art. In common usage, graphs refer to displays of quantitative values of some sort and the relations between them. Graphs in this common usage embody two dimensional quantitative spaces. In graph theory, graphs are not quantitative, and the vertical and horizontal axes are not scales with numerical values. A graph is defined as a mathematical object consisting of nodes (or vertices) connected by edges (or branches). 2 We will call a continuous tracing from node to node along connecting edges a path. As models of particular structures, graphs will often have their nodes labelled, but that is not part of the definition of graph. We will represent them with straight lines connecting black dots. Also, the location and orientation on a page or plane is irrelevant to simple graphs, as are the shape and length of lines that may be drawn to represent the edges. What is fundamental in differentiating graphs are the modes of connection of the nodes of which three are crucial for our discussion:

cyclic directed graph

Figure 6.2
tree with converging paths

Figure 6.3


  • cyclic/acyclic: an acyclic graph has no path which starts and ends at the same node, or which passes through the same node more than once. Directory trees are acyclic, as are all trees, but hypertext site maps are usually cyclic.

  • directed/non-directed: (or ordered/unordered) a graph is directed when its edges are ordered pairs of nodes. In effect the edge is one-way. The streets in a neighborhood constitute an non-directed graph (unless some of them are one-way), but the path of the postman doing his rounds in the neighborhood is directed. HREF links in HTML structures are directed, as are the edges in a genealogical tree.

  • tree: a graph is a tree if all of the nodes are connected and the path between any two nodes is unique. For this to be true, a graph cannot have any cycles in it. In the "converging" graph, the central diamond figure is a cycle (you could trace a path from the top node of the diamond clockwise or counterclockwise around it and back to the top). As a result, there are two paths between top and bottom of the diamond. A tree is an acyclic graph.

Figure 6.4
DAD: Tree Data Structure

DAD defines tree slightly differently:

A data structure accessed beginning at the root node. Each node is either a leaf or an interior node. An interior node has one or more child nodes and is called the parent of its child nodes. More formally, a connected forest . Contrary to a physical tree, the root is usually depicted at the top of the structure, and the leaves are depicted at the bottom.

Here we see a short burst of metaphor, although the metaphors are mixed and less likely therefore to mislead: trees don't usually have parents and children within them and the depicted tree is (usually) inverted. Be that as it may, we can now restate non-convergence as: a graph in which each node has a unique parent.

Turning now to the strongly spatial world of maps, we begin by defining a (literal)map as a surface (or plane) on which some things are represented, and the location of those things on the surface represents the location of those things in the world. A geographical map represents the location of cities, towns, roads, rivers, mountains, etc. and usually (if it is to scale) the distances between the various towns or other things. Distances can be indicated numerically even if the map is not to scale, as is often the case with hand-drawn sketches of "directions" on how to get somewhere. Such maps are prime examples of topographic diagrams. (Kress and van Leeuwen say that only maps drawn to scale are topographic--102) Other things which are maps in this sense are diagrams of an automobile engine, a computer motherboard, and floor plans of apartments. Engines and motherboards are a slight stretch as the objects of mapping since they are too small for geography, but drawn to scale, such "maps" give not only relative position but exact information on measurable magnitudes of location and distance, just as in the case of geographical maps.

Figure 6.5
Topographical UG Map/Diagram

Figure 6.6
2004 Schematic Map of London UG

A map that departs from topographic accuracy is on the way to becoming a (schematic) diagram, such as a circuit or transportation diagram. The standard "map" of the London Underground is a well-known example of either a diagram (so Elkins, Visual Studies, 180-82 and to some degree Kahn and Lenk, 24-25) or a schematized map of point-to-point connections. Harry Beck's "map" was a great success when introduced in 1933 and has undergone subtle modifications (and enlargements, of course) since then, but is still the most popular representation of the system. The London Underground website has many images of this great cultural icon and an excellent Flash animation that shows how the diagram has changed over the seventy some years since it was introduced and also shows the map transforming into one that is topographically accurate.

Kress and Van Leeuwen speak of topological diagrams when what is primarily represented are the connections between things in a system, as in many subway diagrams where stations that are in geographical space bunched together may be spread out for clarity. Sometimes such a topological diagram is superimposed over a schematized geographic map. A completely topological diagram, such as a genealogical tree or a schematic circuit diagram, where space is in no way related to the physical world (i.e. when extension does not signify distance), is not topographic and is scarcely a map at all. Maps anchor down realist epistemology: they are tied to the phenomenal world and can be considered more or less accurate in the correspondence between the things on the map and the things as experienced in the world, as for example by moving about in the world represented in the map.

The Nature of Linguistic Structure- -Halliday, 1978

Figure 6.7
Kress and van Leeuwen, Abstract Topological Map of Language Studies

Our opening move has been to try to distinguish as sharply as possible the topographic, spatialized plane of the map from the blank white space that is the background of a topological display of a data structure graph, but already we have seen cases where the mode of a particular map may be mixed, with topographic accuracy and scale giving way to topological abstraction or schematization for ease of use or reference. A further step in the mixing of modes is a diagram that Kress and van Leeuwen call "abstract topography." This diagram, reproduced at the left, uses distance in a "figurative, yet finely calibrated way" to convey the centrality of particular linguistic pursuits—central, that is, to Halliday's conception of core linguistics as the study of language as a system (which, BTW, is a fairly standard view in the field). Had they not said that, however, I might have entertained the notion that things are laid out for ease of reading, and I am still in the dark as to the exact meaning of boxes and outlines. The dashed oval marks the the boundary of linguistics, I assume, since the terms lying outside it are names of other academic disciplines. The danger of figurative topography is that a particular feature may not signify as you intend for it to. This point will come up several times in the particular analyses.