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Let me get right to the point. A «spatial turn» has recently been announced in Cultural Studies: privileging mapping.  There is a difference between the culturalogical notion of «mapping,» which refers to cartography, and the technological and mathematical use of the term, which means «mapping one content on another.» Do the respective media in which mapping takes place make a difference for the notion of mapping? In digital operations, «space» is nothing but a metaphor. There is space in the physical world, but a map is just a model of space, not space itself. And this model itself is not spatial, but logistical: anaesthetic; not for physical experience, but for cognition, for our mental computing. Why then is there such an abundance of metaphorical representation in cyber‹space›? This is for users, i. e. human use, because the human capacity for data navigation is bound to spatio-temporal metaphors, whereas communication between computers, operating in discrete states, does not need mapping metaphors at all.
I want to insist: the term «cyberspace,» coined byWilliam Gibson in his novel «Neuromancer» in 1984, is less about ‹space› (which is a metaphorical term here) than about cybernetics (the fact that the prefix «cyber-» is derived from Norbert Wiener's «Cybernetics» is a kind of forgotten media-cultural fact today). Cyberspace is not spatial but topological; let us refer to it instead as «cybertope» (analogous to Michael Bachtin's neologism of «chronotope»). Cyberspace is not cartographic but mathematical, e.g. n-dimensional; each 3D navigation on interfaces reduces the n-dimensional potentiality to spatial metaphors. Maps always occur on flat surfaces, depending on their material support for inscription; the crucial quality of digital calculation though is its potential n-dimensionality. The mapping metaphor is seductive but misleading when it comes to computing. And let us remind ourselves: mapping was not invented for aesthetic, but for military and agricultural (that is literally: culturalogical) reasons. Cybernetics, on the other hand, is actually opposed to spatial, cartographic metaphors, since it is linked to shipping. The ancient Greek «kybernetes» (governor) does not steer vehicles on territorial ground, but rather a vessel on the liquid
element (free floating—be it the open sea or the open sky, and only in this case are we dealing with «l´espace strié de type métrique et pulsé» as described by Deleuze/Guattari in «Mille Plateaux.»)  Let us not forget that Norbert Wiener developed his 1948 theory of cybernetics in order for artillery on the ground to predict hostile aircraft movement. The trajectory of a ballistic missile (as described in Thomas Pynchon's novel «Gravity's Rainbow») is no longer a function of space, but a function of numerical tables; in fact, a missile corrects its trajectory during flight with the aid of the numerical feedback of information.
Topological operativity is not multi-sensual but strictly mathematic, more prosaic than poetic. Thus the true «mapping gaze» (Buci-Glucksmann) [LI] is a theoretical gaze. Though the very term «theory» originally stems from theatrical vision, in mediated contexts it means abstraction, and in our context abstraction means mathematization. Just like the Renaissance perspective was for visual arts, mapping and cartography were the first step of abstracting from physical space, representing it symbolically and according to laws of geometry. But the whole modelwas still based on referentiality, i.e. maps representing real or imagined territories. In fact there is a fundamental difference between the representation of geographic space and the spatial visualization of data (just like between the «Media Art Net» information system and genuine Net-based art). It makes a crucial difference if media art is represented, indexed and mapped online or if the Internet itself becomes the material for artistic work, such as art produced by the HTML code itself using ASCII symbols of the source code of homepages. 
Mapping is creating visual metaphors for representing information  , a toll on the reduction of data complexity—which in the digital age means mapping the alphanumeric into the visual. The «Aspen Movie Map» developed in 1978 by the MIT Architecture Machine Group headed by Nicholas Negroponte was still based on input taken by photographs of places in real Aspen, Colorado; thus the resulting interface is a metaphor (or rather simulation) of motion in real space. Jeffrey
Shaw's «Legible City» builds urban architecture using letters, thus rendering spatial data into symbolically readable data; however, this aesthetics of navigable space is based on the idea of an ethics of the virtual, i.e. virtual worlds (should) keep a memory of their corresponding counterpart in the real world, recoding this aura.  But if anything at all of virtual space is rooted in the real world, it is the materiality of the computing device itself;  3D spaces always autopoetically reflect the graphic power of SGI workstations. If there is space, then it is in hardware architecture. Is the wiring of microchips already «mapping»? Flow charts in programming surely mean diagrams; but the autobooting mechanism, which allows computers to start at all, is «burnt into silicon and thus form[s] part of the hardware.» 
An alternative model has been developed by the media art group Knowbotic Research under the idea of non-locatedness online. Their «IO_dencies» project on rebuilding a Tokyo neighborhood is a data-clouded challenge to the mapping paradigm. Truly cybernetic architecture, thought from within the medium of computing and not referring primarily to interfacedesign, no longer represents anything—or rather, for human senses only, remetaphorized in interface design. The digital is neither necessarily coupled to spatial or humanoid metaphors nor is it bound to verticality or horizontality as is the human body (the theatrical paradigm of «computers as theatre,» according to Brenda Laurel). Lev Manovich points to the difference between isotrope space (which is mathematical, Cartesian, logical) and anthropological, relative notions of space (see «Wegzeit» by Dietmar Offenhuber, below  ); most cyberspatial interfaces or computer games try to undo this difference. From a media-archaeological perspective, why not accentuate this man/machine difference in interface design, arriving at a media culture which at last acknowledges that the logic of computing is not alien to human intuition, but rather dis-covers (with Alan Turing) human cognition as a mathematical machine itself?
Cyberspace is fundamentally spaceless, or rather a media theorization of space. It is purely relational, thus not representative in terms of mapping. Geometries are
iconic only for humans, who as underlined by Martin Dodge and Rob Kitchin depend on tangible referents for orientation.  According to William Gibson, cyberspace is a metaphorical expression for networked computing, governed by the so-called «matrix,» a Cartesian datascape. Essential for Cartesian grids is the fact that mnemotechnical images, common since antiquity, have been replaced by numbers on the vertical and horizontal axes. «Mapping» therefore should be taken in its mathematical, topological sense, in order not to confuse imaginary (iconic) with symbolic (indexical) operations in cybernetic aggregates and physical networks. «Any map could be a voyage in thought connecting a passage and a territory» (Christine Buci- Glucksmann) [LI], but this nice metaphor mistakes maps for what they cannot be—free floating. Only computing can actually perform trajectories in n-dimensional calculation. The really relevant maps have always been hidden, kept secret by the power agencies—like the source codes hidden behind cyber-spatial interface metaphors of ‹navigation.› Media-archaeologically seen, cyber‹space› is not about images, sounds or texts, but about bits;thus the cartographic or mnemotechnical approach is misleading. Virtual mapping is a function of mathematical topology. Cyberspace is not a new place of memory, but the transformation of lieux de mémoire into nodes and nets. No longer bound to physical places, the virtual addresses exist in mathematical topologies only. 
Digital mapping opens new horizons for search operations in the «Media Art Net»: mot just addressing and linking images and texts using alphabetical addresses, once again subjecting images and sound to words and external meta-data (the archival classification paradigm), but addressing digital images down to the single pixel from within, in their own medium, allowing for a random search (an apparent disorder as an alternative economy of information—generating the unexpected). Instead of just mapping data banks, computers can be used to make new types of maps that were previously impossible, such as scans of objects organized into networks based on their formal similarity. As long as
the «Media Art Net» defines itself as mediation («Vermittlung») and purely verbal contextualization of items in the Internet, according to McLuhan's law it mirrors an old media in the new one (the content of a new medium is always the previous one): it conservatively sticks to the archival paradigm, which is based on inventories and linkage structures between files. Any archival linking is a freezing of relations, even if it opens up alternative links. Let us instead try to conceive digital data an-archivally.
Search engines as agents of mapping (the generative archive) confront us with the difference between browsing and searching. Crawling through the web using hyperlinks is different than targeted search operations. Maps without textual annotation are of no use to most users. When it comes to the text/image relation in searching-as-mapping, the technical option is to no longer exclusively subject images to the alphabet. Increasing use is now being made of graphical searching devices on maps (as opposed to the subject classification of «Media Art Net», which still follows the encyclopedic order). The ‹iconoclastic› option is a random audio-visual search for links (by similarity, bydigital association, connectivism, fuzzy logic). Which is the model: the ancient ars memoriae (the spatial order of images) or the archive (non-visual, logistical, data-based)? Early catalogues and inventories of works of art in the seventeenth century ordered works of art not according to intrinsic («history of art,» schools of painters), but purely external values: placed in real space (on the walls of a collection) according to formats—pure storage economy. Is the CD-ROM a contemporary form of such non-linear cataloging?  «Media art must be transmitted in a multi-media way» (Dieter Daniels and Rudolf Frieling in their editorial [LI]), since it is by definition timebased or process-oriented. It belongs to this specificity that digital audio-visual data can be addressed (= controlled) down to the smallest sample/bit/pixel; thus resulting in new search options with various access points.
Thus we arrive at literal bit mapping. «Image files contain basically a bit map; that is a long string of bytes […] each of which describes an individual pixel of the image. ‹Metadating the Image› is a subset of data knitting. We can tease out some indications of image structure and semantics (for instance, we can find all
edges in a bitmapped image)» (Lev Manovich)  , or rather «infra-dating»: extracting data from within the image or the sound file. This allows for the administration of bitmapped data objects which are non-linearly related among themselves, while at the same time being parts of an arbitrarily complex network of transcribed information.  Between the classical text-image dichotomy we discover the bitmapped token.
The essential feature of networked computing is its dynamic operativity. Michel de Certeau already separates maps and tours;  space then is an intersection of mobile elements, whereas tours are literally discursive series of operations. In electronic, digital media, mapping means dynamic movement «in flight» as a new quality. Classical maps are neither interactive nor time-critical (feedback). The spatial, that is archival, order might thus be accompanied by «mapping time,» i.e. mapping temporal, dynamic, processual operations, which distinguish traditional from electronic works of art. Static maps differ fromdynamic maps in virtual space, since dynamic maps can be automatically updated (truly «dated» maps). Trace routers are not spatial, but temporal scouts.  Mapping time though is not mapping at all, but sequentializing, time-critical as is a/synchronous communication online; every spatial representation of this process can only be metaphorical—or is it able to show a temporal sequence at one glance?
Whatever is linked to images—to visuality as the privileged channel of information since antiquity—privileges spatial perception, as far as Gotthold Ephraim Lessing was right when in his 1766 «Laokoon or on the Limits of Painting and Poetry» he media-semiotically declared that visual arts belong to spatial perception, whereas linear arts belong to temporal perception. In his reply to René Magritte («This is not a pipe»), Michel Foucault insisted as well on separating the visual from the readable. Buci-Glucksmann interprets maps as readable and visible at the same time, as an «image-index»—a hybrid?  The very term territory, which serves as a referent for any map, privileges spatial rather than dynamic perception (though the Latin term «imperium»
did not originally mean an empire, a territory, but rather the extension, the reach of commanding powers, depending technically on the ways and channels of communication, streets, postal systems). Thus the «mapping gaze» neglects time-based media, that is the von Neumann computer architecture of sequential data processing, dissimulating its operative, temporal character in favor of spatial images, maps. Furthermore, the cartographic paradigm neglects acoustic options of navigating data (a plea for sonification, with sound itself a time-based aesthetic). The question remains: Are there objects which are not mappable at all? What about «mapping time»? What about the option of genuinely «mapping sound» by means of acoustic signals; this is symptomatic for the supremacy of the optical over the sonic. Acoustic mapping would relate to «mapping time.» Sound is already used to create an acoustic shape of the Internet, an alternative to icons. Since the Net is dynamic, processual, it can more readily be mapped acoustically than visually. Heinz von Foerster once named «the enormous task of mapping an almost indefinite variety of signals into a few modes of behavior.» 
In his essay on «other spaces» (heterotopics), Michel Foucault declared that the nineteenth century had been dominated by its concern with time, whereas the twentieth century was concerned with space. The twenty-first century, though, will be concerned with topologies: mathematics, vectorization of space, coupled with the materiality of cables, circuits, processors. Foucault finally acknowledged these implications of computing.  Thus we arrive at n-dimensional data clouds: a complete geometrization and mathematization of all previous metaphorical «archaeologies of knowledge.»
It was Immanuel Kant who once identified the «mathematical sublime»; nowadays data visualization corresponds to the sublime, making visible the invisible of digital data processing (Lev Manovich). Art works like «Polar» by Carsten Nicolai and Marko Peljhan (1991) refer to the changing process of invisible information with an idea stemming from the novel «Solaris» by Stanislav Lem (1972) and the film adaptation by Andrej
Tarkovsky. Here, the «Ocean,» a sea-like substance on the unknown planet Solaris, reflects human emotions, desires, and thoughts. And the media art work by Lisa Jevbratt performs Info-Aesthetics, the spatialization of data flows in the Internet. Today's revival of mapping means «real time dynamization of data structures including the ability to reorganize information in relation to the user's choice and interest»  —the quality of information through algorithmic calculations, mapping movements, not local places (time-slicing by video-recording). This leads to the interactive map, the temporal configuration, the individualization of maps (which is somewhat reminiscent of the Situationist «psycho-geography»).
On a panel called «Mapping the World» as part of the media arts festival Transmediale 03 in Berlin (February 2003), Dietmar Offenhuber explained the permanent decision between representation (iconic links to the real world) and diagrams (which is truly computer aesthetics, numerical abstraction instead of reiterating previous media like maps) for works of data visualization (mapping). Offenhuber's work «Wegzeit» LI displays the geometry of relative distance, anasymmetric, non-Euclidean geometry, ambiguities in time/space distances; this represents relative spatiotemporal dynamics itself (time-critical performances), e.g. phases of traffic within the switching of traffic lights in cities. Man himself is a moving point ruled by laws and rules in traffic/a discourse/a network stronger than himself (Foucault). Svetlana Alpers, writing on «the mapping impulse in Dutch Art,» demonstrates that in Vermeer´s «Art of Painting,» in which a map of the Netherlands figures in the background, this map becomes less rhetorical, more «descriptive»—a function of geometrical instruments in mapping.  In representing imperial spaces, mapping has always been symbolically driven—until mathematics took over.  The 1546 edition of «Cosmographia» by Petrus Apianus contained examples of map design that show how very close by that time European cartography had come to achieving statistical graphicacy. But there was not yet a quantitative abstraction of placing a measured quantity on the map's surface at the intersection of the two threads instead of the name of a city  ; there is a long tradition of cultural resistance of the imaginary as
opposed to symbolic operations by numbers. Non-referential mapping means setting objects into relations (the French physiocrat François Quesnay provided the first such map in his 1758 «Tableau Économique»). «Maps construct—not reproduce—the world.»  Let us then take the digital paradigm as an analytical basis, where mapping as a dynamic operation substitutes fixed archival classification.Is mapping a metaphor compared with mathematical topology? Let us go for the precise technical notion of mapping, which derived from cartography has been transferred to mathematical topology and now suffers from a metaphorical reentry.
Are we talking about mapping or about diagrams?  About cartographies or the rhizome (Deleuze/Guattari)? The Cartesian «cogito» is based on a grid—a grid that is the rationale of the modern state. The mapping impulse corresponds to the very occidental impulse of overview, surveillance, data control. Alphonse Bertillon  once sought to embed the photograph in the archive; Francis Galton LI sought to embed the archivein the photograph. Both mapped out general parameters for the bureaucratic handling of visual documents.  Mapping serves power.
In «Hermes» (1964), Michel Serres describes the communication network «Penelope» as a combinatorial topology, with non-linear, non-hierarchical net-like diagrams involving feedback options. In Buci-Glucksmann´s text «The cartographic view of the virtual» [LI], what is so attractive and puzzling at the same time is a certain undecidedness whether to strictly differentiate or confound maps and diagrams. A diagram is not «already in itself a map or an overlay of maps,» but a genuinely different epistemological tool. The Leibnizian link between numerical continuums and morphogenetic forms fundamentally differs from representational tools like maps. Flashes and vectors, dynamic forces can be represented in painting (like Paul Klee did), but they can only be operative within the calculating machine—which brings us to the notion of «diagrammatic iconicity» coined by Charles S. Peirce (who himself was a practicing cartographer). Peirce divides an overall class of «Icons» into three sub-divisions: «Image,» «Diagram,» and «Metaphor.»
Maps, in terms of this analysis, would be diagrams, not reproducing the «simple qualities» of their referents, but representing «the relations […] of the parts of one thing by analogous relations in their own parts.»  «The Diagram not only represents the related correlates, but also—and much more definitely—represents the relations between them, as so many objects of the Icon.»  For Peirce, the pure Diagram is merely designed to represent and to render intelligible the form of relation: «Consequently, Diagrams are restricted to the representation of a certain class of relations; namely, those that are intelligible.» The diagram equals the statistically based map, as opposed to representation based maps.
The notion of mapping (unless used in its strict mathematical sense as mirroring a given set of data onto another) is associated with metaphorization, visualization, anesthetization, whereas the media-archaeological idea of the diagram is conceptual rather than visual, topological rather than geographical, non-narrative (data-based) rather thannarrative, connective rather than spatial, concerned with code (software) rather than images, numbers rather than sensual perception. The visual display of quantitative information (or quantifiable information) is a by-product of Cartesian modernity; the use of abstract, non-representational pictures to show numbers is a surprisingly recent invention (Edward Tufte). We are still using spatial metaphors for the representation of cybernetic processes. Why not teach the user to apply computation directly to linking data like it is already practiced in data-mining? Even in virtual communication, there is still hardware, not to be reduced to symbolic operations; the Internet topology is a structure of physical (not virtual) links between nodes.  The notion of ‹net,› read against the emerging Internet, has made a career as a metaphor. But hidden behind this romantic surface is the real stream of data: mapping Internet protocols, depending on IPprotocols. Should the Internet be physically or logically visualized? There is a difference in representations of the Internet as a communication tool (logical nodes) and a mapping of the Internet showing its physical nodes (cables, etc.). Internet
engineers focus more on logical connections than on questions of human communication; a map of such connections is not a spatial notion.
Beyond the cartographic metaphor, mapping means the setting-into-relation of data.  Digital navigation systems (GPS) begin to substitute the fixed traditional maps; dynamism replaces rigidity. The navigation metaphor itself implies the sea, which defies cartography (no territory; rather random). Let us redeem the notion of «mapping» from the cartographic metaphor and instead remathematize it (the media-archaeological perspective).