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Part 1: Ordering distinctions

The Territory Construed as the Map: in search of radical design innovations in therepresentation of human activities and their relationships (Part #2)

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In this approach the point of departure is the concept of a completerange of human activity or concern, namely a totality which is dividedup by making distinctions, whether in a series or nested. Aspects of thisquestion have been explored elsewhere (1), especially the relationshipof the act of distinguishing to cognition.

Two much used representations of such breakdowns are the list andthe matrix:

List: A list does not order the relationships between its elementsexcept in relation to nested sub-lists or in the case of a list in seriesform. This does not imply that such relationships are lacking, merely thatthey cannot be reflected in the list form. Note that a list is in fact a seriesof "points", but it is not necessary to conceive of it as such. The pointscould be represented as areas on a surface. It is only in the matrix thatthe manner in which the total area is cut up becomes explicit.

Matrix: The cells of a matrix may be thought of as sub-areas of thearea representing the totality which the matrix attempts to reflect. The sub-areasare of course positioned with respect to column and row communalities. Itis now interesting to ask why the area is bounded in such a limiting manner.For the rectangular/square form is one of the simplest. It provides a (paned)"window" through which the totality may be perceived. But it raises questionsabout the "wall" in which the window is set and the position of the observerin relation to the observed on the other side of the window.

Now to the extent that the matrix is complete in its coverage, therereally should not be any "wall". The matrix should in such cases ineffect "wrap around" the observer; all is window and nothing isimplict, unexplicated or excluded. If this is not so then the wall shouldbe conceived as wrapping around the observer, possibly with other windowscorresponding to other partial views of the external totality to whichthe observer may turn his attention.

From this point of view the conventional two dimensional matrix raisesthe question of the conceptual significance of crossing the encompassingboundary. It is irrational and unmeaningful because the "wall" is unrecognized.There is almost a flavour of danger of "failing over the edge" as sailorsfeared with the early "flat earth" models (quaintly conceived in the Eastas supported on the back of a primordial elephant or tortoise).

If it is assumed that the matrix is complete, then it should be possibleto represent it without such an arbitrary external boundary. If the externalboundary is eliminated then the matrix takes the form of a closed surface(wrapped around the observer). By what procedure can a two-dimensionalmatrix be so modified and to what does it give rise ?

Consider a 2-by-2 matrix. The simplest symmetrical figure which retainsthe same number of areas is the tetrahedron. It provides four "windows"on the external universe for any observer positioned within.

The continuity of surface area of the three dimensional figure emphasizesany functional continuity between the aspects associated with the individualsub-areas or facets (the "panes"). But at the same time it drawn attentionto the discontinuities between the areas associated with the edges. Theyare not smooth transitions but are marked by sharp angles. It may thenbe asked (if reality is continuous in contrast to our conceptions thereof)whether such a representation suggests others which would reflect a lesserdegree of discontinuity between aspects. And indeed there are, for thegreater the number of symmetrically disposed surface areas ("panes"), thelarger the angle between adjacent areas and the closer the approximationto a continuous surface, namely a spheroid.

However, the greater the number of distinct areas (whatever they signify),the more difficult it is to comprehend the totality with any precision.The patterning of the surface area may be readily scanned but it is onlythrough the "distorted discontinuities" of the most unspherical figuresthat it may be grasped to any degree.

A compromise may be considered however. Even a tetrahedron may be projectedonto a circumscribed sphere. This cuts up the surface of the sphere intofour (spherically) triangular areas. More complex figures would of courseresult in more complex patterns on the surface of the sphere. The challengeis of course to maintain continuity but the realities of the discontinuitiesbetween extant conceptual frameworks may suggest that any such goal isidealistic. Disturbing factors are:

(a) Unequal development: Clearly a particular cell of a matrix mayitself be broken down into more sub-cells than is yet possible with its neighbours.such differences would be reflected in the surface patterning of the associatedsphere. (The intermediate three-dimensional figure would of course be asymmetricalto a corresponding degree).

(b) Gaps: Assuming that the original ma-trix was incomplete to theextent of missing one row, for example, then its "presence" could be indicatedby an appropriate number of (shaded) areas on the surface of the sphere -if their "absence" from the total pattern had been remarked of course.

(c) Zones: Assuming that originally there were two or more unrelatedmatrices which each encompassed aspects of the reality to which an observercould be sensitive, then their representation on the sphere surface wouldgive rise to patterned non-contiguous zones separated by unmarked (shaded)areas reflecting the discontinuity between them. (The rules for projectingthe plurality of intermediate three-dimensional figures onto the surface wouldbe more complex than before). The manner in which these disturbing factorsare handled indicates the freedom associated with this representational approach.Clearly distinct matrices could either give rise to distinct spheres or couldbe incorporated onto a single sphere as non-contiguous zones (case c). Onthe other hand, the possible articulation into many nested levels of a particularcell in a matrix (case a), could be handled by representing the latter ona separate sphere if the totality of its special perspective needed to bestressed. List elements, re-presented by areas (see above), could be disposedaround the surface of a sphere on the basis of a projection of a three-di-mensionalfigure with the appropriate number of sides. If the list was not "complete"then gaps in the spherical surface would be required (case b).

Pattern of contiguity

In a matrix it is clear how the cells relate to one another. Once theboundary is eliminated, however, the question of what is contiguous towhat is raised. Also in a two-dimensional matrix there are two types ofcontiguity (row and column) between cells. But, considering the simpleexample of a 2-by-2 matrix transformed into a tetrahedral surface, thevalidity of juxtaposing particular areas may be questioned.

(a) Enantiodromia: A strong objection that may be made to juxtaposingcells at opposite boundaries of a matrix is that they obviously reflect extremepoles of distinction. And yet there is much to suggest the intimate relation-shipof extremes (4). Whether it is the French phrase "les extremes se touchent",traditional Chinese concepts of the continuous transformation from yin intoyang and vice versa, or the classical Greek dramatic notion of enantiodromia(T.S. Eliot, The Four Quartets: We shall not cease from exploration. Andthe end of all our exploring, will be to arrive where we started, And knowthe place for the first time), in all cases there is a functional continuitywhich the matrix form conceals. On the other hand the matrix itself may bemissing rows and/or columns, in which case juxtaposition would be inappropriate.

(b) Valency: In a two-dimensional matrix, all cells have a valencyof 4 (neglect-ing the boundary question discussed above). The better knownthree-dimensional closed figures may have surface elements of valency 3, 4,5, 6, 8 and 10, although not all combina-tions are possible this implies agrea-ter richness than can be adequately captured by a matrix, and a richnesswhose continuity is maintained in its projection onto a spherical surface.

(c) Linkage lines: In a two-dimensional matrix, the links betweencells of the same row or column are clear. Such strings of areas may alsobe present on the three-dimensional closed fi-gures, although partial stringsare then also feasible.

(d) Matrix projection: Although it is acceptable to portray a mapof the globe as a "matrix" of latitude/longitude cells, despite the distortion,a less distorted representation is achieved by using other projections whichde-part from the rectilinear mode. These clarify to different degrees thetime relationship between the areas as projected from the position of theobserver. It is possible that representation of matrices could benefit frombeing seen in this light.

(e) Complementarity: In some matrices, complementary pairs of cellsare evident. Such complementarily may be even more evident in the symmetryof three-dimensional closed figures, in relation to the points raised in Part2.

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