**1. Polyhedral nets**

Nearly all efforts at organization design are based on structures that can be conveniently represented in 2 dimensions with little distortion. This is notably the case for both organizational hierarchies (as charts), so-called matrix organizations, and the many experiments in networking (to the extent that they are represented on maps at all). Most research into the organization of small groups is based of very simple networks (star, ring, y, and the like).

With the increasing interest in "virtual organization" (especially based on e-mail), there is an emerging case for exploring new kinds of organization and agreement that can only be effectively represented by 3 dimensional structures. Of special interest initially may be those that derive their coherence from having some symmetry around a centre. One group of structures of this kind could be based on polyhedral nets. These can be understood as networks ordered in terms of a single attractor.

It should not be forgotten that much of the graph theory approach to social network analysis is based on combinations of very simple graphic forms. These forms are also to be found in the polyhedral nets. Note that these polyhedral nets are in effect different combinations of the small group structures that have been studied in such detail from the graph theory perspective over the years. The important additional factor however is the manner in which these polyhedral nets "curve back" on themselves. The distinction here is the special constraint on the global structure of a network, whether according to the 3-dimensional form from which it derives, or which it can form if curved into 3-dimensions.

The range of such networks of increasing complexity is best understood from Figures. Each 3-dimensional polyhedron effectively generates a network pattern when all its surfaces are rolled onto a 2-dimensional surface.

The polyhedral nets raise useful questions about privileged communication pathways around a global structure. They also highlight the existence of special local regions bounded by particular communication pathways. They point to places where communication pathways can diverge or merge. The global design can be seen in terms of the appropriate configuration of these individual elements.

**2. Vicious and serendipitous cycles**

The possibility of isolating vicious cycles from the networks of problems that was discussed in an earlier note needs to be related to privileged communication pathways around such polyhedral forms, notably the so-called great circles by whose interlocking they are effectively structured. Hopefully it would be possible to map interlocking vicious cycles in such a way as to determine the nature of the higher dimension construct which they form. It is this construct which would give visual form to the global problematique. Clearly, as the differing sizes of the vicious loops indicated by preliminary searches would suggest, some of these loops are likely to be "local" rather than "global". This characteristic is also evident in some features of the polyhedral nets.

**3. Patterns of sustainability**

These possibilities suggest that the concept of sustainability could well be given a geometric representation based on such polyhedral structures. Such characteristics become even clearer, as Buckminster Fuller has demonstrated, when the negineering constraints on the construction of polyhedra are confronted. Certain features are vital to their sustainability, namely to prevent them from collapsing. The engineering challenge, properly interpreted, may prove to be a very useful metaphor of the social challenge in designing sustainable social systems.

**4. Towards a new language of social organization**

There is the possibility of a new "language" through which policies, programmes and institutional structures can be articulated. In theconventional language, it is typical to see "points" made in a "line" of argument about a subject "area". Such points may be enshrined in policy principles, in programme mandates, or in the departmental structures of organizations intended to implement them. Normally such nested point structures are embedded in linear text (possibly with the aid of an outliner on a text-processor). Typically the only organization structure expressed in graphic form is the hierarchical (pyramidal) organization chart, with all that that has implied.

In what follows, there is an attempt to look at the possibility of expressing points and lines in geometric form on the assumption that a coherent policy or organization takes a coherent geometric form. By extending the language beyond "lines", to "areas" and "volumes", it is hoped to enrich the range of possibilities that can be clearly distinguished and discussed. It is also hoped to introduce an extra degree of rigour into understanding of debate, especially where it is common for people to "go on and on" without contributing to the articulation of a larger strategic vision.

Organization charters (constitutions), policy documents and programmes are the result of considerable effort at articulating and integrating understanding. It should be possible to represent/interpret/translate such structures into geometrical form, especially by adapting any pattern of nested points.

**5. Points and lines**

It takes several points to establish a line of argument

It takes the intersection of two or more lines of argument to establish a point of higher order than those defining any one line of argument. A 2-order point is defined by 2 lines, a 3-order point is defined by 3 lines, etc

Points may be (or be held by): individuals, nodes (in a network), concepts, principles, etc

Lines may be (or be held by): bonds between people, lines of communication, arguments, lines (courses) of action, lines of authority, etc

Duals: point and line structures can be inverted as a standard geometric operation.

**6. Elaboration of structure**

Once a line of argument has been defined, a point in the line may be used as a departure for another line. This process may continue without any effort at closure, namely reference back to earlier points or lines in the debate. The debate is then unconstrained, but does not define **a structure transcending the linear processes of the debate**.

In "developing" a line of argument, references may however be made to other "supporting" points. A distinction should be made between the case where such points form part of the line of argument, and the case where they define the context within which the argument is developed. In the latter case, subsidiary lines link the developing line to the reference points.

A structure may be understood as acquiring coherence when cross-reference between points effectively triangulates the pattern of points and lines.

**7. Areas **

Enclosure of areas: lines of argument may be structured in relation to one another so as to refer back to earlier lines, intersecting them, and thus creating closure. Closure typically takes the form of an area, of which the simplest is the triangle. The range of relevance of the lines is thus constrained relative to that area. Pursuing a line of argument beyond the points at which it borders a particular area makes it "irrelevant" to that area -- typically associated with the phrase "going on" unnecessarily (about a point).

An area is defined by three (or more) lines and in its simplest form these lie in the same plane (whose existence they define)

Areas may be understood as: (co)missions of nodes with a shared concern

Types of area (polygons): triangle, square, pentagon, hexagon, etc

**8. Contiguous areas**

A line may be common to two (or more) areas. The areas will not necessarily lie in the same plane and there is then a form of **conceptual discontinuity** between them. Different subject areas may then be considered "related". In a debate, the accusation of haing "changed the subject" may be understood as having shifted from one area, through discontinuity, to a neighbouring area. Accusations of "missing the point" are of similar significance.

Elaboration of structure: contiguous areas may define a complex surface of which the simplest is the grid pattern. Typical examples are the many 2x2, 2x3, 3x3, etc matrices of categories common to academic texts and to presentations by consultants. More complex is a tiling pattern, or arbitrary structures such as might emerge from network maps.

**9. Symmetry and integration **

Integration of more complex structures is achieved by ensuring the presence of dimensions of symmetry. The integrative potential increases by constraining contiguous areas to define volumes characterized by such symmetry. Coherence and integrity are properties of such volumes. It is through such properties that significance can be attributed. The volume thus becomes a "container" for meaning.

**10. Enclosure of volumes**

Volumes are defined by systematically constraining areas in relation to one another, accepting the necessary degree of conceptual discontinuity between the areas of common concern. This discontinuity is proportional to the angle at which contiguous areas intersect (possibly understood as a torsion on the common line of intersection). The areas must be constrained so as to enclose a volume. Thus a 2x2 matrix of categories might be folded into a tetrahedron, a 3x2 matrix could be folded into a cube.

**Minimal discontinuity occurs when the volume approximates to a sphere.** Here the challenge is to interrelate the many necessary areas required for such a structure

**Maximal discontinuity occurs with a minimum number of areas**. Here the challenge is to constrain the areas, given the high degree of discontinuity. Exploring an area beyond its line of intersection with a particular volume makes it "irrelevant" to that volume.

Types of volume (polyhedra): a distinction must be made between regular and irregular polyhedra. A regular polyhedron is a more idealistic structure and presupposes that all areas are of the same pattern. With an irregular polyhedron, areas may be of many patterns. As the number of patterns diminishes, the polyhedron approximates more to an ideal structure. Buckminster Fuller made the point that all systems may be understood as polyhedra.

Regular polyhedra: a point in a polyhedron is usually defined by the intersection of three (occasionally four) lines which together define three (or occasionally four) planes. A line is thus common to two contiguous areas, whereas a point is usually common to a minimum of three areas. A point can then "tolerate" the discontinuity between three domains.

**11. Fluidity of the language **

One purpose of the language is to channel, and give rigour to, the fuzziness of normal policy debate. This raises the question of when to "code" a sequence of words (whether verbal or in documents) as a point, a line or an area. Often the point structure of a document is a major guideline. This tends to be absent in verbal debate.

It is important to recognize than there are geometric operations which are a useful guideline to the ways in which an existing structure can be "manipulated" during a debate. "Points" may be converted into "lines" and "lines" into "areas", etc. through operations such as projection and rotation. Items can be expandedout of points and contracted back into points. These processes could be effectively tracked with the aid of such a geometric approach to "minute" writing.

There would be considerable advantage in being able to provide a visual display of an evolving debate in such terms. Under the best circumstances participants will in effect be endeavouring to construct a coherent and integrated form so that feedback on its intricacies would prove of value. At the same time, and under less favourable circumstances, some participants will be endeavoring: to extend lines outside the structure, to expand areas into volumes (as execresences on the structure), or to deny the existence of parts of the structure already established.

**12. Developmental and transformational possibilities**

The fluidity noted above is basic to the potential for representing developmental and transformational pathways without loss of continuity.

Growth and extension can be tracked by allowing a full range of geometric extensions of a structure. The principal methods of generating polyhedra are:

Folding the two-dimensional net,

Creating the dual of an existing polyhedron,

Truncating features (faces, vertices, and/or edges) of a polyhedron. This may be understood as flattening the vertex or edge to obtain a new faces, vertices or edges. The reverse process is to pull the centre of a face away from the centre of a polyhedron to obtain a new vertex and hence new faces and edges. Truncating the five platonic polyhedra generates all but two of the semi-regular polyhedra.

Rotating-translation involves: (a) rotation of each face of a polyhedron about its central axis, (b) translation of each face along its axis by a motion that holds the face perpendicular to that axis, (c) maintenance of one connection (whether vertex or edge) between two faces during the roation/translation process, and (d) creation of new faces defined by the voids which appear as a consequence of the process.

Vertex fusing and separating involves: (a) disappearance of faces and edges, (b) fusion and separation of vertices , (c) continuous alteration in face angles, and (d) continuous change in face positions.

Also of great interest is the range of manipulations of the cuboctahedron with flexible joints, as discovered by Buckminster Fuller.