Patterns of Conceptual Integration

Annex: Symbol Generation

Anthony Judge

1. In a system with P terms, it should be possible to identify by analysis (with computer assistance and graphic output) configurations of the P terms (linked by Q relationships), selected in order of their degree of symmetry for a given value of P. Constraints on the maximum and minimum value of Q in each case could also be partially determined in terms of symmetry requirements. Tables of such configurations, without considering symmetry, have been produced by Frank Harary ( 124). The less symmetrical structures, for a given P value, should then prove to be those of less probable value in the representation of the central concept -- although possibly of more value in representing an aspect of it. And indeed the "traditional" diagrams are those which are likely to be prominent in the results - although valuable new ones may well be discovered by this procedure.

2. The same procedure may now be applied for the representation of P-term systems in 3 dimensions. Here the symmetry constraints are more severe. This procedure should preferentially select the regular and semi-regular polyhedra (when P is even) or less well-known structures (when P is odd) (22), (23), (125).

3. The procedure may be made more powerful if, for a given P-term system the structure selected is based upon P equal to:

  • either number of edges of the structure
  • or number of sides of the structure
  • or number of vertexes of the structure (as above)
  • or number of axes of symmetry.
For, in terms of representation, it may be as meaningful to associate an aspect of the P-term system with any such numerable features. The emphasis is on ordering structures in terms of probable iconicity, with the expectation that families of decreasing iconicity will be distinguished by the procedure for a given value of P. Such families may be more valuable for representing aspects of the central concept, although the highly asymmetric structures in any family probably reflect the various forms of pre-comprehension, mix-comprehension or non-comprehension of the concept. Information gaps of this kind in education have been modelled in graph theory terms (126 129) [83]

4. A variation on the procedure in 2 dimensions is to allow each term to be represented:

  • by, the same simple shape (circle, square, etc) and to select syrntnetric configurations in which the relationships are represented either by the points of contact between shapes or from implicit symmetry features (see (22), (30), and (36) on net diagrams for example).
  • or by different simple shapes, each characterising a different aspect.
This procedure should select out many well-known symbols (130).

5. Again this variation may be applied in 3 dimensions using simple solids instead of flat shapes. As mentioned earlier the possible configurations are then governed by well-known packing constraints (22), (23).