Projects Overview (Explanations)
Transformative Approaches Project (Explanations)

## Patterning: Global configuration from number theory

### Transformative Approaches Project

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A unique computer software package, Chryzode, has been developed to enable the visual exploration of complex patterns generated using all the riches of the mathematical theory of numbers as the relate to projective geometry. It uses hitherto little explored mathematical properties to handle the presentation of large amounts of information in a comprehensible manner. The package is being developed by Jean-Pierre Sonntag (Résonances Transdisciplinaires, Les Gîtes, Revest des Brousses, F-04150 Banon, France). Its use in encouraging the interest of children in mathematics is currently funded in part by the French Ministry of Research.

Examples of the results of this programme are given on the following pages. Whether in colour or not, their significance is more than purely aesthetic. The user-friendly program enables users to explore the manner in which numbers generate patterns of many different kinds. Many of the patterns reveal hitherto unrecognized forms of harmonic relationship.

Like the management tool NETMAP, described earlier (see also the inside covers of this volume), Chryzode generates patterns of relationship by positioning nodes on the circumference of a circle. In the case of NETMAP, the positions can be determined by a clustering algorithm to facilitate comprehension. The patterns generated by linking these nodes may or may not be comprehensible as whole, as would be expected with an essentially empirical approach.

In the case of Chryzode, however, the subdivision of the circumference of the circle is based on a whole range of options in number theory. One of its innovations results from the utilization of graduated circle onto which are written the numeric results of arithmetic operations such as addition, multiplication, division and the like. The name Chryzode given to these graphic images by their inventor derives from Greek terms for writing in gold on a circle. The images are generated from combinations of points and lines based on a circle. Despite the number of relationships between nodal elements, which may become a very large number (as with NETMAP), a sense of coherence is nevertheless maintained through an emphasis on globality. This globality emerges from the pattern of relationships rather than through being artificially imposed upon it.

It is tempting to argue that what is required is some kind of marriage between the approaches of NETMAP and Chryzode, between the empirical and the abstract, in responding to the needs of comprehending the complexity of the world problematique in new ways, as well as to the complex relationships in the information of human values and human development in this volume. One of the advantages of Chryzode, especially in colour, is that it offers clues to ways of using the interference effects (in the form of moiré patterns) of large number of relationships to provide understanding at a new level. As with NETMAP, colour values can also be varied to ensure more striking displays.

The developer of Chryzode sees it as indicative of the emergence of a veritable cybernetic, permitting both the visualization of logical abstractions as well as the creation of new aesthetic forms. Many of the forms are basic to hitherto unvisualized developments in a range of sciences.

The programme offers a means of configuring line items so as to bring out significant interrelationships between groups of lines. In every case the emphasis is placed on the pattern of line items as a whole, whatever the degree to which some line items contribute mainly to detailed aspects of the overall pattern.

In contrast with more simplistic approaches to the conceptual challenge of managing sets of line items, such as in nested hierarchies, the CHRYZODE package can handle a wide variety of configurations with very large numbers of line items.

Despite the number of line items handled, a sense of coherence is nevertheless maintained through an emphasis on globality. This globality emerges from the pattern of line items rather than through being artificially imposed upon it.