**Implications of WH-questions for self-reflexivity and dialogue (Part #1)**

Introduction

Cyclic patterning of WH-questions: vital cognitive self-reflexivity in a "Kekulé resonance" model

Interrelating the three umbilic catastrophe forms: a "Grail chalice" proto-model

Mnemonic significance

Explanatory frameworks

Symbolism

Toward a new typology of dialogue -- based on the "Grail chalice" proto-model

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As a human response to the perception of a cognitively chaotic situation, WH-questions (when, where, which, how, what, who, why) might be considered to lend themselves to analysis with the tools of catastrophe theory as developed by Rene Thom and others. Thom had developed differential topology into a general theory of form and change of form as a mathematical way of addressing the work on morphogenesis done by C.H. Waddington in the 1950's. Thom's Classification Theorem culminates a long line of work in singularity theory. The term "catastrophe theory" was suggested by C. Zeeman (1977) to unify singularity theory, bifurcation theory and their applications. The crucial theorems rigorously establishing Thom's conjecture were proven by Bernard Malgrange (1966) and John N. Mather (1968). Its essential concern is change and discontinuity in systems (cf Robert Magnus, *Mathematical models and catastrophes*). WH-questions may be considered as triggered and formulated in response to discontinuity -- when habitual adaptive responses to change are inadequate.

It is possible therefore that the set of WH-questions may in some way be mapped onto elementary catastrophes. This is partially suggested by mathematical techniques of conformal mapping where, for example, the "cognitive flow field" around one known shape (as with an elementary catastrophe) might be mapped onto the flow field around a particular WH-question -- preserving the "angles". Conformal mapping notably makes use of complex variables as combinations of real and imaginary numbers. [applet]

This exploration develops aspects of earlier work on WH-questions (*Functional Complementarity of Higher Order Questions: psycho-social sustainability modelled by coordinated movement*, 2004; *Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness*, 2004).

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