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were behaving linearly and smoothly whereas within this region we observe global and dramatic changes via a forced discontinuity in what Thom called a catastrophe and others use related words such as bifurcation or phase transition. The transitions from painful fatigue to running rage and then to ecstatic transcendence feels like the gifts from two kinds of Gods, the first, bearing the righteous lawfulness of the Old Testament, the second bringing the empathic forgiveness of the New Testament Jesus. Catastrophe and bifurcation theories predict and keep track of these transitions using mathematically describable changes in global characteristics of the “motion” using technical descriptors such as eigenvalues, germs and jets.

Thom taught me my first catastrophe, called the cusp, in words during our late afternoon walks along a shadowed green wooded path on the grounds of the Institute des Hautes Etudes, outside of Paris. My homework consisted of trying to visualize his verbal descriptions. It was not until weeks later that he drew the geometric object being discussed on the blackboard. With eyes twinkling and in his provocatively playful style, he said,

“Imagine an empty rectangular box with the front edge of its roof buckled into an `S’ and the back edge, an unfolded, left-to-right gradually rising simple smooth curve. If one moves the causal force from low to high, from left to right along the back of the box, the changing effect (represented by height) would be smooth; moving from left to right in the front encounters a sudden drop off at the S shaped buckling, a discontinuity in roof height indicating a discontinuity in effect. The energy equivalent height of the roof graphically indicates the amount of result. The roof is the manifold upon which the result of causal change is portrayed. The two dimensional floor of the box represents a graph of the two causal parameters, the increasing amount of normal factor going left to right along the `x’ dimension, the increasing amount of splitting factor (taking one from the back to the front to the region of the buckling) going back to front along the `y’ dimension.”

He gave me some examples of systems that showed cataclysmic changes in effect from smooth changes of normal and splitting factors. About the onset of a war: “At the back of the top surface of the box, the manifold, the normal factor increasing from left to right is the amount of the perceived threat. The splitting factor

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