We will not discuss here the beauty of biological forms, but two kinds of “catastrophes”. These are not necessarily disasters; in the Catastrophe Theory of mathematician Thom, the term “catastrophe” indicates merely sudden discontinuous transitions. From the point of view of our ethical valuations, these could be either good or bad.

According to Thom, there are only seven “elementary catastrophes”: fold, cusp, swallowtail. butterfly, elliptical umbillic, hyperbolic umbillic, and parabolic (Zeeman).

Not all change in nature (as in mathematics) is smooth or continuous; the sudden jumps may be due to accumulated stresses finally breaking through (as in hysteresis or in revolutions), or be as capricious and spontaneous as quantum jumps or bursts of radioactive decay. Yet the capriciousness and spontaneity may be only apparent, due to our ignorance of the underlying mechanism. Possibly all is hysteresis.

In mathematics, sudden breaks in curves occur outside catastrophe theory. A hyperbola is a discontinuous curve, and so is the graph of the tangent of an angle. Some would say that these curves jump through infinity, whatever that may mean. If plus and minus infinity meet somewhere in the nether world on the back side of the universe, then these are not really discontinuities. But let us stay with the discontinuities that belong to catastrophe theory.

Catastrophes may be good or bad. The event could be a sudden conversion by a beatific vision of God (as a delayed reaction to a previous accumulation of smaller insights that stayed just below the threshold of consciousness), or a sudden total breakdown of a highly technological communications system, because of an accumulation of overload and smaller errors. It can be a “peace crisis” as well as a war crisis” – an unexpected sudden change in the “normal relations range” between two nations. For example, on November 9, 1989 the Berlin Wall came down. The uncertainty was overwhelming; if West Gennany had come to the aid of the East-Gennan revolt, we could have jumped from a peace crisis to a war crisis in an instant. Uncertainty is chaos, and what new order emerges from the chaos is up for grabs.

Among the mathematical models of catastrophe, only the fold and the cusp can be pictured in the three dimensions of ordinary space. The swallowtail, the butterfly, and the others can be conceptualized only as projections. It is rumoured that some highly talented mathematicians can think directly in 5 or 7 dimensions, but I would not know anything about that. The 7 catastrophes are listed in Table 1 arid illustrated in Table 1 (both taken from Zeeman’s article).

A fold is a simple hysteresis curve, like the graph of a cubic equation with a minimum and a maximum. It can be pictured in two dimensions on a sheet of paper. A cusp (the type of catastrophe usually discussed) is a folded curved surface (rather than a folded curve in a plane) and can be pictured in three dimensions.

Hysteresis means that, as a point moves along one branch of the curve or surface, it persists on that branch even though there are two possible admissible values when the folded part is reached; the jump to the other curve or surface occurs only when the edge is reached. In nature or society, hysteresis (delayed change) is due to inertia: staying with the existing system or the status quo beyond its range of unique applicability. Such a delay means that change, when it necessarily comes because the status quo is now totally untenable, will be sudden and not gradual.

A consequence of hysteresis in natural systems is that the state of the system depends on the direction of approach, i.e. on past history. This means a lack of reversibility, or at least a lag in reversibility. On a fold or a cusp, approach from one end differs from approach from the other end. Physical examples are the magnetization/demagnetization of iron and the stretching/relaxation of rubber. A social example might be the nationalization/privatization of enterprises.

Path-determined system states are quite common. Partly filled glass capillaries present a different picture when the liquid is being introduced (a steadily advancing meniscus) and when it is being withdrawn (walls remain wetted while the centre becomes empty). A growing organism differs from an aging one. For the universe (if it is closed), the Big Crunch will be qualitatively different from the Big Bang.

In a fold there is one independent variable that varies smoothly, and one dependent variable that exhibits jumps. In a cusp, there are two smoothly varying independent variables and one discontinuous dependent variable. A cusp has been used to model such phenomena as the following:

1. Sudden flip- flops in political regime in situations of intense ideological polarization (bimodal distribution of left and right opinion with hardly any middle or political centre) and high intensity of political involvement and passion. At lower values of polarization and involvement, regime change can be slow and gradual; but with increasing intensity of both independent variables, the system enters a region of “bifurcation” where the catastrophic cusp model becomes operative.
2. Phase change in physical systems, e.g. ice to water or water to steam, where the two independent variables are temperature and pressure.
3. High technology and interconnectedness producing sudden breakdowns, e.g. the New York black-out. These are often unpredictable, because they are quasi-chaotic. According to Bereanu, the closely linked US-Soviet nuclear system could “self-activate” to an accidental nuclear war in this manner.

If a third independent variable is added to the cusp catastrophe, we get a swallowtail, so called because its three-dimensional projection has the shape of a swallowtail. There is still only one independent variable. The third dependent variable is called “the splitting factor”, because it accentuates and emphasizes the inherent antagonism which the two independent variables of a cusp can produce. The bifurcation region is bigger and the jumps steeper if the splitting factor assumes high values.

The butterfly has four dependent variables and one independent variable. The new fourth dimension is a compromise-seeker to alleviate the stress introduced by the splitting factor. The projection looks somewhat like a butterfly, but this is really a projection of a projection, because the full butterfly catastrophe has five dimensions. The main feature is that, at the crossroads of the bifurcations, there is a pocket of stability — like the eye in the middle of a hurricane, or an oasis in the desert, or the Ark bobbing in the Flood waters. Thus a highly stressed and split system, subject to multiple catastrophes in the form of sudden jumps, can find respite in the midst of tumult, a shelter from the winds of change.

The butterfly pocket is a stable steady state, but a narrow one, like a rock ledge in raging waters. Stay put and you can survive, or at least rest for tomorrow’s struggle. The pocket could be the model for the homeostatic state of living structures, temporarily stable while rapid flow-through of matter and energy is managed, but always on the edge of falling into chaos.

We spoke of the jumps as “sudden”, mentally collapsing the time duration to zero. But possibly. even an electron cannot jump from orbit to orbit instantaneously, without consuming time, though we have no unit small enough to measure or conceive it. During the jump, the system is in limbo, neither fish nor fowl, neither up nor down. In a stressed region of many and frequent jumps, we approach a state called “chaos”, highly unstable and supersensitive to small changes. (Interestingly enough, this supersensitivity in weather forecasting has been called “the butterfly effect”, though this has no connection with our present usage of the term.) Presumably some regions of the swallowtail hypersurface are approaching the state of chaos. And yet, like a miracle, a region of order emerges from the midst of chaos in the butterfly catastrophe.

Is the swallowtail like the escalating fluctuations that Prigogine (Jantsch) postulated, fluctuations that can destabilize a dissipative structure system and dash it to bits — or, as if by a miracle, cause it i: snap to a higher, more complex, more ordered configuration? And is the pocket in a butterfly an example of such a more ordered, highly evolved system?

Did organisms evolving on Earth have to go through swallowtails of chaos each time a new species was formed — i.e. many millions of times? Was a butterfly pocket occasionally found, but usually not? Is breakthrough very rare compared to breakdown?

If indeed humanity is in a major transition now, the swallowtails and butterflies contend, andthe search for an Ark is intense, even desperate. Look for a lighthouse to point the way in the dark. Take our one-in-a-million chance — there is no other way.

Table 1
Catastrophe	Control Dimensions	Behavior Dimensions	Function	First Derivative
Fold	1	1	1/3x 3 -ax	x 2 -a
Cusp	2	1	1/4x 4 -ax -1/2bx 2	x 3 -a-bx
Swallowtail	3	1	1/5x 5 -ax -1/2bx 2 -1/3cx 3	x 4 -a-bx-cx 2
Butterfly	4	1	1/6x 6 -ax -1/2bx 2 -1/3cx 3^1/4dx ^4	x 5 -a-bx-cx 2 -dx 3
Hyperbolic	3	2	x 3 +y 3 +ax+by+cxy	3x 2 +a+cy 3y 2 +b+cx
Elliptic	3	2	x 3 -xy 2 +ax+by+cx 2 +cy 2	3x 2 -y 2 +a+2cx -2xy+b+2cy
Parabolic	4	2	x 2 y +y 4 + ax + by + cx 2 + dy 2	2xy+a+2cx x 2 +4y 3 +b+2dy

SEVEN ELEMENTARY CATASTROPHES describe all possible discontinuities in phenomena controlled by no more than four factors. Each of the catastrophes is associated with a potential fimction in which the control parameters are represented as coefficients (a,b.c.d) and the behaviour of the system is determined by the variables (x,y). The behaviour surface in each catastrophe model is the graph of all the points where the first derivative of this function is equal to zero or when there are two first derivatives, where both are equal to zero.

GRAPHS of five of the elementary catastrophes suggest the nature of their geometry. The fold catastrophe is a transverse section of a fold curve of the cusp catastrophe, and its bifurcation set consists of a single point. The cusp is the highest-dimensional catastrophe that can be drawn in its entirety. The swallowtail is a four-dimensional catastrophe and the hyperbolic umbilic and the elliptic umbilic catastrophes are five-dimensional. For these graphs only the three-dimensional bifurcations sets can be drawn: the behavior surfi1ces are not shown.

CUSP MODEL of the catastrophe machine erects a pleated behaviour surface over one segment of the bifurcation set, such as the cusp nearest the disk. Each point on the top and bottom sheets of the behaviour surface gives the position of the disk having minimum energy for the position of the control point. Within the bifurcation set, where there are two stable positions of the disk. there are likewise two local minimums, one on the top sheet and the other on the bottom sheet. The middle sheet represents the local maximum in the energy function. Catastrophic changes in angular position are observed whenever the control point moves all the way across the cusp.