Air University Review, May-June 1981

The Conflict Manifold

Lieutenant Colonel John T. McGarth

The so-called spectrum of potential conflict is used today as a model of warfare and a basis for national strategy. Professional military education courses depict it as a continuum ranging from political agitation on one extreme to total thermonuclear war on the other.1 Unfortunately, a "spectrum" is woefully outdated as a model on which to base our future lives and fortunes and should be replaced by a multidimensional "conflict manifold."

The word spectrum is defined in terms of the light and radio spectrum, which emphasizes the orderliness (by wavelength) and the continuous aspects of our everyday use of the word spectrum. But spectra come in many shapes and sizes; they may be continuous or discontinuous, linear or nonlinear. For example, ordinary sunlight refracted by moisture in the air forms the rainbow, whose colors, blending into one another, comprise the familiar continuous spectrum discovered by Sir Isaac Newton. On the other hand, light from individual atoms forms a discontinuous spectrum; these spectral lines are evident in the different colors of "neon" signs (each color of which is accounted for by a different atomic gas). Our number system is a linear spectrum, with identical spacing between adjacent elements. But the response of the human ear, which can detect sounds ranging from the dropping of a pin to the roar of a jet engine, represents a decidedly nonlinear spectrum.

The spectrum of conflict, as portrayed in most readings, is single-dimensioned, linear, and continuous. It is probably the simplest model possible, but it is just too na´ve for the complexities of modern warfare.

Evidence for discontinuity in the spectrum of conflict comes from several areas; the most obvious is the nuclear versus conventional aspects of war. Whereas there is a clear progression from 500- to 750- to 1000-pound conventional bombs, or even from one to two to three megatons in nuclear weapons, the jump from high-explosive to nuclear detonations is anything but continuous. Indeed, this quantum jump is the central point of issue in the debate over the use of tactical nuclear weapons. The use or nonuse of chemical and biological weapons, herbicides, and their ilk also constitutes a decided discontinuity. Targeting, which can be industrial, military, or population oriented, also represents distinct subsets that cannot blend into one another as do the colors of the rainbow.

The linearity of warfare is likewise open to doubt. The incredible range of available weapons, from stones and slingshots to smart bombs and cruise missiles, can be ordered in degree of complexity. But it would be folly to assume that the step from muzzle-loader to repeating rifle is identical in impact to that from gravity bomb to cruise missile. The issues of war are similarly nonlinear in scope, ranging from simple occupation of territory to complex economic subtleties. The countries involved vary widely in size, ideology, and capability.

In noting the diverse aspects of targeting, weapons, issues, and political constraints, one is hard pressed to conceive of war as being one-dimensional. An example that comes to mind is the use of an Olympic boycott to meet the Russian challenge in Afghanistan. Without going into its ultimate efficacy (on which the jury is still out), the decision to use the Olympics is clearly an example of multidimensional thinking. The grain embargo added a third dimension, that of economics. Had the President chosen to match military intervention with a military response, i.e., one-dimensional thinking, we would have reverted to the brinkmanship of the Dulles era, with an increase in the unpredictable probability of war.

One can cite more examples, but the simple fact is that modern conflict is neither continuous, linear, nor single-dimensioned.

Consequently, the spectrum must be replaced by a multidimensional model, perhaps nonlinear and discontinuous. Topological mathematicians would call it a manifold; hence, the name conflict manifold. Its primary characteristic is multidimensionality. Instead of having nuclear and conventional weapons as part of the same dimension, each would have its own. The degree of independence between the two factors would be represented by the degree of orthogonality between the two dimensions. Completely independent factors would be represented by dimensions that are totally orthogonal or perpendicular. The conventional and nuclear weaponry dimensions might be continuous; within each of these dimensions, there is a smooth gradation in size and efficacy of weapons employed.

A third dimension, targeting, might then be added; this would likely be a discontinuous dimension with a discrete point for each type of target, namely, industrial, military, etc. A fourth dimension might contain the issues involved, a political dimension. A fifth could be the cost or economic dimension, etc.2

The number of dimensions is immaterial; it does not matter that it is virtually impossible to visualize five or more dimensions as being mutually perpendicular. The important point is that each independent factor in conflict can be assigned its own dimension; this dimension is then tailored to that factor’s continuity or discontinuity, linearity or nonlinearity. Interactions between factors would be represented by a hypersurface in this multidimensional space.

One result of such a model of conflict is that choices are no longer restricted to sliding in one direction or the other along a unidimensional scale. This unfortunately complicates matters. However, the real world is indeed very complicated and seldom conforms to simple models. This complexity of choice was always present but perhaps masked by the choice of model. By conceptually freeing our model from a linear spectrum, we enlarge the range of our strategies. We can be more specific in response, with greater delineation of purpose, as opposed to the simple strategy of "upping the ante."3

Our multidimensional manifold has another striking benefit; it can accommodate the "catastrophe theory" of RenÚ Thom.4 This is a mathematical theory that can be used to explain unpredictable events, hence the term catastrophe. Unpredictable behavior is characteristic of many systems, whether mechanical, physical, biological, or social. The example commonly used to illustrate this theory is the "flee or fight" choice.5 When confronted by an adversary, an animal has two general choices, fleeing or fighting. Its first inclination is to flee or avoid the conflict if possible. But, if pushed further, the fleeing animal reaches a point where it turns suddenly and fights; we commonly say it has been pushed too far. If the adversary retreats, the animal will continue to fight, even in a regime where it had previously resorted to flight. Further retreat by the adversary will cause the animal to break off the attack just as abruptly as it started.

The concept is diagrammed in Figure 1. As the encroaching variable moves to the left along the horizontal axis, the tendency is to continue to flee, along the path A to B. But a point is reached, B, where the fleeing one turns and fights. This is represented by an abrupt drop to point C. Further encroachment keeps the animal fighting in going from C to D. If the encroachment variable is eased or retreats (moves to the right along the horizontal axis), the animal continues to fight, shown by the path C to E. At E it breaks off the attack as abruptly as it started. This behavior exhibits a double-valued function in the region of concern; the term hysteresis is normally used to describe it.

Figure 1. Encroaching variable

The value of Thom’s work lies in seeing this graph as part of a surface, as shown in Figure 2. As the path from A to D is traced, the subject falls off a ledge, so to speak, from the area of fleeing behavior to the area of fighting behavior. In returning to A along the same path, the fight behavior continues to point E, and then "falls up" to the upper surface. This abrupt "falling up and down" is the catastrophe part of the theory. More important for us, note that it is part of a fold or pleat in a two-dimensional surface; it is possible to go from point A to D via point F and never encounter the catastrophic behavior that occurs at the fold.

Figure 2. The catastrophe fold

This exemplifies only one of Thom’s seven elementary catastrophe figures, called the cusp, but illustrates the point. The theory includes surfaces of many dimensions, representing behavior controlled by several variables.

Avoidance of the catastrophe requires moving into an additional dimension, over the single one used in Figure 1. By going back to the proposal to place each aspect of conflict in its own dimension, perhaps targeting changes would be a better path to take than a switch to nuclear weaponry in a given situation. Perhaps negotiation would be the additional dimension to be used. For example, in Afghanistan reliance on a purely military dimension could possibly carry us to the brink of a "catastrophe." Open military counterforce might place the Russians in a flee or fight position with its inherent unpredictability. Since it is so difficult to determine the parameters controlling the situation and the behavior changes are so drastic, the situation is virtually out of control. In contrast, the boycott puts the Soviets in the more predictable dimension of world opinion. Their actions (such as overpreparation for the games, internal and external propaganda, and the use of American tourists as unwitting pawns) are what we would normally expect of a nation put in their position—and these actions do not lead to war.

The Iranian situation is a second example. The Iranians were certainly using multidimensional thinking in taking international hostages. Our consideration of such alternatives as blockade, embargo, and military rescue were apparently multidimensional. The catastrophe fold comparable to flee or fight that we are trying to avoid was putting the Iranians into a kill or release the hostages position.

In the Russian case, we want the Russians to leave Afghanistan. In Iran, we wanted freedom for the hostages. But we have not wanted to force either country into a behavior region where a slight change in conditions would abruptly trigger war or death of the hostages.

The intent is to find a way (dimension) to achieve the end without crossing a catastrophe fold. In either case, the single-minded pursuit of only one dimension, e.g., invade/not invade or rescue/not rescue, would likely be disastrous.

We are certainly not constrained to two dimensions or combinations of two dimensions, although these cases are easily pictured. Perhaps we rely too often on pictures or visual aids to clarify and represent our conceptual thinking, to the extent that we fail to generalize sufficiently or include ideas that defy direct visualization. The conflict manifold, as opposed to the spectrum of conflict, is thus a step in the right direction, removing some of our conceptual barriers.

It is relatively straightforward to look at a portion of reality and devise a model to account for its characteristics. The linear, continuous spectrum of conflict was a decent first try at a representation of warfare, but a real gain is made when the model furthers our knowledge of reality. Here, the spectrum of conflict is too limited—it lacks power. A model of conflict as a multidimensional manifold, with the idiosyncrasies and pathology of a mathematical hypersurface, increases our predictive power tremendously. It remains to refine and modify the manifold as predictions are compared with experience.

The point to be made here is not the adoption of topological mathematics or any other scientific area into political thinking. The emphasis is on a shift in viewpoint from the linear, one-dimensional spectrum of conflict to a multidimensional manifold. This multidimensional approach allows the incorporation into decisions of such worthwhile ideas as catastrophe theory, allows greater flexibility in planning, and probably conforms better to reality. If this shift in viewpoint is seriously pursued, the final model may not look like anything proposed here, but it is a start.

United States Air Force Academy


1. Fundamentals of Strategy, Air Command and Staff College, Course 1F, Lesson 3 (Gunter AFS, Alabama: Extension Course Institute, Air University, 1976), p. 21; National Security Strategy, Air Command and Staff College, Course 1F, Lesson 4 (Gunter AFS, Alabama: Extension Course Institute, Air University, 1976), p. 30.

2. E.C. Zeeman, "Catastrophe Theory," Scientific American, April 1976, pp. 76, 80.

3. Air Force Policy Letter for Commanders, 1 October 1978, p. 1.

4. R. Landauer, "Stability in the Dissipative Steady State." Physics Today, November 1978, p. 23; RenÚ Thom, Sturctural Stability and Morphogenesis: An Outline of a General Theory of Models, translated by D.H. Fowler (Menlo Park, California: W.A. Benjamin, Inc., 1975).

5. Zeeman, p. 65.


Lieutenant Colonel John T. McGrath (B.S., University of Notre Dame; M.S., University of Wisconsin; Ph.D., University of Wyoming) is Associate Professor of Physics, USAF Academy, and a master navigator. He was an Outstanding Educator of America in 1974 and is an Air Force nominee for space shuttle mission specialist.


The conclusions and opinions expressed in this document are those of the author cultivated in the freedom of expression, academic environment of Air University. They do not reflect the official position of the U.S. Government, Department of Defense, the United States Air Force or the Air University.

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