The world is a fascinating combination of fragility and resilience. In the midst of terrible wars and atrocities and countries being torn apart, we also see images of children playing soccer in the street and young couples starting futures together at wedding celebrations. The signs of past collapses are all around us, but we keep building new lives, structures, and institutions to take their place.

This is a book about collapse. It's intended to help some of us manage collapse a little better, promoting it when we want to and protecting ourselves from it when we need to. The first step is to understand it better, using different tools and different points of view. Much of this book is about history and experience — the anatomies of past collapses in many different subject areas from finance to fighter jets, networks to nanostructures. In surveying these diverse cases, we find extraordinary commonalities. The same kinds of dynamics occur over and over again.

Let me explain why my background as a mathematician gives me a particular perspective on this subject. Some people think that math is all about solving equations, but they're dead wrong. Math is much more about identifying common features and describing them in a way that captures their essence. Mathematical symbols and equations are basically a language that helps to clear away some of the detail so we can focus on those fundamental underlying features or principles.

What is a collapse?

Can you name ten occurrences that you would regard as “collapses”? Just think about this for a moment before reading on.

In case you're having any trouble getting to ten, let's think about categories: civilizations, empires, governments, economies, technologies, industries, companies, species, fads, styles, banks, buildings, bridges, cranes, just to name a few. No problem getting to ten now, right? And if you're getting up in years like me, you can probably recall seeing or hearing about at least ten in almost every single one of these categories, as well as others.

Can you think of any good, i.e., beneficial, collapses? If you grew up in a Western country during the Cold War, then the collapse of the Soviet Union might be one of the first items to come to mind. And then there's the collapse of major diseases such as smallpox and polio, or even the occasional mysterious collapse of cancerous tumors, often not fully understood. There are of course many more.

We may all have slightly different definitions of a “collapse,” but let's use the term loosely to refer to some relatively rapid process that leads to a significant reduction in quantity, quality, or level of organization. For example, we speak of the collapse of the dinosaurs, a process that may have taken up to several thousand years after a likely calamitous meteorite impact that significantly changed the Earth's environment; but this time period is still almost instantaneous when compared with the 160 million year period during which the dinosaurs were the dominant terrestrial vertebrates.

Sharing an electric blanket and other challenges

I happen to have a lot of experience with electric blankets. It comes from owning that old unheated New Hampshire summer getaway I've mentioned earlier, and from making an occasional winter foray into it on snowshoes to see if it's still standing and to retrieve one or another forgotten item. The best way to survive the night there in freezing conditions (both inside and out) is to turn on an electric blanket several hours before getting into the bed and letting things warm up a bit. But invariably, when you get into the bed and lie there for a while, you find that some adjustment of the temperature control is needed. If it's too high, you wake up sweating to death, and if it's not high enough, you wake up wondering why you decided to torture yourself by staying overnight. With adjustments during a single night or a setting based on experience gained over several years, it's not too hard to find a “stable equilibrium setting that keeps things at or close to a comfortable, constant temperature. Fine so far.

The situation is slightly more complicated when my wife accompanies me as we prefer somewhat different temperature settings for the blanket. Thus there have been occasional middle-of-the-night adjustments by one or the other of us to suit our own tastes, hoping that the other person is sufficiently sound asleep not to notice.

The elephant's toenail

I remember a silly gimmick of one of my high school teachers back in Brooklyn, and the fact that I remember it so vividly after about fifty years makes me think it's worth retelling. I'm finally beginning to really appreciate it.

He showed us a huge blown-up dirty white picture of something that had no strong distinguishing features, a bit of nondescript texture, and that extended right to the boundary of the photo frame. Then he asked us to guess what it was that we were looking at. The whole class really got into this exercise with an impressive degree of creativity and energy. Everyone wanted to be the person with the correct guess. The teacher was being bombarded with suggestions. Hands were waving in the air. The suggestions probably included things like a garage floor, a subway platform, old kitchen linoleum, or maybe even the surface of the moon.

But no one got it. After all, who would have guessed that we were looking at an elephant's toenail? No one was thinking that this was the image of something such that if we had access to just a small additional amount of the photo field, everything would change drastically from what we were thinking about. Built into our way of thinking was that this image should extend off to the sides in a fashion similar to the part we could see and thus that there was a certain homogeneity to the structure of the object.

How's your networking?

Working as a college professor, I spend lots of time talking with students about their future — what they might like to do and how to find the right path to get into it. Hardly anyone emerges from one of these conversations without the word “network” reverberating in their minds. Who do you know that works in this field and who might tell you what it's like? Who do you know who might know someone else in the field who could help identify an internship? What connection might I have through alums or business acquaintances? Who else on campus might I refer them to for leads or advice? Once we start looking at our collective networks, all kinds of connections turn up that could have been so easily overlooked. Of course it doesn't all have to be personal, although that can be especially valuable. We can start following pathways on an electronic network like the World Wide Web and by following a few links also identify useful resources and opportunities. This is networking, and it's almost essential to success in business, government, academia, and practically anything else. In fact, our very survival, let alone success, depends every day on networks: food distribution, electric power, communications, fuel, roads, even blood flow and nerve transmissions in our bodies. We're so tied up with so many networks that it can sometimes seem like a miracle that things run as smoothly as they do.

My New Hampshire

The only wildlife I encountered regularly in my Brooklyn youth were pigeons, squirrels, and rats. I quickly learned not to look up when walking close to apartment buildings with fancy cornices where pigeons would roost, and we never knew if we would be greeted by beady eyes and whiskers when opening the dumbwaiter door. Thus you can probably well imagine the interest I brought to a fairly large tract of New Hampshire woodland that my family and I bought many years ago as a summer getaway and, as it turns out, an endless source of projects. Here was rich nature in a new setting ready to be engaged at a different level. But it also came with collapse themes at almost every turn, from broken electric lines to trees falling on the well house, crushed culverts under the access road, and the accumulated effects of over a century of snow and wind loads on a poorly designed building. An academic like myself can turn these experiences into fascinating learning opportunities and intellectual (and physical!) challenges, while at the same time gaining a rock-solid rooting in reality that's a good balance to too much time in university ivory towers. In this chapter I hope to tell you something that you may not already know about evolution and collapse, or at least something that you may not have thought about very much. It derives from this New Hampshire experience. But first a brief review of evolutionary principles.

A quick review

We started in Chapter 1 with a broad survey of types of phenomena subject to collapse, surely not all inclusive but certainly representing a wide range of subject areas, time frames, spatial dimensions, and dynamics. You may recall from Table 1.1 in that chapter that these ranged from empires to fads, galaxies to tumors, companies to civil order. There are certainly many ways to study and learn about these phenomena and about their vulnerability to collapse. These may variously include, depending on the particular subject matter: historical analysis, scientific data collection or experiments, logical or theoretical argument, expert opinion, or other methodologies. I asserted that the use of mathematical models is a particularly useful method in that it enables us to see commonalities across a wide range of cases from different subject areas and thus to get better insight on the underlying dynamics of many collapses. It has the additional advantage of offering a language that lets us depict these common dynamics efficiently and clearly in symbolic and graphical ways, as have been utilized throughout the previous chapters.

In the subsequent six chapters, we looked at six mathematical frameworks and how they might be applied to seemingly diverse areas of collapse. These are the “six sources of collapse” referred to in the title of this book and the six dimensions suggested in the title of this chapter. They include low probability events, group behavior, evolutionary processes, instability, nonlinearity, and networks. A brief summary of each is given in Table 8.1.

Like a thief in the night?

If your house has ever been burglarized, as has mine, I'm sure you share with me the recollection of surprise and shock, along with other emotions like anger or sadness. But if you read in the newspaper about a burglary in another town, you're likely to shrug your shoulders and think, well, there are lots of burglaries all the time. So here we have this occurrence, a burglary, which is quite common, but the trigger that raises our eyebrows is when it occurs to us. The dynamics that control whether it occurs to us in particular may be sufficiently vague that we characterize it as a totally random event, or they may be more specific if we can relate the burglary to our own individual circumstances, such as leaving the house unlocked, hiring lots of occasional workers, displaying wealth, etc. In this latter case, these dynamics may well admit a deeper level of understanding.

I don't know when the next really big earthquake is going to occur along the San Andreas fault, but when it does, it's going to shock a lot of people. And yet, we all know that over time, earthquakes along this fault are quite frequent and occur with some regularity. I also know that the next really large earthquake to hit the Boston area, where I live, is going to catch even more people by surprise, even though geologists tell us that we have had big ones in the past (e.g., the Cape Ann Earthquake of 1755) and we will definitely have them in the future [114].

Fire!

A famous Supreme Court opinion by Oliver Wendell Holmes once cited the example of falsely shouting “Fire!” in a crowded theater as unprotected free speech, given the well-recognized danger of an unnecessary stampede to the exits [156]. Numerous human stampedes have been documented over the years, some for false alarms, as in his example, and others for a wide variety of causes. Indeed, we've all seen headlines like these:

• Death toll reaches 100 in Station Nightclub fire

• Iroquois Theater fire claims 602 victims

• 251 trampled to death in Hajj crush

• Sixty-three injured in Dutch Remembrance Day event

• Deadly stampede at Yemeni political rally

• 95 crushed to death in Hillsborough stadium disaster

In many historic cases, the collapse of orderly egress or crowd control can be easily understood, such as with the all too common blocking of emergency exits in order to prevent unauthorized access or theft. This was the case, for example, with the Cocoanut Grove nightclub fire in Boston in 1942, which claimed almost 500 lives largely because of emergency exits that had actually been welded shut, as well as other exits that opened inward and became quickly jammed by the crush of the crowd. However, in other cases, adequate physical means for egress or crowd movement may have existed but were not well utilized. For example, in the Station Nightclub fire, there were several available emergency exits, but the panicked crowd members moved toward the doorway through which they had originally entered, which quickly became blocked by the crush.

In this chapter we explain how to construct a discrete network for nonlinear high-contrast densely-packed composites. We use this presentation to demonstrate the so-called perforated medium technique in the analysis of high-contrast composites. We also use this investigation to demonstrate how the discrete network approximation arises from the interplay between geometry and asymptotic analysis. More specifically, the key mathematical feature of partial differential equations that describe high-contrast densely-packed composites is that their solutions exhibit asymptotically singular behavior, when particles are close to touching (high concentration). The singularities of the solutions occur exactly in the necks between almost touching particles. The location of these singularities can be characterized naturally by the geometric patterns of the distribution of the particles in the materials. Thus a geometric construction of a network is completely natural. As it is illustrated in this book, a rigorous mathematical justification is based on geometric and asymptotic arguments. These two arguments are coupled together. As a result, most of the constructions of asymptotic discrete network approximations for high-contrast composites are complicated, thus they are not attractive for practitioners. It is possible, however, to separate the geometric and the asymptotic arguments. It makes the construction of the network more transparent, and allows us to strengthen some of the previous results. In particular, it turns out that the validity of discrete network approximations could be verified for a class of composites, which is larger than the one that satisfies the δ-N close-packing condition.