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DOTTED around in the Central American jungle lie the ruins of once bustling cities. Birds chirrup on the steps of crumbling palaces, and creepers twist around the remnants of magnificent temples. Around 1000 years ago the city of Tikal in modern-day Guatemala stood at the heart of the Maya civilisation. It had 60,000 inhabitants, an extensive trade network and a thriving artistic culture. Stretching out in all directions from Tikal, the Maya dominated the area from southern Mexico through to Honduras. They developed astronomy, hieroglyphic writing and a complex calendrical system. At the time, they were the most advanced civilisation in the world. Then, almost overnight, this sophisticated society vanished.

By the time the conquistadors arrived, in the 1540s, the cities were deserted, and the Spaniards found only primitive tribes living in thatched huts in the forest. What had happened to the people who built these magnificent cities? Scientists have speculated about famine, warfare, climate change, epidemics, invasion and even mass suicide to explain the sudden collapse of the Maya civilisation, but no one has ever found a convincing explanation.

Now two archaeologists believe they may have uncovered a way to solve the mystery. The Maya, they claim, were living on a knife edge.

Perhaps rather surprisingly, Clifford Brown of Florida Atlantic University and Walter Witschey of the Science Museum of Virginia reached this conclusion simply by analysing maps of the towns and cities that comprised the civilisation. They discovered that the maps showed a particular form, known as a fractal pattern (Journal of Archaeological Science, vol 30, p 1619). A fractal pattern is one that repeats itself at smaller and smaller scales. Ferns are one example. They are made up from branches that also look like individual ferns, and in turn each of those is made up from even smaller branches that also look the same – and so the pattern goes on. Another example is the Sierpinski triangle (see Graphic).

Archaeologists find fractal patterns useful because they can help quantify settlement patterns. But those patterns can also reveal things about the society that – consciously or unconsciously – made them.

Many of the Maya sites are hidden under dense layers of jungle undergrowth, making it difficult to gather data over large areas. So Brown and Witschey chose to study the area that had been most completely surveyed: the state of Yucatán in Mexico. They drew up a list of more than 1000 Maya sites in this region that are thought to have been inhabited during the late and terminal Classic period (AD 800 to 1000), and noted their locations. Using a computer program that scanned a digitised map, they calculated the size of each settlement.

Armed with the number of settlements of each size, and their geographical distribution, Brown and Witschey were able to calculate a “fractal dimension” for the area. This is a measure of how quickly the self-similarity pattern scales up: the smaller the jumps between each new level of the self-similar pattern, the higher its fractal dimension. The Sierpinski triangle has fractal dimension of 1.58. The fractal dimension that Brown and Witschey measured from the Maya settlements of Yucatán is 1.51. And this led them to a suggestion of what might have happened to the Maya.

Brown and Witschey soon discovered that other phenomena, such as the devastation caused by forest fires and war, can be characterised by a very similar fractal dimension. Forest fires are known to follow behaviour patterns known as “self-organised criticality”: this has also been recognised as the fundamental driving force behind avalanches, earthquakes, solar flares and even the muscle contractions of a woman giving birth (91av, 17 November 2001, p 30).

The main characteristic of self-organised critical systems is an instability that rises progressively until, on occasion, the system suddenly “resets” itself to a more stable configuration. The classic example is a pile of sand that grows as more sand is added, one grain at a time. As the gradient of the slope increases, there may be several small avalanches that slightly reduce the gradient. Eventually, however, one added grain will trigger a major, critical avalanche that sends the slope back to a much lower gradient. In self-organised critical systems, seemingly insignificant causes can provoke dramatic effects – maybe even the fall of a civilisation.

Brown and Witschey found suggestions of self-organised criticality not only in the large-scale distribution of Maya settlements. When they investigated the layout of rooms and houses within one of the cities, Mayapán, they found it there too. Mayan houses have an unusual, clumpy appearance and archaeologists have always found it difficult to describe and quantify the complex patterns. For example, small groups of buildings tend to cluster around plazas. These groups of houses often join into clusters of different sizes that are like neighbourhoods or quarters.

By carefully measuring the sizes and location of rooms, houses, courtyards, temples and palaces in Mayapán, Brown and Witschey were able to demonstrate that this clustering pattern was fractal, with a fractal dimension of 1.35 – again, similar to other self-organised critical system.

Brown and Witschey are not the first to suggest that self-organised criticality could collapse a society. In 2002, for example, Gregory Brunk, a historian and economist made the same claim (Journal of Theoretical Politics, vol 14, p 195). But is there any evidence to suggest that the Maya civilisation could be related to the instability of an avalanche system? Brown and Witschey believe there might be. Joyce Marcus, an anthropologist at the University of Michigan at Ann Arbor has carried out extensive studies of Maya civilisation, and argues that the civilisation passed through repeated cycles of geographical growth and fragmentation: regional states frequently merged and then split again. Brown and Witschey suggest that the Maya’s fractal dimension could be a reflection of such activity, and that Marcus’s account is a description of self-organised criticality.

Along with politics and economics, warfare is likely to have had a significant influence on Maya settlement patterns, too. David Webster, an anthropologist based at Pennsylvania State University at University Park, has shown that the Maya were almost incessantly at war – but war on many scales, from minor raids to major conflicts between large states. This too, Brown and Witschey argue, is a hallmark of self-organised criticality: earthquakes, for instance, happen over all possible size scales, each with varying degrees of frequency.

Ezra Zubrow, from the State University of New York at Buffalo, one of the first archaeologists to use fractal analysis, says it is highly plausible for the fractal dimension of a settlement to reflect the aggressiveness of that society. “The structure and layout of buildings and their defensiveness or offensiveness will translate into different fractal patterns.”

Of course, the instability associated with geographical faults or snow-laden slopes seems quite obvious, but what could make a society so intrinsically unstable? One answer might simply be progress. There seems to be a correlation between fractal dimension and some form of “efficiency”.

When fractal patterns on a plane are so intricate that they almost fill the entire space, the fractal dimension rises inexorably towards 2, the maximum it can have. For a society, a low fractal dimension means the use of land, trade, government and labour is far from optimised – there is plenty of slack in the system.

Scientists from the University of Nottingham in the UK have shown this correlation in studies of an area of ancient Greece. William Cavanagh, an archaeologist, and the late Robert Laxton, a mathematician, characterised land use in Laconia, which lies to the east of the modern city of Sparta, and worked out its associated fractal dimension. This held at around 0.7 from 600 BC until around AD 600, when it started to rise – eventually reaching around 1.

They compared the numbers with historical records on politics, agriculture and society, and it became clear that the fractal dimension rose in response to changes in government and equated to the generation of a more stable and fairer society as land was better shared out. This, in turn, led to greater efficiency in the use of land, labour and other resources. “The higher fractal dimension corresponds to more stable politics and a modern economic structure,” Cavanagh says.

But in a self-organised critical system, the fractal dimension cannot keep growing indefinitely because of the system’s tendency to “reset”. The pile of sand will have a maximum gradient, determined by the size of the sand grains and the frictional forces between them, beyond which it will suffer catastrophic collapse. Since sand is always being added, an eventual collapse is inevitable.

Similarly, when a society grows ever more efficient – optimising and re-optimising the use of its resources as conditions change – it too might reach a critical point where all the available land is dedicated to agriculture or buildings, and yet ever more food and housing is needed. That could precipitate crises such as war and famine.

And so perhaps there is something in Brown and Witschey’s suggestion that the Maya civilisation was fundamentally unstable – like living on an increasingly stressed geological fault. For more than 2000 years the society may have suffered numerous crises of differing magnitude before things finally collapsed, never to recover. “It may not be necessary for archaeologists to identify a major change to explain why the Maya civilisation collapsed: a minor perturbation may be all that is necessary,” Brown says.

Just as one apparently insignificant flurry of snow on a critically loaded slope can trigger a major avalanche, some relatively common feature of Maya life – a single war, an unfortunately timed change of government, or an economic slump – may have finished them off. Of course, if the society was always undergoing upheavals of various types and severity, the collapse might even have been caused by an unlucky coincidence of a number of these events – maybe a war combined with a bad harvest.

It is an intriguing hypothesis but, at the moment, not much more than that. Ron Eglash, an anthropologist from Rensselaer Polytechnic Institute in Troy, New York, has reservations about drawing too many conclusions from the Maya data. “I think they are on to something when they note that the large-scale fractal patterns could be the result of a process that can be modelled by self-organised criticality,” he says. However, he is worried about applying fractal dimension as a tool. “Correlation doesn’t always mean causation, so we need to be cautious when applying fractal analysis across cultures.”

Brown and Witschey agree: they are keen to stress that they have simply put forward a suggestion to be investigated in further archaeological research. They certainly do not claim to have proved that Maya society was a self-organised critical system. For one thing, they point out, their map data is far from ideal. “Unfortunately there will be some data missing from areas that have not been surveyed extensively,” Brown says. “Also, the smallest sites are likely to be under-represented because they are harder to find in areas of thick bush.”

Nevertheless, this could prove a useful way to investigate the political, military and economic history of ancient societies where the archaeological evidence is more reliable. “I would like to see fractal analysis being used as a tool to encompass all of culture, from transport to trade, family to state and ideology to religion,” Zubrow says.

Fractal analysis is also providing a mathematical perspective on life in modern societies (see “Welcome to fractal city”) – and that, Brown says, might raise another set of questions. Can life get simply too efficient, or too civilised? Is our society due for a major collapse in the near future?

Self-organised critical systems are inherently unpredictable, so it might not be a question we can answer with confidence. But maybe we should still take those ancient cities, whether they are buried under creepers or submerged in Guatemalan swamps, as a timely warning. Otherwise, if the fractals are to be believed, the relentless march of progress might also take us somewhere unexpected.