Branches: Nature's Patterns: A Tapestry in Three Parts
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As part of a trilogy of books exploring the science of patterns in nature, acclaimed science writer Philip Ball here looks at the form and growth of branching networks in the natural world, and what we can learn from them.
Many patterns in nature show a branching form - trees, river deltas, blood vessels, lightning, the cracks that form in the glazing of pots. These networks share a peculiar geometry, finding a compromise between disorder and determinism, though some, like the hexagonal snowflake or the stones of the Devil's Causeway fall into a rigidly ordered structure. Branching networks are found at every level in biology - from the single cell to the ecosystem. Human-made networks too can come to share the same features, and if they don't, then it might be profitable to make them do so: nature's patterns tend to arise from economical solutions.
centre, typically by some relationship in which this density is proportional to the inverse of the distance raised to some power. A similar relationship describes how the number of settlements (cities, towns, villages, hamlets) in an urbanized area depends on their size (in population or area, say): there are many more small villages than there are towns, and still fewer cities, and the power law quantifies that fact. Planners and geographers could measure these relationships, but they could not
regarded with awe, as though uncovering a profound natural regularity. But in 1962 Luna Leopold and Walter Langbein showed that randomness alone is enough to ensure that these relationships hold for any branching network. They proposed a model of stream formation based on that developed by Horton himself, according to which networks emerge as rain falls onto a gently undulating surface. Wherever rain delivers more water than can be removed by filtering down through the rock bed, water accumulates
understood within the context of the network topology in which they occur. Once the pattern of the network is taken into account, it might be possible to tailor an appropriate response strategy. Dirk Helbing at the Dresden University of Technology and his co-workers have proposed that in such cases the best strategy is to reinforce the most highly connected nodes first against failure. That makes intuitive sense and is what you might have guessed anyway—but only once you appreciate the way the
chemical and mechanical engineers who wanted to gauge the performance of their machines. But it offered a rather artificial view of the world in which everything happens in a series of jumps between stable states that do not otherwise alter over time. That is not very like the world we know. Thermodynamics was silent in the face of the uncomfortable fact that some processes never seem to reach equilibrium. A river does not simply empty itself into the sea in one glorious, ephemeral rush—the water
minimum entropy production rule can break down. Where do they come from? During the 1950s and 1960s, Prigogine and his colleague Paul Glansdorff attempted to extend the treatment of non-equilibrium thermodynamics to the more interesting far-from-equilibrium situation. They were able to show that, as the force driving a system away from equilibrium increases, the steady state of minimal entropy production reaches some crisis point where it breaks down and becomes transformed to another state.