Professor Stewart's Cabinet of Mathematical Curiosities
Triangle of Cards
Turnip for the Books
The Four-Colour Theorem
Shaggy Dog Story
Shaggy Cat Story
Rabbits in the Hat
River Crossing 1 – Farm Produce
More Curious Calculations
Extracting the Cherry
Make Me a Pentagon
What is π?
Legislating the Value of π
If They Had Passed It …
Much Undo About Knotting
To Find Fake Coin
Mathematical Jokes 1
An Age-Old Old-Age Problem
Why Does Minus Times Minus Make Plus?
How to Unmake a Greek Cross
How to Remember a Round Number
The Bridges of Königsberg
How to do Lots of Mathematics
Euler's Pentagonal Holiday
Who Was Pythagoras?
Proofs of Pythagoras
A Constant Bore
Fermat's Last Theorem
A Little-Known Pythagorean Curiosity
Squaring the Square
Squares of Squares
Ring a-Ring a-Ringroad
Pure v. Applied
How Old Was Diophantus?
If You Thought Mathematicians Were Good at Arithmetic …
The Sphinx is a Reptile
Six Degrees of Separation
Duplicating the Cube
Curves of Constant Width
The Stolen Car
The Square Wheel
Why Can't I Divide by Zero?
River Crossing 2 – Marital Mistrust
Wherefore Art Thou Borromeo?
Kinds of People
The Sausage Conjecture
Tom Fool's Knot
What is the Golden Number?
What are the Fibonacci Numbers?
The Plastic Number
Don't Let Go!
The Most Likely Digit
Why Call It a Witch?
Möbius and His Band
Three More Quickies
Miles of Tiles
Why No Nobel for Maths?
Is There a Perfect Cuboid?
When Will My MP3 Player Repeat?
The Poincaré Conjecture
Pig on a Rope
The Surprise Examination
Mathematical Jokes 2
Why Gauss Became a Mathematician
What Shape is a Crescent Moon?
What is a Mersenne Prime?
The Goldbach Conjecture
Turtles All the Way Down
A Puzzling Dissection
A Really Puzzling Dissection
Nothing Up My Sleeve …
Nothing Down My Leg …
Can You Hear the Shape of a Drum?
What is e, and Why?
May Husband and Ay …
Many Knees, Many Seats
What Day is It?
Logical or Not?
A Question of Breeding
The Sixth Deadly Sin
How Deep is the Well?
What is the Square Root of Minus One?
The Most Beautiful Formula
Why is Euler's Beautiful Formula True?
Your Call May be Monitored for Training Purposes
Archimedes, You Old Fraud!
Fractals – The Geometry of Nature
The Missing Symbol
Where There's a Wall, There's a Way
Constants to 50 Places
Are Hard Problems Easy? or How to Win a Million Dollars by Proving the Obvious
Don't Get the Goat
All Triangles are Isosceles
If π isn't a Fraction, How Can You Calculate It?
Let Fate Decide
What Shape is a Rainbow?
The Riemann Hypothesis
Disproof of the Riemann Hypothesis
Murder in the Park
The Cube of Cheese
The Game of Life
Drawing an Ellipse – and More?
Mathematical Jokes 3
The Kepler Problem
The Milk Crate Problem
Spherical Sliced Bread
Professor Stewart's Cunning Crib Sheet
Complexity Science // 239
Complexity Science Complexity science, or the theory of complex systems, came to prominence with the founding of the Santa Fe Institute (SFI) in 1984, by George Cowan and Murray Gell-Mann. This was, and still is, a private research centre for interdisciplinary science, with emphasis on `the sciences of complexity'. You might think that `complexity' refers to anything complicated, but the SFI's main objective has been to develop and disseminate new mathema- tical techniques that could shed light on systems in which very large numbers of agents or entities interact with one another according to relatively simple rules. A key phenomenon is what is called emergence, in which the system as a whole behaves in ways that are not available to the individual entities.
An example of a real-world complex system is the human brain. Here the entities are nerve cells neurons and the emergent features include intelligence and consciousness. Neurons are neither intelligent nor conscious, but when enough of them are hooked together, these abilities emerge. Another example is the world's financial system. Now the entities are bankers and traders, and emergent features include stock-market booms and crashes. Other examples are ants' nests, ecosystems and evolution. You can probably work out what the entities are for each of these, and think of some emergent features. Anyone can play this game.
What's harder, and what SFI was, and still is, all about, is to model such systems mathematically in a way that reflects their underlying structure as an interacting system of simple compo- nents. One such modelling technique is to employ a cellular automaton a more general version of John Conway's Game of Life. This is like a computer game played on a square grid. At any given instant, each square exists in some state, usually repre- sented by what colour it is. As time ticks to the next instant, each square changes colour according to some list of rules. The rules involve the colours of neighbouring squares, and might be 240 // Complexity Science
something like this: `a red square changes to green if it has between two and six blue neighbours'. Or whatever.
Three types of
pattern formed by a
(blocks of the same
(the spirals), and
chaotic (for example
the irregular patch at
It might seem unlikely that such a rudimentary gadget can achieve anything interesting, let alone solve deep problems of complexity science, but it turns out that cellular automata can behave in rich and unexpected ways. In fact their earliest use, by John von Neumann in the 1940s, was to prove the existence of an abstract mathematical system that could self-replicate make copies of itself.* This suggested that the ability of living creatures to reproduce is a logical consequence of their physical structure, rather than some miraculous or supernatural process.
Evolution, in Darwin's sense, offers a typical example of the complexity-theory approach. The traditional mathematical model of evolution is known as population genetics, which goes
* There is now a lot of interest in doing the same with real
machines, using nanotechnology. There are many science fiction
stories about `Von Neumann machines', often employed by aliens
or machine cultures to invade planets, including our own. The
techniques used to pack millions of electronic components on to
a tiny silicon chip are now being used to build extremely tiny
machines, `nanobots', and a true replicating machine may not be
so far away. Alien invasions are not a current cause for concern,
but the possibility of a mutant Von Neumann machine turning
the Earth into `grey goo' has raised issues about the safety, and
control, of nanotechnology. See en.wikipedia.org/wiki/Grey_goo Complexity Science // 241
back to the British statistician Sir Ronald Fisher, around 1930. This approach views an ecosystem a rainforest full of different plants and insects, or a coral reef--as a vast pool of genes. As the organisms reproduce, their genes are mixed together in new combinations.
For example, a hypothetical population of slugs might have genes for green or red skins, and other genes for a tendency to live in bushes or on bright red flowers. Typical gene combina- tions are greenbush, greenflower, redbush, and redflower. Some combinations have greater survival value than others. For example redbush slugs would easily be seen by birds against the green bushes they live in, whereas redflower slugs would be less visible.
As natural selection weeds out unfit combinations, the combinations that allow organisms to survive better tend to proliferate. Random genetic mutations keep the gene pool simmering. The mathematics centres on the proportions of particular genes in the population, and works out how those proportions change in response to selection.
A complexity model of slug evolution would be very different. For instance, we could set up a cellular automaton, assigning various environmental characteristics to each cell. For example, a cell might correspond to a piece of bush, or a flower, or whatever. Then we choose a random selection of cells and populate them with `virtual slugs', assigning a combination of slug genes to each such cell.
Other cells could be `virtual predators'. Then we specify rules for how the virtual organisms move about the grid and interact with one another. For example, at each time-step a slug must either stay put or move to a random neighbouring cell. On the other hand, a predator might `see' the nearest slug and move five cells towards it, `eating' it if it reaches the slug's own cell so that particular virtual slug is removed from the computer's memory.
We would set up the rules so that green slugs are less likely to be `seen' if they are on bushes rather than flowers. Then this mathematical computer game would be allowed to run for a few 242 // Complexity Science
million time-steps, and we would read off the proportions of various surviving slug gene combinations.
Complexity theorists have invented innumerable models in the same spirit: building in simple rules for interactions between many individuals, and then simulating them on a computer to see what happens. The term `artificial life' has been coined to describe such activities. A celebrated example is Tierra, invented by Tom Ray around 1990. Here, short segments of computer code compete with one another inside the computer's memory, reproducing and mutating (see www.nis.atr.jp/~ray/tierra/). His simulations show spontaneous increases in complexity, rudi- mentary forms of symbiosis and parasitism, lengthy periods of stasis punctuated by rapid changes even a kind of sexual reproduction. So the message from the simulations is that all these puzzling phenomena are entirely natural, provided they are seen as emergent properties of simple mathematical rules.
The same difference in working philosophy can be seen in economics. Conventional mathematical economics is based on a model in which every player has complete and instant information. As the Stanford economist Brian Arthur puts it, the assumption is that `If two businessmen sit down to negotiate a deal, in theory each can instantly foresee all contingencies, work out all possible ramifications, and effortlessly choose the best strategy.' The goal is to demonstrate mathematically that any economic system will rapidly home in on an equilibrium state, and remain there. In equilibrium, every player is assured of the best possible financial return for themselves, subject to the overall constraints of the system. The theory puts formal flesh on the verbal bones of Adam Smith's `invisible hand of the market'.
Complexity theory challenges this cosy capitalist utopia in a number of ways. One central tenet of classical economic theory is the `law of diminishing returns', which originated with the English economist David Ricardo around 1820. This law asserts that any economic activity that undergoes growth must even- tually be limited by constraints. For example, the plastics industry depends upon a supply of oil as raw material. When oil is cheap, Complexity Science // 243
many companies can move over from, say, metal components to plastic ones. But this creates increasing demand for oil, so the price goes up. At some level, everything balances out.
Modern hi-tech industries, however, do not follow this pattern at all. It costs perhaps a billion dollars to set up a factory to make the latest generation of computer memory chips, and until the factory begins production, the returns are zero. But once the factory is in operation, the cost of producing chips is tiny. The longer the production run, the cheaper chips are to make. So here we see a law of increasing returns: the more goods you make, the less it costs you to do so.
From the point of view of complex systems, the market is not a simple mathematical equilibrium-seeker, but a `complex adaptive system', where interacting agents modify the rules that govern their own behaviour. Complex adaptive systems often settle into interesting patterns, strangely reminiscent of the complexities of the real world. For example, Brian Arthur and his colleagues have set up computer models of the stock market in which the agents search for patterns genuine or illusory in the market's behaviour, and adapt their buying and selling rules according to what they perceive. This model shares many features of real stock markets. For example, if many agents `believe' that the price of a stock will rise, they buy it, and the belief becomes self-fulfilling.
According to conventional economic theory, none of these phenomena should occur. So why do they happen in complexity models? The answer is that the classical models have inbuilt mathematical limitations, which preclude most kinds of `inter- esting' dynamics. The greatest strength of complexity theory is that it resembles the untidy creativity of the real world. Paradoxically, it makes a virtue of simplicity, and draws far- ranging conclusions from models with simple but carefully
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