One of the great things about mathematics is its ability to bring together apparently completely unrelated phenomena, and explain them all with the same simple concept. It turns out that a very basic mathematical idea — power laws — describes the economics of pizza, limitations on the sizes of insects, income inequality, and countless other things.
We recently looked at why it makes sense to buy a large pizza rather than a small. As mentioned in that post, the area of a pizza increases as a function of the square of the radius. Formally, the area of a pizza with radius R is, as we all saw in high school, given by the formula A = πR2.
The area of the pizza — the actual amount of pizza you get — grows much more quickly than the width. As the NPR piece on pizza prices points out, a 12 inch pizza is 50% wider than an 8 inch pizza. Plug those values into the area formula, and the 8 inch pie gives you about 50 square inches of pizza, but the 12 inch pie gives you about 113 square inches of pizza — a 125% increase, or more than twice as much pizza.
The relationship between the area and width of a circle is an example of a power law: the area is proportional to the width raised to some power, in this case the power of 2.
The Power Law
Power laws are somewhat counterintuitive at first, because we are used to linear relationships. If you work for an hourly wage, you get paid the same amount of money for each hour you work, and so the total amount of money you make in a week is linearly proportional to the number of hours you work in that week.
Power laws grow non-linearly. If you were paid according to a power law, you would get more money per hour for each hour you work, just as you get more pizza per inch of width the wider the pizza is.
Power laws show up in many different contexts. Just as the area of a circle is proportional to the width of the circle squared, the volume of a sphere is proportional to the width cubed, or raised to the third power. The formula for the volume of a sphere with radius R is V = (4/3)πR3.
Suppose you have two balls, both made out of the same material, but with one ball twice as wide as the other. Then the larger ball will not weigh twice as much as the smaller ball, but will instead weigh eight times as much.
This has serious implications in biology. Take a six foot tall human, and double his height to twelve feet, keeping all of his other proportions the same. His mass, which has a cubic power relationship to his height, will be eight times more than when he was six feet tall. However, the cross-sectional area of the muscles and bones in his legs is proportional to the square of his height, and so at twelve feet tall, he will have only four times as much bone and muscle area, putting far more pressure on his legs.
So, in nature, we see differently sized animals developing different proportions for anatomical features. Rhinoceroses have disproportionately larger leg bones and muscles than do mice. The study of these differences, which tend to follow power laws, is called allometry.
Allometry and the Six-Foot Long Arthropleura
Allometry can help explain one of the stranger moments in the history of life on Earth. In the Carboniferous period, a little over 300 million years ago, the planet was covered with vast rain forests. This caused the amount of oxygen in the atmosphere to be far higher than it is today.
The Carboniferous also featured enormous insects and other arthropods, like the six foot long centipede-like Arthropleura to the left.
Scientists have long speculated that the gigantic arthropods and plentiful oxygen of this period are related. Arthropods do not breathe using lungs and transferring oxygen with blood the way vertebrates do. Instead, they have a series of small breathing tubes, or trachea, running along their bodies, directly bringing air to internal tissues.
A 2007 study of modern beetles showed that the sizes of these breathing tubes grows following a non-linear power law with the size of the insects. As the bugs get bigger, they have to dedicate proportionally more and more of their internal volume to their respiratory systems.
This limits the size of modern insects — at some point, they get so big that, if they kept following the power law governing breathing tube size, they would run out of room for other muscles and organs. Back in the oxygen-rich Carboniferous period, arthropods might have been able to get away with relatively smaller breathing tubes, following a more forgiving power law relationship between overall size and the amount of internal space needed for breathing. This might have allowed them to grow to the absurd sizes the fossil record shows.
Power laws also show up in economics and sociology. The Pareto distribution, which shows, among other things, how wealth and income in a society are distributed, follows a power law. The percentage of the population that makes more than a particular amount of money is proportional to the reciprocal of that amount, raised to some power. This means, as you go up the income scale, you very quickly get fewer and fewer people making at least that much money.
This explains the wide inequalities we see in income and wealth — the largest part of the population falls on the lower end of the income spectrum.
Power laws — relationships between two quantities where one quantity is proportional to the other, raised to some power — describe a huge number of phenomena in our universe. In addition to the examples above, power laws describe the sizes of cities, the strength of gravity on other planets, the frequencies of earthquakes of different magnitudes, and much more.
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