Posts tagged with power laws

The radiolab story “It’s Alive” made vivid the claim of Geoffrey West and Luis Bettencourt that a city’s size determines how fast people walk in that city.

West & Bettencourt have written that people earn more in large cities, waste less, file more patents, and commit more crimes — and that city size is the main determinant of all these things.

The charming Cosma Shalizi has recently published a 15-page paper that rebuts them. From the abstract:

Re-analysis of the gross economic production and personal income for cities in the United States, however, shows that the data cannot distinguish between power laws and other functional forms … and that size predicts relatively little of the variation between cities.

The striking appearance of scaling in previous work is largely artifact of using extensive quantities (city-wide totals) rather than intensive ones (per-capita rates).

(Sorry if that’s hard to read. Horizontal axis = log( city population ). Vertical axis = pedestrian speed in m/s, give or take a standard dev. Solid & dashed lines are two fits proposed by Bettencourt & West.

Why don’t children get hurt when they fall down?  Because mass is proportional to the cube of height.  So not only does somebody half your height fall half the distance to the ground, and therefore incur 1/4 of the speed (gravity accelerates as time squared).  They also weigh merely ~1/8 of what you do (~1 / 2³).  So all in all, toddlers hit the ground with ~1/32 of the force.

$\large \dpi{200} \bg_white \begin{matrix} \rm{mass } \approx \rm{volume} \approx \rm{length}^3 \\ \ \rm{surface\ area} \approx \rm{length}^2 \end{matrix}$

For example, a 3’ tall boy probably weighs 30 pounds (and is 2 years old).  But a 6’ tall boy probably weighs 160 pounds, and a 6’ tall man may be 180 pounds.

According to Gould, much of what we humans take for granted as facts of life, in fact depends on being one of the largest animal species on Earth:

The relative strength of … gravity … varies with size in a regular way….  Simply by growing larger, any object will suffer continual decrease in relative surface area…

since volume and mass are proportional to height³, while surface area is proportional to height².

$\large \dpi{200} \bg_white {\rm{surface\ area} \over \rm{volume}} \sim \rm{height}^{2/3}$

To which we responded by developing internal organs — like the lungs, which are “richly convoluted bags of surface area”.  Moreover,

at our large size, we are ruled by gravity….  But gravity is negligible to very small animals….  An insect walking up a wall … the small gravitational force pulling it down … is easily counteracted by surface adhesion.

Large size does have some perks, though.

An ant-sized man might don some clothing, but forces of surface adhesion would preclude its removal.  The lower limit of drop size would make showering impossible….  He could pour no liquid, light no fire … surface adhesion would prevent the turning of pages [of a tiny book].

Not only that, but being half our size would reduce the force of spear throws by 1/16 to 1/32.  Which would mean that we couldn’t hunt the even-bigger-than-us woolly mammoth.  And you want to hunt the woolly mammoth.