e. j. wang

On being the wrong size


There’s an old paradox that goes something like this:

P(1, 3). Suppose that someone has gone and tied a rope tightly around the earth’s equator. If the rope were lengthened by a meter and uniformly raised off the earth’s surface until it became taut again, how far would it be off the surface?

The answer, polymathe lecteur, is 1/2π meters, or a little less than a sixth of a meter.

For those who haven’t been inoculated to this sort of thing by prior exposure, this is a counterintuitive result. The origin of our intuition is this: we know full well that if, instead of lifting the rope off the surface of the earth, we were to expand the earth itself to fill the rope, we would require 511 trillion cubic meters of dirt. The catch is that this intuition is irrelevant. We are not expanding the three-dimensional earth. We are merely tugging the one-dimensional rope.

There are several quantities at play here: the earth’s three-dimensional volume, its one-dimensional circumference, and the one-dimensional radius that governs both. As the radius grows, the volume grows quickly; the circumference, not so much. Following this general form, we can generate more paradoxes, like the following:

P(2, 3). Suppose that someone has coated the surface of the earth in a uniform layer of ice, a meter thick. By how much would its surface area increase?

The answer is forty square kilometers, a comparatively tiny area less than half the size of Manhattan. Those familiar with cell biology will recognize in this version of the paradox the concept of SA:V, which constrains the size and shape of cells. Beyond a certain size, a cell cannot support itself; the demands of the organelles within its volume cannot be sustained by its ability to transport across its surface. A single, undifferentiated cell that grows too large becomes structurally compromised.

Finally, let us consider a third variation on the paradox:

P(1, 2). Suppose that the equator has a certain importance in the history of the world. Suppose that, since time immemorial, every great civilization has extended through the earth’s midsection. Suppose that only a privileged few live within a narrow strip along the center of the world, where the arts, letters, and sciences flourish, where kings build their palaces and priests their temples. And suppose that in the hinterlands to the north and south dwell colonized peoples — shepherds, farmers, textile workers and the like — hitherto and heretofore irrelevant to the churn of history.

No doubt, intelligent and ambitious strivers will still be born in these backwaters. Those who aspire to make a name for themselves will be able to make their way down to the equator; if you can make it there, you can make it anywhere. After all, why be a textile worker in the hinterlands if you can create so much more value at the equator, allocating resources between hinterland textile factories? In our increasingly globalized, entirely hypothetical society, the equatorial capitals swell with the world’s brightest, who make their living understanding and optimizing the activities of the hinterlands.

The march of progress continues throughout the centuries. With successive improvements to the standard of living, and with the ongoing completion of man’s dominance over the elements, the world’s population has started growing exponentially. (The fertility rate at the equator is markedly lower than elsewhere, but the ongoing influx from the hinterland more than offsets this.) The capitals grow dense; skyscrapers appear; but the truth is that there is only so much real estate at the middle of everything.

Living conditions begin to worsen, at first imperceptibly. It becomes harder and harder to secure one of the prestigious resource-allocation jobs. Educated would-be elites, now choking the cities of the equator, have nowhere to turn but inward, becoming disaffected with their lives and scoffing at the stories of easy wealth with which they were regaled the day they gained admission to an equatorial university. And in the hinterlands appears an utter hopelessness as even the faraway cities cease to be the centers of dynamism and social mobility that they used to be.

Suppose, finally, that the leading scientists of this world have discovered a solution that will solve all their social ills. Their proposal is a complicated one, involving esoteric particle physics and a good deal of algebraic topology, but its core is that the surface of the earth will begin to swell, virtually exponentially, at around the same rate that the population is growing. There will be enough real estate for everyone.

The economists are quick to point out that with an exponential increase in the size of the earth’s surface, the equator will grow exponentially too, bringing room for more and more new elites. Certainly, the expanding resources of the earth will bring more jobs allocating those resources. Growth solves all problems. Rather than malaise, stagnation, and decline, we can even look forward to a future where everyone lives at this expanding equator, working a meaningful, fulfilling white-collar job, allocating resources, and playing their own important role in the course of world history.

So do you believe them?