Chief Justice Roberts Knows Law, Doesn't Know Math OR The Difference Between Polynomial and Exponential

In yesterday's New York Times online, I read the recent news that the Supreme Court Rules for Navy in Sonar Case. For those of you who aren't familiar with the case: environmental groups challenged the legality of the Navy to conduct training excersizes using mid-range SONAR (an underwater object detection and localization method that uses pulses of sound) while aquatic mammals were in the vicinity (and some other circumstances).

As I understand things, their main argument was that these tremendous pulses of sound cause damage to marine mammals (much like someone setting off a firecracker right next to your ear, possibly multiple times in a row) that could potentially result in outcomes such as failure to reproduce, beaching, and death, being that these mammal rely strongly on their ability to hear in order to figure out where they are going.

Meanwhile the Navy argued that (1) it doesn't hurt the marine mammals so much and (2) national security is more important. While lower courts put a hold on the Navy's ability to conduct exercises while the mammals where nearby (agreeing with the environmentalists), our Supreme Court ruled that courts should not second guess the military, or at least not overmuch.

I honestly don't know all of the details of precisely how often the restriction the environmentalists argued were called for by the National Environmental Policy Act and other statutes. Although, my bias is to guess that the Navy probably could have managed to conduct their exercises under these constraints. However one part of the majority opinion text authored by Chief Justice Roberts floored me for both it's lack of precision and misleading nature.

Quoting the aforementioned NY Times article:
Chief Justice Roberts took issue with both restrictions. The Navy had agreed to shut down its sonar if marine mammals were sighted within 200 yards. The appeals court’s requirement that it increase the zone to 2,200 yards, Chief Justice Roberts said, would “expand the surface area of the shutdown zone by a factor of over 100,” given “the exponential relationship between radius length and surface area.”

Now our Chief Justice may be a brilliant legal mind, but he could use a primer in college math 101. The relationship between surface area (A) and radius length (r) on a plane is actually:

A=pi*r^2

(where pi is approx 3.14 and ^2 denotes the operation of taking a number and multiplying it by itself - e.g. 5^2 = 5 * 5 = 25). This relationship is what is know as a polynomial relationship, this one in particular being of degree 2.

What this means in plain English is that as the minimum legal distance allowed between the Navy ships and the aquatic mammals for Navy SONAR use increases, the amount of potential area in which a mammal can show up and disrupt exercises (shutdown zone) grows increasingly faster.

And what the Chief Justice wrote is that this increase is not polynomial, but rather exponential, which is a whole different ball game in terms of fast as can be seen by the graph above. Namely exponential growth works essentially like a doubling process. If that were true in this case, of every single yard increase in distance, the size of the shutdown zone area would double! After a 100 yard increase or so, the shutdown zone's size would far exceed the surface area of the earth (and possibly the milky way, I'm not going to bother checking on this). Anyway, it really gets my goat - particularly given how precise with their language these Justices are supposed to be (I mean for the past 50 years at least, folks have been arguing whether the 2nd Amendment protects your right to a semi-automatic based on the presence and positioning of a comma)!

But mathematical correctness aside, there's a rhetorical issue here that bothers me. Yes, by increasing the distance at which the Navy needs to be away from an aquatic mammal (which is the right way to argue the restriction since the level of the sound is related directly to how far you are from its source), you increase the shutdown area by a larger factor (in geek-speak we call this super-linear). So ten times as far means 100 times more shutdown zone. So what?

The rate of growth, be it linear, polynomial, or exponential only matters if you are presupposing that there is a growth process in the first place. If the assumption is that if the envionmentalists win then they immediately go back to court demanding an even larger safety distance be imposed, and that if they've won once they will win again, then I guess I see how the rate of growth matters. But otherwise, it's empty rhetoric.

The only real issues are (1) what distance won't cause damage to our aquatic friends (2,200 meters claimed by the environmentalists), (2) what the physical size of the shutdown zone will be in order to accomodate that (just around 5 square miles according to my calculation), and (3) whether that is compatible with our national security interest (my guess being yes, given that there is something like 20,000,000 times that amount of oceanic surface area today). The geometric properties relating distance and surface area matter only insofar as they allow us to calculate the value in (2) from the value in (1). Aside from that the relationship really doesn't provide any insight or value into the decision making process. The only other use you could find is a rhetorical one: "look at how quick this function grows, giving up more than one yard for one yard is a bad deal". It sounds good, but has no intelligence to back it up.

So to summarize: Chief Justice Roberts' opinion was both factually incorrect, being ignorant of basic college math, and misleading as well, using mathematical relationships to make a point that was essentially a non-sequitur (10 times increase in distance = 100 times increase in area) to the real question (If 2200 meters is needed for the safety of marine life and that implies a 5 square mile shutdown zone: can a 5 square mile shutdown zone be compatible with our national security interests?).