This is the second time I have come across this particular anecdote in a few months, so I thought I would weigh in on it as well. Last night I saw on Reddit that “The Earth is relatively smoother than a billiard ball.” This was a reference to a Wikipedia article about the shape of the Earth, quoted here in case someone gets around to fixing it:
“Local topography deviates from this idealized spheroid, though on a global scale, these deviations are very small: Earth has a tolerance of about one part in about 584, or 0.17%, from the reference spheroid, which is less than the 0.22% tolerance allowed in billiard balls.”
A few months ago, I also came across a Discover Magazine blog post from 2008 titled “Ten things you don’t know about the Earth.” The first two “things” on the list dealt with this same issue of the shape of the Earth compared to that of a billiard ball. The article correctly recognized the need to distinguish between smoothness and roundness, but in my opinion still came up with a wrong answer on both counts.
Before getting started, why do we care about any of this? The motivation for this anecdote begins with the fact that the Earth is not a perfect sphere, viewed on either a large or small scale. On a large scale (think roundness), because the Earth is spinning, its shape is best approximated not by a sphere but by an ellipsoid– specifically, an oblate spheroid, with a larger radius at the equator than at the poles. On a smaller scale (think smoothness), the Earth is more obviously seen to not be a perfect sphere, nor is it a perfect ellipsoid, since it has all sorts of ridges, grooves, etc. corresponding to mountains, rivers, ocean trenches, etc.
The question is, how “non-spherical” is it? The comparison with a billiard ball would be interesting if it were true, because to us a billiard ball seems to be a nearly perfect sphere, at least to the naked eye. The source of this comparison– and, I think, the source of much of the confusion– is what the World Pool-Billiard Association has to say about a regulation billiard ball:
“All balls must be composed of cast phenolic resin plastic and measure 2-1/4 (+/-.005) inches [5.715 cm (+/- .127 mm)] in diameter.”
The key observation, I think, is that this description has nothing whatever to say about the smoothness of a billiard ball. It does not mean, as the Discover article states, that a billiard ball “must have no pits or bumps more than 0.005 inches [sic] in height.” Even such a small pit, bump, or groove would be easily noticeable on a billiard ball; manufacturing capabilities and requirements for smoothness are on the order of microns, much less than 0.005 inch. (I emailed the WPA asking for clarification on this requirement, but have so far received no response. I wonder if they get a lot of questions about this.)
The Wikipedia entry makes the same mistake, comparing the 0.22% (0.005/2.25) “tolerance” of a billiard ball to the 0.17% figure corresponding to the largest deviation of the Earth’s surface from the reference ellipsoid. The latter almost certainly refers to the Mariana Trench, 10,911 m below sea level. (Actually, this figure should be 0.0855%, not 0.17%, since the referenced billiard ball tolerance is relative to its diameter, not its radius.)
In any case, before we can comment on the smoothness of the Earth compared with a billiard ball, I think we require more information on either WPA rules or manufacturing standards. [Edit: Thanks to commenter Mark Folsom for providing the following clarification of just how smooth a billiard ball is:
“125 microinches rms is a really rough surface–much more so than any billiard ball I have seen. In my estimation, a new billiard ball has a surface finish no worse than 32 microinches…”
Comments on the Bad Astronomy post give similar estimates. And “Dr. Dave” provides some actual measurements with photos and plots showing deviations of approximately 20 microinches. In comparison, at the scale of a billiard ball, the Mariana Trench is a groove almost 2000 microinches deep. So it seems the Earth is nowhere near as smooth as a billiard ball.]
So let us move on to roundness. The following is quoted from the Discover blog:
“If you measure between the north and south poles, the Earth’s diameter is 12,713.6 km. If you measure across the Equator it’s 12,756.2 km, a difference of about 42.6 kilometers. Uh-oh! That’s more than our tolerance for a billiard ball. So the Earth is smooth enough, but not round enough, to qualify as a billiard ball.”
First, a minor nit: neither of the quoted diameters is correct to the given number of significant digits. But that will not affect our calculations here. What the article seems to miss is that the stated tolerance of a billiard ball diameter is plus or minus 0.005 inch. That is, the diameter may be as small as 2.245 inches, or as large as 2.255 inches. Enlarging this 0.01 inch difference to the scale of the Earth, the allowable difference in diameters is about 56.6 km, more than the actual difference of 42.8 km. So the Earth is indeed as round as a regulation billiard ball.
Having said all this, I think this entire analysis abuses the spirit of the law, so to speak. The WPA probably does not intend to allow such ellipsoidal billiard balls onto pool tables around the world, but rather to allow some variability in the size of nearly-spherical balls. That is, the intent of the regulation is more likely that a ball should be spherical with a fixed diameter, but that diameter may be 2.245 inches for one ball, and 2.255 inches for another ball.
- Is the Earth as smooth as a billiard ball? Answer: I’m not sure. The question may be rephrased by comparing again with the Mariana Trench: can one detect a 49-micron groove in a billiard ball, and if so, would it be acceptable to play with? [Edit: As mentioned in the edit above, I think the answer is now a definite No.]
- Is the Earth as round as a billiard ball? Answer: technically, yes… but you probably wouldn’t want to play with it.