Earth vs Cue Ball

It is often stated that if the earth were the size of a cue ball, it would feel smoother than one.  No less a scientific giant than Neil deGrasse Tyson has referenced this “fact” in his excellent book The Pluto Files (p. 39).  But is this true?

The answer is a big NO.  This “fact” has become a scientific urban legend, passed along without being examined.  Debunking it takes us on a wild goose chase to billiard halls before finding truth in the mountains of Colorado.

Perusing the web, I found multiple “earth-is-smoother-than-a-cue-ball” writers referencing the World Pool-Billiard Association regulations for a cue ball:

“All balls must be composed of cast phenolic resin plastic and measure 2 ¼ (+.005) inches [5.715 cm (+ .127 mm)] in diameter and weigh 5 ½ to 6 oz [156 to 170 gms].”

Comparing the stated tolerance to the diameter, these writers get a maximum variation of 0.22%.  Discover Magazine’s blog does it this way, for example.  That maximum variation is then multiplied by the earth’s diameter to give a maximum variation of 28km, far more than the height of Mt Everest or the depth of the Marianas Trench, implying that the earth is indeed smoother than a cue ball.  Put another way, the height of Everest plus the depth of the Marianas Trench represents 0.17% of the Earth’s diameter, which is less than the 0.22% WPA number.

However, there is a problem with this reasoning.  It lies in the interpretation of the WPA rules, and more fundamentally, in the reality of how smooth a cue ball is.  The WPA rules are ambiguous as to whether they are giving maximum variation in diameter measurements over a particular cue ball (thus defining its roughness and/or roundness), or the maximum variation in diameter between various cue balls (thus defining their maximum variation in size).  If the latter, then the maximum variation in size says nothing about the roundness or smoothness of an individual cue ball.  Several web sites point this out, such as the Possibly Wrong blog.

(I must admit, I don’t think the WPA is very much at fault.  They are pretty clearly defining the size of a cue ball, while assuming it will be round and smooth to the tolerance of modern manufacturing.  They just didn’t anticipate a bunch of amateur geophysicists descending on their web site looking for interplanetary standards of roundness and smoothness and finding them where they didn’t necessarily exist.)

Without clarification from the WPA, it’s impossible to know for sure what the WPA rules mean.  But common sense dictates that if cue balls are actually much smoother than 0.22% then the actual smoothness is the real test.

Fortunately a reputable academic has used (public) university equipment to settle this debate!  Dr Dave at Colorado State University finds that a real cue ball has a variation of about 100 parts per million (or 0.01%) while the Marianas Trench represents a variation of 1700 ppm (or 0.17%), 17 times as big.  Unfortunately, Dr. Dave misses a decimal place and states a figure of 17000 ppm, which is obviously wrong, but fortunately he showed his math, so he gets full credit.  A+!

So if the earth is not nearly as smooth as a cue ball, what is a good comparison for the “roughness” of the earth?  Well, clear tape is about 1/20th to 1/10th of a millimetre thick, and a dispenser roll is usually about 5cm in diameter, about the same as a cue ball.  So the end of the tape, which we all know can be hard to peel, but definitely possible to feel, represents a height variation of about 1/1000 of the diameter.  That’s about the same as the height of Everest compared to the Earth’s diameter.  Put another way, if the earth were the size of a cue ball, the Himalayas would feel as rough as a bit of clear tape stuck to it.

If you tried to use such a cue ball in a WPA event, I’m sure you’d be disqualified, and probably beat up as well, for playing dirty pool!

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