The Earth resembles an oblate spheroid - an ellipsoid that is flattened towards a disk-like shape: the points on the equator are further from the center of the planet than the poles - by about 20 kilometers.
Just for the sake of completeness, the other ellipsoid would be a "prolate spheroid" that resembles a long stick.
Needless to say, the Earth is oblate because it is spinning around its axis. Mark Eichelaub has seemingly made a very sensible - and "nearly correct" - calculation of the "height difference" between the poles and the equator.
The sea level should have a constant gravitational potential. He assumed that this constant arises from a variable term "h.g" where "h" is the height as a function of the latitude - a deviation from the spherical shape - and from another latitude-dependent term, the centrifugal potential "-omega^2.R^2.sin(theta)^2/2".
By demanding that their sum is constant, he obtained the equator-pole height difference (Earth's equatorial bulge) as
h = (omega.R.sin theta)^2 / 2g = 11 kilometerswhich is about 1/2 of the actual bulge, close to 21 kilometers. Now, this looks like a cool discrepancy: the sea level (its difference between the pole and the equator) is 10 kilometers higher or lower than it should be!
Now, when I saw this question at the Physics Stack Exchange, two answers were already posted. One of them didn't get to 10% of the calculation by Mark - it just "suggested" that some deviations could depend on the cosine of the latitude, and proposed to do a calculation "for the equator only" which is a completely nonsensical comment. All the problems in this task are three-dimensional and depend on a comparison of the equator and the poles. So I downvoted the answer.
There was one more answer that was longer but even worse. After saying many irrelevant things about phase transitions, it essentially said that Mark's calculation of the equatorial bulge would produce the observed result of 21 kilometers if it were done with the parameters a few billion years ago. And the Earth just froze, so that the bulge is twice as large as the result of the calculation with the current values of the rotational frequency etc.
Now, just try to appreciate how stunningly idiotic such an answer is. The guy is saying that the calculation of the constant potential says that the sea level of the world ocean is 10 kilometers higher or lower than it should be - but the water just stays there, 10 kilometers above or below the right sea level, because it got used to do so in the recent billions of years.
Meanwhile, in the real world, if some volume of water happens to be 30 meters above the right level somewhere in Japan, a giant tsunami wave immediately spreads across the world. Within half a day, it reaches the opposite side of the world. Within a day or two, the new equilibrium is reached across the globe. If there is a non-uniformity of the sea level that exceeds the typical fluctuations common for ocean waves, it takes a day to fix the imperfection. And this guy thinks that a 10 kilometer high column of extra water just stays there for billions of years.
So of course, I downvoted this answer as well and viewed the situation as an alarming one, so I immediately posted the right answer as well. Astronauts are not dying in space shuttles but I found the situation urgent, anyway: it's my instinctive behavior. ;-) The only mistake in Mark's otherwise valid calculation is that he didn't appreciate that an ellipsoid has a spherically asymmetric gravitational field, too. If you're measuring the gravitational potential on a sphere composed of points of a fixed distance from the origin, the potential won't be constant if the matter distribution is a uniform ellipsoid - a non-spherical one.
It may be at least numerically calculated that this effect changes the linear term in the gravitational potential from "hg" to "hg-hg/2", to one half of the original value, and the calculated "h" will therefore be twice as large as Mark's value, in a good agreement with observations. With this halving of Mark's "hg" potential, you may say that a surface of a constant potential is located "exactly in the middle" between the "R=const" sphere and the actual surface of the ellipsoid - a compromise. (Extra corrections arise from the fact that the Earth is not an exact uniformly dense ellipsoid, either. And from higher-order terms in the (bulge/Earthradius)^N expansion. As numerical calculations showed, the correcting factor for the "hg" term is almost exactly 1/2 and quite possibly, it is exactly 1/2, but I don't have an analytic proof at this moment because that seems to require a double integral.)
Once I downvoted the other answers, the author of the "water stays 10 km above the ground for billions of years" answer immediately attacked me - correctly guessing that I gave him the first "-1" vote. It must be a great sin to downvote his answer! Well, it's not a sin: it's exactly the very purpose of the down arrow to label junk answers - such as a huge portion of answers from this moron - so that they disappear at the bottom and the original posters or other people who are interested in the question are more likely to get the more correct answers first. And the votes should actually be anonymous, for a good reason - a policy he has repeatedly violated.
Needless to say, I have already learned who this moron is. It's a guy who has recently completed his graduate studies at Penn State University - the world's headquarters of the hockey sticks and loop quantum gravities. So he may not deviate too much - it's normal for the hacks at that school to think that water stays 10 km above the ground for billions of years - and that a climate doomsday is imminent unless Republicans are defeated in all elections (Michael Mann).
But despite my rich experience with this sort of people, I am always stunned by the breathtaking stupidity coupled with arrogance of these hacks. This guy's name is Deepak Vaid and he would write his PhD as a collection of footnotes for papers by Bilson-Thompson who "constructs" the world out of knitted octopi inspired by loop quantum gravity. Bilson-Thompson himself or herself would be considered a crackpot even among physicists of a kindergarten caliber - and now imagine that someone wants to build his career by writing irrelevant confusing footnotes to this infinitely infantile nonsense. It's just amazing.
You can demonstrably get a PhD for this complete nonsense.
But it's possible in the Penn State. In a natural world, the likes of Deepak Vaid would be starving to death in the middle of India. In the politically correct world, such crackpots are brought to the U.S. and allowed to pretend that they're physicists even though - unlike many other superbright Indian thinkers I know - they don't have the slightest glimpse of knowledge or intuition or talent needed for the discipline. And for years, people are discouraged from telling such Vaids that they just suck. You would surely be a racist if you dared to inform an Indian chap that he sucks as a scientist, wouldn't you? The fact that the Indian folks belong to the same race can't change anything about it.
Just to get some context: in a graduate quantum mechanics course I once taught at Harvard, I and my TA have found out that two students were just plagiarizing all the homework. They waited until the official answer is posted on the web, they copied it with minimal modifications, and handed it in, pretending that they were just a little bit late. One of those two students was an undergraduate girl and the other student was a male graduate student from India. My TA in particular was very offended - but we eventually avoided any investigations and reports because I got convinced that the PC Nazis would surely accuse me of racism and sexism combined - because the two fraudsters in my class happened to be female and Indian.
What a surprise that many of those people just formally end up as physicists and they're convincing the mankind that water stays 10 kilometers above the ground for billions of years and that loop quantum gravity with knitted octopi is a good candidate theory of everything. Or that there is a climate threat for the mankind. That's just a direct result of the political correctness, the lack of will of many people to show the door to similar Deepak Vaids. The U.S. Academia is literally flooded with such know-nothings who simply shouldn't be there but who, because of the very large number of similarly incompetent hacks who have gotten into the system, just got very aggressive and effectively overtook the system.
By the way, the "fat Earth" wasn't the first context in which Deepak Vaid tried to attack me - and others - for the downvotes. Someone had asked why the pseudoscalar "E.B" wasn't the action for electromagnetism. The author of the question mentioned that Sean Carroll (rightfully) wanted his students to explain why "E.B" in the action didn't matter etc.
Deepak Vaid gave an answer in which he combined almost all conceivable mistakes you can make. First, he declared that a "pseudoscalar" is given by the behavior under time reversal (instead of the correct transformation, namely parity). Also, he apparently wrote that "E.B" did matter in the electromagnetic action (despite its being a total derivative that doesn't influence the equations of motion).
To add an idiosyncratic twist to his silliness, he confused the action with the right hand side of some equations of motion, and started to link "E.B" in the action to "E.B" on the right-hand side of the continuity equation for the electromagnetic current (which fails to vanish if there are anomalies). Needless to say, "E.B" in the action has nothing to do with gauge anomalies - for example, the former is perfectly harmless while the latter is perfectly lethal: he only wrote "random impressions" about anomalies (which were wrong in details, anyway) because he saw "E.B" at several places - and didn't understand any of the different roles that "E.B" could play. Again, he would scream after getting negative votes.
