I.

A recent breakthrough in pseudoscience: the location of the Great Pyramid of Giza encodes the speed of light to seven decimal places.

This is actually true. The speed of light in a vacuum is 299,792,458 meters per second. The coordinates of of the Great Pyramid are 29.9792458° N, 31.1342880° E (you can confirm with Google Maps that this gets you right on top of the Pyramid). The speed of light and the latitude number there have all the same digits. That’s a pretty impressive coincidence.

You might think this is idiotic because the meter was invented by 1600s French people. If ancient aliens or Atlanteans built the pyramids, why would they encode their secret wisdom using a unit of measurement from 1600s France? But there’s a way around this objection: the 1600s French people defined their meter as 1/10,000,000th the distance between the Equator and the North Pole. If the aliens also thought that was an interesting way to measure length, then they could have encoded their secret wisdom in it. So you wouldn’t need aliens who could predict the thoughts of 1600s Frenchmen. Just aliens who thought exactly like 1600s Frenchmen.

(actually, a different group of 1600s Frenchmen proposed a different version of the meter, defined as the length of a pendulum with a half-period of one second. This turned out to be 99.7% of the 1/10,000,000th-the-way-to-the-North-Pole definition, so either one works unless you want super-exactness. I think a much more interesting conspiracy theory would be that aliens designed the Earth to encode secret wisdom about the periods of pendulums.)

But realistically, aliens who think suspiciously like French people probably weren’t involved. So how do we explain the coincidence?

II.

The following is indebted to user mrfintoil’s great explanation on metabunk.org.

First, it’s not a coincidence to seven decimal places. Yes, that particular nine-digit sequence lands you atop the Great Pyramid. But that gives you way more precision than you need – cutting off the last three digits actually gets you closer rather than further from the center of the Pyramid. The only numbers that are doing any work are the 29.9792° N. So you really only get four decimal places worth of coincidence.

On the other hand, matching six digits is still pretty good. That’s literally a one-in-a-million chance.

So here the explanation has to go to how hard the pseudoscientists worked to find a coincidence of this magnitude; in other words, how many degrees of freedom they had.

Here’s an obvious example; as far as I can tell, the longitude of the Great Pyramid doesn’t encode anything interesting at all. So it’s not the equivalent of winning a one-in-a-million lottery with a single ticket. It’s the equivalent of winning a one-in-a-million lottery with two tickets.

A second issue: if the latitude of the Great Pyramid had been 10.7925 N, that would be the speed of light in kilometers per hour, which would be an equally impressive match.

So just taking these two degrees of freedom, we have four lottery tickets:

1. The one where the latitude is the speed of light in meters/second
2. The one where the longitude is the speed of light in meters/second
3. The one where the latitude is the speed of light in kilometers/hour
4. The one where the longitude is the speed of light in kilometers/hour

In other words, the number of lottery tickets increases exponentially as we get more degrees of freedom.

Let me list out all the degrees of freedom I can think of and see where we end up. I am going to try my best to be as fair as possible to the ancient aliens. For example, I was considering saying that since there are three pyramids at Giza, we have to multiply by three, but to be honest the Great Pyramid is clearly greater than the other two, and it would be less elegant if Menkaure’s pyramid encoded some amazing cosmic constant, so I won’t raise that objection. I am going to try to be really fricking fair.

1. Latitude vs. longitude (2 options)

2. Speed of light in meters/second vs. kilometers/hour vs. cubits/second vs. cubits/hour. I’m avoiding using feet/miles, because that’s even more arbitrary than meters. But I think it would actually be even more convincing if the calculation actually used the real Egyptian unit, which I understand is the cubit. So let’s go with (4 options)

3. Great Pyramid vs. Sphinx. Like I said before, the other two pyramids at Giza are noticeably less impressive than the Great Pyramid. But the Sphinx is pretty impressive, and the ancient aliens folks talk about it just as much as the Pyramid, so I think that would be an equally good hit if it had been true. (2 options)

4. Use of a 90 degree latitude system vs. use of a 100 degree latitude system. I’m a little split on this one, because it wouldn’t look anywhere near as impressive if the pseudoscience sites had to explain that they found a really cool coincidence but it only worked if you converted normal latitude into a different hypothetical latitude system that had 100 degrees. But since we know the aliens/Atlanteans use base 10 anyway (they’re encoding their wisdom in the base 10 representation of the speed of light) it makes more sense for them to use a base 10 latitude system instead of replicating our own bizarre custom of using base 10 for everything else but having latitude go from 0 to 90. On the other hand, if these were Earth-based Atlanteans, they might have gotten the custom of dividing the circle into 360 parts for the same reason we did – there are about 360 days in a year. And if they were aliens, maybe we got our bizarre latitude convention from them – the idea of 360 degree circles is really old and lost in the mists of time. Overall I can see this one going either way, so I’m going to give it as (2 options)

5. Decimal point placement. The latitude 29.9792 N matches the speed of light exactly, but so would the latitudes 2.99792, 2.99792 S, and 29.9792 S. I checked these other sites at the same longitude as the Pyramid to see if there were any mysterious features. But they seem to be, respectively, a perfectly ordinary field in Uganda, a perfectly ordinary field in Tanzania, and a perfectly ordinary patch of ocean. But a world where the pyramid was in Uganda and the ordinary field was in Egypt would be just as much of a hit as our current world. Therefore (4 options)

From these really simple things alone, we learn we’ve got 2 x 4 x 2 x 2 x 4 = 128 lottery tickets, reducing our 1/1 million chance of winning to something more like 1/10,000. Progress!

There are a few other degrees of freedom that I think are a little harder to judge, but still important:

6. What aspect of the Pyramid we’re looking at. That is, it would have been equally interesting (maybe moreso!) if its height or width matched the speed of light exactly. So that’s another (3 options). I guess if the ancient aliens were really good at what they were doing, they could have given the pyramid 299,792,458 sides, but I won’t hold that against them. This should really make the multiplication more complicated because I can no longer use all the different ways of representing latitude vs. longitude, but I’ll stick with the simple method for now.

7. Which site we’re looking at. This one is hard, because I don’t know if anywhere else has the ancient alien-related credibility of the Great Pyramid. The only equally mysterious site I can think of is Stonehenge, and maybe the Nazca Lines. I don’t feel comfortable saying it would be equally impressive if Tiwanaku or Yonaguni had the right coordinates. I’ll just say (2 options) for Pyramids and Stonehenge.

8. Which constant we’re looking at. Sure, the Pyramid encoding the speed of light is pretty cool, but what about the Planck length? Avagadro’s number? I’m split on whether I want to include mathematical constants like pi or e in here. I think if it encoded pi to some number of decimals places then I would just think that the Egyptians were more advanced at math than I thought but it wouldn’t necessarily be earth-shattering. The Egyptians knowing e would be pretty shocking but still maybe not worth believing in ancient aliens over. There really aren’t that many physical constants as cool as the speed of light, so I might just arbitrarily call this one (4 options).

So now we have a total of 128 x 3 x 2 x 4 = 3072 lottery tickets, for a 1/300 chance of winning the one-in-a-million lottery.

I would like to say “Ha ha, I sure proved those dumb conspiracy nuts wrong”, except that a 1/300 chance is still a pretty impressive coincidence – what scientists call p < 0.01. And now I've used up all my excuses. I think what’s going on here is that I’m still accepting the terms of the game – comparing only the exact categories used in the original calculation. Suppose that the latitude of the Great Pyramid was exactly 30.0000? That too would be impressive – it would prove that the pyramid builders knew the exact size and shape of the Earth and were able to build their Pyramid one third of the way between Equator and Pole. Suppose that the Great Pyramid was latitude 19.69724. That’s the date humankind first landed on the moon in yyyy/mm/dd format – clearly the Pyramid was built by a time-traveling Nostradamus! Suppose that the Pyramid was built of stones of four different colors, with blue stones always paired opposite red stones, and yellow stones always paired opposite green stones. Then the ancient Egyptians were trying to tell us about the structure of DNA. What if the Pyramid, viewed from above, looked like a human brain?

Is it fair to take all of that into account? If so, does the remaining coincidence go away? I wish I were able to give these questions a more confident affirmative answer.

III.

I still believe that pseudoscience is helpful for understanding regular science. The loopholes that let people discover proofs of ESP or homeopathy are the same ones that let them discover proofs of power posing and ego depletion.

In the same way, numerology is helpful for understanding statistics. You can see the same factors at work, free from any lingering worry that maybe the theory you’re investigating is true after all.

Andrew Gelman writes about the garden of forking paths. The idea is: the scientific community accepts a discovery as meaningful if p < 0.05 - that is, if equally extreme data would only occur by coincidence 5% of the time or less. In other words, you need to win a lottery with a one-in-twenty chance if you want to get credit for discovering something absent any real effect to be discovered. But if a scientist forms their hypothesis after seeing their data, they might massage the precise wording of their hypothesis to better fit their data. If there are many different ways to frame the hypothesis, then they have many lottery tickets to choose from and a win is no longer so surprising. Gelman discusses a study claiming to find that women wear red or pink shirts during the most fertile part of their menstrual cycle, which sometimes involves red or pink coloration changes in primates. The study does detect the effect, p < 0.05. But there were a couple of different ways the researchers could have framed the problem. They could have looked at only red shirts. They could have looked at only pink shirts. They chose days 7-14 as most fertile. But they could also have chosen days 6-15 without really being wrong. They could have looked only at the unmarried women most likely to be trying to attract mates. A recent paper listed 34 different degrees of freedom that can be used in this kind of thing. Add up enough of them, and you have more than twenty tickets to the one-chance-in-twenty lottery and success is all but certain.

I used to call this the Elderly Hispanic Woman Effect, after drug studies where the drug has no effect in general, no effect on a subgroup of just men, no effect on a subgroup of just women, no effect on a subgroup of just blacks, no effect on a subgroup of just whites…but when you get to a subgroup of elderly Hispanic women, p < 0.05, apparently because it's synchronized with their unique biological needs. This is pretty obvious. The lesson of the Pyramid-lightspeed link is that sometimes it isn't. It just looks like some sudden and shocking coincidence. The other lesson of the Pyramid is that I cannot consistently figure this kind of thing out. I threw everything I had against the correlation, and I still ended up with p = 0.003. I don’t think this is because the Pyramid really was designed by aliens with a suspicious link to 1600s France. I think it’s because I’m not creative enough to fully dissect coincidences even when I’m looking for them.

This is always happening to me in real studies too. Something seems very suspicious. But their effect size is very high and their p-value is very significant. I can’t always figure out exactly what’s going on. But I should be reluctant to dismiss the possibility that I’m missing something and that there’s some reasonable explanation.

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A somewhat moderate nitpick, a lot of your degrees of freedom don't multiply with each other. For example, #4 and #6. We have 2 types of lattitude systems, and we have 3 types of measurements (location, height, width) But this only gives us 4 possible combinations, not 6 (lattitude 1, lattitude 2, width, height). And a lot of your other degrees of freedom don't multiply like longitude/lattitude, or location also don't multiply with width/height. Similarly, #3 and #7 don't multiply. You only have two degrees of freedom corresponding to different monuments if every alternate location also has two valid monuments.

Some of this can be fixed by fusing some categories and treating them as additive, but in general this means your number of combinations should be smaller than you think (and more mathematically complicated to compute).

Thanks for pointing that out; I noticed that too. This is perhaps partly made up for by the fact that he doesn't count feet or the mouth of the Nile river as possibilities.

Just a small nitpciking correction : the metric system wasn't invented in the 1600s, but in the late 1700s during French Revolution.

There is another significant way you can increase the search space. If we're only considering either latitude or longitude and throwing out the other variable, we get to draw lines all the way around the planet when looking for a site of interest:

That is, we can search:
- a line around the whole planet at latitude 29.9792458° N
- a line around the whole planet at latitude 29.9792458° S
- a line around the whole planet at longitude 29.9792458° E
- a line around the whole planet at longitude 29.9792458° W

Then you could search another 4 lines at 2.99792458° N/S/E/W.

With this we drastically increase the places we can look for meaningful / coincidental intersections.

But that line has an infinitesimal chance of intersecting with any point as significant as the Great Pyramids? Scott's argument about Uganda and Tanzania seems wrong for the same reasons 

Agreed, there is an infinitesimal chance of intersecting something with such large significance as the pyramids. My point was to show that by leaving 1 of only 2 chosen variables open AND having a variable that can be set to many different values along a continuum, you significantly increase optionality for how you could retrospectively jump through hoops to assign importance to what is just a coincidence.

One further thought would be this example chose a single hypothesis a conspiracy theorist may try to prove, when in reality they will be looking for these 'proofs' in many different places. If they don't find the connection in one place they'll just keep looking.

There's a very good reason not to consider longitude vs. latitude as a degree of freedom. Latitude is measured from the Equator, which is objectively defined by the motion of the Earth. But longitude is measured from the Greenwich meridian, which was defined by nothing more than where the creators of the coordinate system were located.

There's one factor to explain this coincidence that is not referenced here and I couldn't find it mentioned on the SSC post either: polar motion.

As a recap, latitude is the angle between a given point (like the tip of the Pyramid) and the Equator. The Equator is the points at the surface that are equidistant from both poles. And the poles are the points where the rotation axis intersects the surface. They're the points the Earth rotates around, sort of.

Well, it turns out that the axis of rotation is not fixed with respect to the surface. This is independent of plate tectonics, the fact that some parts of the surface move with regards to each other. The Earth's surface could be perfectly immovable and we could still have polar motion. The scale of the motion is that, per Wikipedia, it has moved 20m since 1900, and recently the direction has changed from 80 degrees west towards the Prime Meridian.

To illustrate this, imagine that some cosmic force made the exact point where I'm currently sitting writing this comment be a pole. That is, the Earth revolves around this very point in my bedroom. (I guess it's a good thing I'm snuggling under a blanket.) Then the Equator would be this line between the brighter and darker parts of the map (I used the nearest airport, São Paulo-Congonhas, as the pole); it runs somewhere near San Diego, just barely includes all of Great Britain and Antarctica, and crosses Egypt suspiciously close to the Pyramids. They're actually 209 km from it on the opposite hemisphere as me, so their latitude would be just shy of negative two degrees.

Now, of course at the time the aliens Khufu's slaves built the Pyramids the North Pole was somewhere fairly close to its present location, and not in tropical South America. But it'd be very unlikely if it was at precisely its current location! (Or wherever it was at the time when the version of WGS84 Google Maps uses was made.) And since the pole can move 20m in 1.2 centuries, it could have moved way more than the size of the base of the pyramid, which measures just over 200m, since the 26th century BCE.

Another degree of freedom comes from the number of different topics there are where aliens are claimed to have meddled. I admit I can't think of too many off the top of my head (crop circles? maybe 9/11?) but I'd be willing to bet there are at least 5 popular, serious claims to be had. That would bring the odds up to 1/60, or p = .017.

Maybe another degree can be squeezed out by saying that there are topics in pseudoscience besides aliens, and topics to house crazy coincidences besides pseudoscience. I'm not sure how this could be counted, though.

One last thought, it's not necessary to get p up to anything like 1. Coincidences do happen, after all.

Pseudo-Science can be useful if you lack the time or the tools to do the real science correctly. Anyone who's ever studied the Navier Stokes Equation (NSE) knows you can more or less describe a nice, warm summer breeze in mathematical terms without solving all the way down tot he 4th order terms. The Coriolus Force term in the NSE will not be large enough to bear any relevance to how strong or pleasant a breeze you experience. Some would say you are pseudo-scientific if you just shorthand your calculation and only take temperature and pressure differences into account, but in reality, you are solving in about twenty seconds of computational time what could take all day if you really wanted to know how much every term of the equation mattered.