# Could the Earth really be flat?

So, you’ve been told you’ve been lied to your whole life, and the idea of a flat earth made more sense everyday, and before you knew it, you were a true believer. Those bastards are never gonna fool you again.

THE. EARTH. IS. FLAT.

OK then! So… would you be interested in looking into some geometric properties on this flat Earth with me?

Unfortunately there is no official flat Earth map that can help us calculate distances (you may have heard that building one would be too expensive), so we have to believe Google Maps for that (for now, at least). I’ve found this website distancefromto.net that uses Google Maps under the hood to calculate distances between cities. As long as those distances don’t cross an ocean, we can probably believe there’s some truth to them. If they were too wrong, people would start noticing, right?

So, I picked 3 pairs of cities:
* San Francisco — Washington DC, on the North
* Quito — Macapá, on the Equator
* Perth — Sydney, on the South
And those are the flying distances I got. And it’s OK, if you don’t believe for a second that those distances are correct, keep reading and I promise you’ll still get something useful out of this article! ;)

Now, you may also notice that I’ve picked “horizontal” distances, meaning, the cities I picked have roughly the same latitudes.

Speaking of which, maybe I should talk a little bit about what latitude and longitude are, just to level our understanding about it. So you probably heard about latitudes and longitudes. It’s those numbers that if you feed into a GPS, they can locate any point in the world. Which one is which again?

* Latitude is the thing that changes if you move up and down (North and South). It goes from -90º to 90º.
* Longitude is the thing that changes if you move left and right (East and West). It goes from -180º to 180º
So… maybe it’s something like this?

Well… no. On the flat Earth, North is not a fixed direction, it’s rather always pointing to the center of a circle, so this would be a more accurate representation:

And if you combine the two, the intersection of a longitude straight line with a latitude circle, you can still locate a single point in the world. Brilliant!

Now, let’s examine longitudes more carefully. Longitudes seem to be tracking the angle to the North Pole, right? So, for example, if you get city A at longitude 15º and city B at longitude 45º, then the angle between A, B and the North Pole should, in theory, be 30º, right? Here’s an exercise you might wanna do: find some cities in the world and mark then in a printed flat world map. Then see if their longitude differences match the angles that you can measure on paper (they should!).

There’s also an interesting relationship between longitude and TIME. The sun should be completing a full 360º worth of longitude in 24h. This means that it must be going at 15º/hour. So we should be able to observe a 2h difference between the hypothetical cities A and B above, which have a 30º longitude difference. Maybe we can find some concrete examples of that?

Looking at the table above, we can see that time differences follow more or less a 15º/hour ratio. This may not be exactly the case when the longitude difference is not that big (see San Francisco — WashingtonDC). That’s because of timezones, which truncates the time differences to the next full hour. Nice!

OK, so did you know that common people like you and me can actually measure those kinds of longitudinal angles? We can use the shadow of the sun to tell time, then for example, if
* Shadow in city A says it’s noon
* 2h pass
* Shadow in city B says it’s noon
THEN we can conclude that the longitude difference between A and B is 30º. Cool huh! There are even devices that can be used to tell the time using shadows. They’re called sundials. I’ve read here that a practical precision target for a sundial is 10 minutes, but there are expensive, hard to set-up sundial models that can get as accurate as 1 minute. So, here’s a fact: sundials are basically “shadow watches”, but they can also be used to calculate longitude differences.

Now, let’s take a look at an interesting geometric property of the flat Earth. It’s what I call the “expanding South effect”.

The expanding South effect is very easy to grasp: If Anna and Max at points A and B in the picture above start walking South, they get further apart. Their longitude difference remains the same (because longitude is the thing that doesn’t change if you move N-S), but the actual distance between them increases. It’s also easy to demonstrate that the distance between them is proportional to their distances to the north pole (if they double their distances to the North Pole, then they’ll also double the distance between them).

So let’s see an example of what those distances would look like in a flat map.
Let’s say that a 1º arc is an 80km walking distance at latitude 45º.
On equator (latitude 0º), you’re twice as far from the North Pole, so the same 1º arc would be a 160km walk.
On latitude -45º, you’re 3 times as far from the North Pole, so the same 1º Arc would be a 240km walk.

Those walking distances are also proportional to the size of the angle. It we were looking at a 5º angle instead of 1º, those distances would be five times bigger. This is how geometry works in a circle. So if the flat Earth model is correct, the distances that we can see and measure in the actual world should conform with those rules.

Now, we have those 3 real world distances. Let’s see if the numbers add up. First we should try to understand what those distances mean. distancefromto.net is saying that if we fly east from San Francisco, that will be the shortest flying distance to Washington DC. But we know the earth is flat, so that can’t be exactly right… right?

So Google Maps must really be bending the map so that that distance looks like a straight line, but in reality, since the earth is flat, we know that flying east will cause you to to fly over an arc rather than a straight line. So maybe it’s safer to assume that 3928Km is the flying distance on that arc (and also there is a shorter straight-line flying distance that Google Maps is hiding from us, but that’s not relevant right now).

Good! Now let’s try to use some of the circle geometry rules that we learned above to calculate the distances Quito — Macapá, and Perth — Sydney. In other words, let’s try to use circle geometry to calculate the green numbers below, given the orange numbers.

SanFrancisco— Washington DC tells us that, at latitude 38.3º, we should always see a Km/ºlongitude rate of 3928/45.4 = 86.5Km/ºlongitude.
Circle geometry says that at latitudes -0.1º and -32.95º, we should see that ratio grow in propoortions to 90.1/51.7 and 122.95/51.7, which gives us:

Uh-Oh... So if we start with the known distance of San Francisco — Washington DC, and apply circle geometry to calculate distances that are more to the south, we get numbers that are wronger and wronger the more south we go.

OK, just for comparison, what would be the results if we applied sphere geometry? What’s the geometry formula to calculate “horizontal” distances like those again?

It’s easy to demonstrate the formula above in a sphere, but I want to keep the math in this article super accessible, so I’ll ask you to just believe me that that’s how things work on a sphere (and if you don’t you can maybe try to demonstrate yourself, or ask a friend, but this is what you’ll get). Let’s see what results we get.

That’s completely outrageous! Someone is manipulating those numbers, OBVIOUSLY! >:[
There, there, take a breath and calm down, we’ll get to the bottom of this.
We learned that the statements:
* The Earth is flat
* Distances, latitudes, and longitudinal angles between those 6 places are given by the table above.
We learned that the statements above cannot both be true at the same time.
So, being good flat earthers that we are, we should now decide not to believe those numbers that Google Maps is feeding us.

* horizontal distances
* longitudinal angles
* latitudes

But… where is it lying, exactly? Well, if you keep repeating those kinds of experiments with different points of the world, this is what you’ll get:

It’s easy to see this, right? On ball Earth, the expading south effect does happen, but only in the northern hemisphere. What you get on the southern hemisphere is the opposite “shrinking south” effect. And we can repeat the calculations above pretty much anywhere on earth and get the same kind of results. So, Google Maps should be lying EVERYWHERE! And the more south we go, the bigger a liar it gets! And, as long as it is lying on land, we can bust it!

If it’s lying about latitudes, the only way that horizontal distances can be shorter than distances on equator is if those distances are located above (or inside) the equator, but that would put Australia in the norther hemisphere, so that’s just absurd.

If it’s lying about distances — and it should be lying everywhere — all we need to do is find a road that is aligned with East-West direction somewhere in the Southern hemisphere. Bring a compass and 2 or 3 different GPS devices just to make sure. Then drive maybe 10Km or 15Km. See if the distance you drive and the coordinates on GPS matches the formula from Figure 12 (which will match the distance predicted by Google Maps). The expected result is that the Google Maps (or calculated) distance should be a lot less than the actual distance that you drove. But thinking about it, maybe that result is a bit unlikely because, you know, there’s people down there using that kind of software and predicting distances and arrival times just fine. But, anyway, it’s worth trying. It’s lying everywhere, we just need to run ONE simulation.

If it’s lying about longitudes, sundials to the rescue! Maybe you can even find someone in a different place of the world who can help you with that! Building a sundial as also a fun activity, specially if you involve children :-)

But in the end, it will all match the expected results from sphere geometry (and if you’re still not sure, PLEASE, go ahead make your measurements, now you know how). Google Maps *is* right about those distances. And it’s not Google Maps actually (or NASA), it’s really the latitude/longitude coordinate system, something that was invented a long long time ago, long before Google and NASA.

I know that the lat-lon system works because I work with geoprocessing for agriculture in Brazil (you can have a look at my resume if you want). A small part of my work involves calculating distances from coordinates and match it back with different sources of information that come both from the ground and from the sky. The only way to make it work is to use sphere geometry.

This is the reason why there’s no accurate flat map. It’s not expensive, it’s impossible. The distances and angles that we can see and measure on this earth match what you’d expect from a sphere, and you can’t make that fit in a circle.

Now, your senses tell you one thing. And reason and logic tell you another — and you know how to check that for yourself.

What are going to believe now? That’s up to you.