## Tuesday, September 13, 2016

### Three Tests, Part One: Shadows At Sunset

Flat-Earthers will, it seems, go to great lengths to prove that the Earth is not a globe. They will try to raise money, unsuccessfully, to travel to Antartica, perhaps in order to demonstrate that they will be turned away at gunpoint for the attempt.

They will buy fairly expensive cameras with powerful zoom lenses and use the zoom to somehow prove that the ships don't go over the horizon, or that faraway skylines are visible when they shouldn't be, or that stars are really blurry undulating masses.

They will shoot lasers across lakes in an attempt to debunk a similar (but notably not identical) experiment performed on the series Genius recently.  And they will measure the temperature, so they say, of moonlight compared to moon shade which, somehow, is supposed to prove that the Earth is flat. But what they won't do is perform a few simple, definitive tests that show whether or not the Earth is flat.

So, if you seriously think that the Earth is flat, forget about high-tech, GoFundMe campaigns, and vague, badly designed experiments. Be prepared to get out, reach out, and find out. Warning: there's math involved.

Test One: Measure a Sunset, Watch the Moon Set
I've talked about sunrise and sunset before, and gotten a lot of excuses about perspective. But perspective is a visual phenomenon, and we're going to ignore it. Instead, we are going to cast and measure some shadows, and do some trigonometry. If you're not particularly good at trig (it's okay, I'm not either), I'll point you to an online calculator that you can use to help out.

It helps to be on an ocean beach that faces west for this. A beach that faces east also works, but then you should watch the moon rise and measure the sunrise. And if you don't have access to a beach, any big field without any really tall surrounding mountains will work almost as well. I'm going to assume a west-facing beach.

Pick a clear evening when the moon is full or nearly full. It will be easier to see. For tools you'll need a yardstick, a tape measure, a compass, a small level, binoculars, and paper and pencil. Get yourself a spot at least an hour before sunset. Plunge the yardstick into the sand until two feet are sticking up. Use the level to make sure it's straight up and down.

Half an hour before sunset, measure the shadow cast by the stick, and use the compass to work out the direction of the shadow. Believe it or not, that's all the data you need to gather. After the sun has fully set and no longer casts any usable light, try to bring it back into view with the binoculars.

Bring the binoculars back to the beach half an hour before the moon sets. Watch the moon sink below the horizon, note it's apparent size, and whether the horizon cuts it off as it sinks. When it's out of sight, try to bring it back into view with the binoculars.

Now it's time to go home and do some evaluation of what you just did.

Evaluation
Get out your simple data on the shadow. First, let's start with the length of the shadow. Open this triangle calculator. Erase any data already provided. In the box for side x, put the height of your stick, or two feet. In the box for angle a, put 90, since your stick formed a right angle with the ground. In the box for side z, put the length of the shadow, and click on "Calculate."

The key number is the angle that will show as opposite side 1 (the calculator should show side x and angle b; I don't know why it doesn't). This is the angle above the ground that the sun was when it cast it's shadow. In my test, it was 6.71 degrees. What does this say about the flat Earth? Well, let's apply this math in reverse.

The most common flat-Earth model has the sun 3000 miles above the "plane." Let's plug that back into the calculator: put 3000 into side x, erase side z, and put 6.71 into angle b. Calculate. Look at the figure for side 3. In my test, the stick cast a 17-foot shadow; if the sun were 3000 miles over a flat plane, it would be over 25,000 miles away at sunset. In other words, off the disc of the flat-Earth model.

Not a problem, you say? We know that when it's sunset where you're standing, it's noon somewhere else. Where else on Earth is 25,000 miles away? The model is breaking down.

Okay, what if the sun is not 3000 miles high? If the sunset happens when the sun is about 6000 miles away, which is one of the implications of the model, then how high would it be? Erase side x, put 6000 in side z (leaving the original figures for angles a and b). Calculate.

In my test, the sun would have to be 706 miles high to create that shadow. If you think that's possible, then it's time to go out and try to measure the height of the sun. I give you some idea of the difficulties of that task if you assume a flat Earth in this blog post. But before you go to the trouble, let's think about your other observations.

Get out your favorite azimuthal equidistant map projection, or another flat-Earth map if you have one. You can get the Gleason Map here.  Find yourself on the map. Then chart the angle of the sun based on the shadow angle you jotted down. Does it make any sense? In some lucky circumstances it might, but in most locations on most days, it won't match up.

When you watched the moon go down, it didn't shrink in size. It may have even looked a little larger as it came close to the horizon. And it moved down into the horizon, the horizon cutting through it until it sank out of sight. And you couldn't bring it back into view with the binoculars. So it wasn't just moving away from you. Relative to the ground, it was actually behind the ground. Same with the sun. Behind the curve. Because the Earth is curved.

I know of no other way to explain these observations. You're welcome to bring your own interpretation, but if it's vague, prepare to be challenged. Because on something as important as the truth, especially when you deign to rebuke a couple of thousand years of established scientific thought, you'd better be specific, or you'd better go home.

And if this doesn't convince you, the next experiment (in this post) is designed to directly indicate that water does, contrary to flat-Earth myth, curve over the surface of the Earth.