measuring a flat earth

Say that you have levelled a very large field of dirt.  You or your interlocutor have personally inspected enough of it with a level to agree that it is completely flat with respect to gravity.  (Do flat-earthers have fun ideas about what direction gravity pulls toward?)

You (or they) sketch out a grid system across the entirety of this field of dirt.  Squares about 1 mile wide by 1 mile long would match a lot of the farmland covering middle america, but they might be a little small for the height we'd need to attain.  This grid is completely straight and all the lines are either parallel and non-intersecting or perpendicular (incidentally, if this field were to cover the entire earth, what would happen to those parallel lines on a flat earth?)

Now, we rise high into the air, as high as we need to, perfectly perpendicular to our completely flat field (which, again, was confirmed to be completely flat at every point according to our level; also of note is the fact that the atmosphere right now is so clear that your grid is perfectly clear at any height you wish to view it from).  Hold up one of those right-angle rulers (preferably one that you and they used to construct the grid in the first place) with inches or centimeters, or really any regular spaced interval, align the bottom with the closest line of your grid you see, then tilt the ruler outwards until the next tickmark aligns with the next line of the grid.

Better double check your initial point of reference, they're still aligned right?  So you have one inch on your ruler lined up with one square of your grid.  How does inch 2 look?  Pretty close, right?  How about inch 12?

You and your companion may find that your carefully constructed grid seems rather short when it gets further out, how strange. The ruler gets shorter looking too, because the effects of perspective and foreshortening are fairly well understood, but still, not as fast as the grid seems to be shrinking.

When you go back down from your height to double check the spacing of the lines, take the opportunity to install a mirror at the intersections of grid lines.  After you've satisfied yourselves that the grids are in fact equally spaced, head back up to your previous vantage point. This time, take a laser and a stopwatch.

Now, we know that the grids are equally spaced, regardless of what we thought we saw up here before.  And to prove it, we will measure the distance between us and each of those vertices where we placed mirrors using this laser (do proponents of flat-earth theory have any doubts as to the speed of light in our very clear atmosphere in non-relativistic settings?).  By the way, those mirrors are all connected via 5G so that we can maintain a perfect angle of reflection, just in case some stray animal has knocked them around while we were ascending, or we decide to ascend higher or lower than before.

You send a request for all of the mirrors to point to your position by providing them the height you'd reached and they already know their position in the grid as well as where you went up from. Trigonometry easily describes your exact position-- your height divided by their distance from your grid coordinate equals the tangent of whatever angle theta they need to aim themselves to be looking straight at you.

Your companion has the laser and is getting frustrated that it doesn't seem to be working--the light is being reflected, but other than the first couple of mirrors, it seems to be reflecting lower and lower the further out they aim, instead of at the same place each time. No worries, these mirrors have an auto-adjust feature, and with one command, begin to align themselves as your companion shines the laser at them.

Satisfied that they are all capable of reflecting light back at you, you hold a very precise stopwatch; it can measure a millionth of a second!  Which is good, because we have reasonable measurements that the speed of a laser approaches 182,000 miles per second, and while your grid is indefinitely large, 182,000 miles is still rather a lot. Too bad the air has to be so clear for you to see the grid, otherwise it might have slowed the photons down a bit.

The distance between each mirror and your viewing point again is a matter of not even trigonometry, but mere geometry.  Your height (h) times itself plus the distance of each mirror (m) times itself will equal the distance your laser beam will travel times (l) itself.  Oh, and the laser has to actually go there and back(2l), so the time it takes to complete a round trip should be about equal to the square root of (h<sup>2</sup> + m<sup>2</sup>) * 2 divided by 182,000 (to whatever precision you can agree on).

The laser and stopwatch are coordinated so that as soon as the laser fires, the timer begins, and the timer stops the instant the return beam arrives. The times for the closest mirrors are very good, and you take a few minutes to review the angles that the mirrors have indicated they have self-adjusted to.  Some of them seem rather steep, actually, surely they would be aiming too high, but your companion reports that they are still getting measurements, although the values are a little less accurate now, when suddenly they express consternation and reveal that the latest mirror, the furthest out so far, didn't return a beam.

You share your concern about the steep angle of adjustment, and the two of you agree to ascend a bit until this mirror responds to the laser.  Once you do, they resume aiming at the next mirror out, but it too is not responsive.

You check the program for the auto-adjustment, but there are no errors in the math, other than it thinks you are much higher than you know you are.  Your companion admits they've started having difficulty, even with their great visual acuity and the clarity of the atmosphere, discerning the individual mirrors, and they were really aiming at where they assumed the next ones would be.

You decide that the time for experimental methods is over.  It is time to take matters into your own hands.  You hated to do it, but you had been saving a very special filament, infinitely long and malleable, yet extremely light weight and resistant to tear or sagging.  You ask your companion to stay aloft with one end of the filament, and you set off to check the orientation of this mirror yourself.

When you reach the mirror, you see your companion up where you had just come from--it's funny, the distance makes their height seem so much less, and you smile and wave to them, then check the inclination of the mirror.  It matches what the program said it was.  You perceive a flash of monochrome light across your vision, and you laugh. You begin to pull the filament taut.  It is actually marked with measurements itself, just like the ruler was, that match the increments of the grid. When you have it taut and begin to take the reading, you realize that the mirror is mounted on the ground, while you stand a few feet high. So you lie down next to it and take the measurement which seems to be a little longer than trigonometry had told you, but perhaps there was a little sag in your filament, after all.  You roll over and start to get up, but to your horror, your companion has vanished.

All you see are your grid lines, rising away (but they were supposed to be flat), blocking all of your view of your companion (they were so high up, though).

You pull out your level, and the grid field still claims to be perfectly flat.

You pull out your ruler, and the lines in the distance seem to mock you, in every direction you look, insisting on being closer together than they really are.

You look back at the line in your hand, drawn taut, and realize that it seems to be catching on something; you can draw it tighter if you raise it a little ways above the ground.

Late in the evening, you have slumped back to the origin of your grid system.  Your companion is there, weeping.  They scream when you arrive--they thought you had died when you vanished.  Privately, you had had the same thought.  The two of you huddle together in the cold and dark, at the center of a desolate plane of not-quite squares, at the foot of a dizzying obelisk, alone on a planet that obstinately refuses to be flat.  Too disturbed to sleep, you anxiously trade theories that could explain the lunatic observations made on this day.  Tomorrow, you will build a flatter field, a straighter grid, a more precise ruler, more sophisticated mirrors, a narrower laser beam, and a filament that doesn't sag in the middle.

When sleep takes you both regardless, you dream of an existence that would allow itself to be modeled and justified more easily.