Ah, the internet. The signal-to-noise ratio is atrocious. Make no mistake, I am not in the Flat Earth camp, but I like to test claims and explore how they can be supported or disproved. My background is in drafting and design, so I am reasonably skilled in the practical application of geometry and trigonometry. Let's take a look at the argument that the reason Polaris can't be seen south of the Equator is because of "perspective."
Diagram the Theory
Polaris, also known as the North Star, appears fixed in place while other stars sweep across the sky over the course of the night. According to the globe model, this star is in line with the axis of Earth's rotation as it spins, causing night and day. According to most flat Earth theories, this is the center of a dome covering the circle of the earth. I know all of this is still abstract, but geometric proofs do not rely on scale.
My apologies for the pixels. CorelCAD, AutoCAD, and similar programs can look quite rough in drawing space no matter how precise the calculations are behind the scenes, but the point it illustrates should hopefully be crystal clear even if the image is not. Download LibreCAD or break out the drafting tools and create your own diagram with pen and paper if you like. If you have suggestions for improvement, feel free to let me know in the comments.
The green line at the bottom is the plane of a hypothetical flat Earth. Elevation variations are comparatively negligible, so I have omitted them for simplicity. Hash marks are equidistant, and I marked them with degree indicators for reference. There are three different arcs for the purported celestial dome, since descriptions vary. The center dome is a hemisphere. The lower dome is an elliptical arc with the secondary radius equal to one half the primary radius. The upper dome is an ellipse with the secondary radius set to 1.5 times the primary radius.
The nearest major city to me is Spokane, Washington, so I have added a marker for my latitude on the horizontal line. I also added a secondary marker for Buenos Aires, Argentina, because it is a significant city well south of the Equator and was relevant to the discussion which inspired this post. The dashed purple lines represent the angle from these locations to Polaris on each of these potential domes, and measurements are rounded to the nearest whole degree. Feel free to duplicate this drawing, check my measurements, adjust the dome curve, and substitute cities from anywhere in the world, including south of the Equator.
Crunch the Numbers
According to the flat Earth model, the angle to Polaris will get shallower as the observer is further south, but will never drop below the horizon even if the height to Polaris is far lower than shown. My diagram should help visualize the situation even if your specific flat earth model suggests a different dome center height.
Perhaps you remember the trigonometry formulas based on the ratios of the sides and angles of all right triangles, including,
tanθ = (opposite side)/(adjacent side)
where θ is the angle measured between the adjacent side and the hypotenuse, which can be rearranged to,
(adjacent side) tanθ = opposite side
So, if you know your distance south of (or outward from?) the North Pole, and you can measure the angle to Polaris, you can find the height of the peak of that dome with a basic scientific calculator. You could even build a crude sextant with a protractor by attaching a sighting tube and plumb bob. The mathematical principles here should be beyond dispute. Show the geometry of your theory. Test it against real observations.
Observation and Testing
Even from the alleged Antarctic ice wall on the typical flat Earth model, Polaris would still be above the horizon, and never fall below it. We would, however, expect the inverse square law to kick in, and the star would be increasingly dim as the observer moves further away.
In contrast, according to the globe model, Polaris should not be visible south of the Equator. Well, technically, under ideal conditions thanks to its slight divergence from the north pole and atmospheric lensing, it can be seen perhaps slightly further south by a degree or so, but even then, barely over the horizon. In addition, if the star is so far away that the radius of the entire Earth is insignificant in comparison, the apparent brightness will be the same no matter where the observer is in the northern hemisphere where it remains visible at all, and light pollution or cloud cover will be the main factors affecting visibility. The angle to Polaris above the horizon will approximately equal the viewer's latitude, with allowances for instrument precision, atmospheric factors, etc.
The usual counterclaim I have seen at this point is perspective, and the assertion is that since things look smaller in the distance and converge toward the horizon at a vanishing point, distant objects always vanish toward the horizon. Not so. Objects do not merely vanish toward the horizon. Three-point perspective drawings also account for tall structures and objects appearing to shrink upward from the observer on the ground, or toward the ground from an observer above, and not just toward one or two vanishing points on the horizon. Stand at the base of a skyscraper and observe how all parallel lines appear to converge toward some point in the sky, or look down at the street from the observation deck of a tall building and observe how everything seems to converge toward some point below the ground. This optical effect and the horizon are simply not relevant to the topic at hand.
Conclusion
In short, what does a given model predict, and how can you test the idea? What challenges might disprove it, requiring it to be either refined or discarded? A scientific claim is one which can be subjected to testing for verification or falsification. If good test methodology breaks your model, good science requires reassessing the model itself. If your idea cannot be subjected to analysis, it isn't science. That is not to say truth can only be discovered in a laboratory, but claims which are based on measurement and analysis can be tested by measurement and analysis. In fact, feel free to comment if there is a fundamental flaw in my own analysis or diagram. It is very late as I write this, so for all I know it is completely incoherent after my insomniac editing.
In addition, many other problems also arise on the flat earth model. For example, constellations would morph over the course of a single night as the varying angular relationships between the stars shift from the viewer's perspective. People would see different sides and phases of the Moon from different places. This only scratches the surface of analyzing what flat earth models would predict when examined more closely against observations at night. I did not want to distract from the simple illustration regarding Polaris and trigonometry, but if you want to take your analysis to the next level, opportunity abounds. Remember, a crazy idea can still be a serious opportunity to explore mathematics and science!
