Correct Hidden Formula

Flat-Earthers like to use the erroneous “8 inches per mile squared” to calculate the height hidden by Earth’s curvature. The following is the correct equation for the purpose, accounting for the observer’s height & atmospheric refraction

$h_\mathrm{h} = { R \over \cos\left[ { s \over R } - \arccos\left( { R \over R + h_\mathrm{O} } \right) \right] } - R$

Refraction Effect on the Lake Pontchartrain Power Transmission Line

The strength of atmospheric refraction is not constant but varies depending on the weather condition. As a result, a distant object usually obscured by Earth’s curvature can sometimes be visible if the refraction is strong enough.

Lake Pontchartrain transmission towers demonstrate Earth’s curvature. Flat-Earthers like to cherry-pick moments where the towers appear straight, ignoring other cases. In reality, they look straight only because the refraction was strong enough.

Flat-Earthers’ laser tests are misleading. They pointed the laser from near the surface toward a distant observer. If the observer sees the beam, they conclude Earth is flat. It was a misunderstanding about laser & how atmospheric refraction affects it.

Refraction of Laser Beam

Lasers are distinguished from other light sources by their coherence, or that their frequency and waveform are identical, and the output being a narrow beam. But lasers are still a form of light and exhibit the same characteristics as other light sources.

Flat-Earthers assume lasers always travel in a straight line, not affected by the medium. They use this incorrect assumption in their “experiments” to “prove” a flat Earth. In reality, like other light sources, atmospheric refraction can bend laser beams.

Blood Moon: Why The Moon Turns Red During a Total Lunar Eclipse

The Moon becomes red during a total lunar eclipse because Earth’s atmosphere acts like a lens and refracts sunlight toward the Moon. It also scatters bluish components of sunlight more strongly and lets more reddish components reach the Moon.

The Moon does not become totally dark during a total lunar eclipse, and flat-Earthers use it as “evidence” the Moon is not in Earth’s shadow during a lunar eclipse. In reality, it occurs due to optical phenomena involving Earth’s atmosphere.

Experiments Involving Flashlights, Laser, & Reflected Sunlight Over Water Surface

Atmospheric refraction is stronger near the surface. Additionally, a thermal inversion can form a duct guiding light ray to follow the curvature of Earth. These phenomena allow light rays to reach the viewer far behind the horizon, but the rays will be compressed, distorted, and scattered, and will not form the true image of the objects. However, a nondescript, intense point light source will stand out against a darker background.

Flat-Earthers like to perform experiments involving lasers, flashlights, and mirrors over the water surface. If the light is visible from the other side, they would wrongly conclude Earth’s curvature does not exist. In reality, atmospheric refraction causes the result they observed.

Curvature App

Curvature App is a web application we can use to create a simulation of the horizon’s shape and the visibility of distant objects on a flat and spherical Earth. Curvature App is available at walter.bislins.ch/Curve.

When determining a distant object’s visibility, flat-Earthers often use wrong calculators that fail to account for all the important variables and give them incorrect results. Curvature App accounts for all of these variables, and not only it gives us the numbers, but also the simulated view.

Refraction and Water

Refraction is the change in the direction of light due to the change in the medium’s refractive index traveled by the light. It is convenient to use water in a demonstration of refraction. But water alone is not the cause of refraction.

Flat-Earthers falsely claim that just because our atmosphere has water vapor in it, it will produce the same effect as any demonstration of refraction involving water. In reality, it requires far more reasoning than just that water is involved.

The Impossible Eclipse

A selenelion is a rare lunar eclipse where the Sun and the Moon are both visible at the same time. An even rarer form of selenelion occurs if it is a partial lunar eclipse, and the upper part of the Moon is eclipsed. Some call this an “impossible eclipse.”

Flat-Earthers claim that such an eclipse should not be possible to occur because the Earth’s shadow is in the wrong position. In reality, it is possible to happen because the observer is looking slightly downward due to the dip of the horizon and atmospheric refraction.

Observation of Chicago From Michigan

Chicago’s skyline is visible on clear weather across Lake Michigan from Michigan as long as the observer is on high ground, or if atmospheric refraction is strong enough. This observation is consistent with the expectations if Earth is a sphere.

Flat-Earthers claim that the visibility of Chicago from Michigan is not possible if Earth is a sphere. In reality, they failed to account for the height of the observer or atmospheric refraction and ignored the fact that only the upper parts of the buildings are visible.

Inferior Mirage

An inferior mirage is a phenomenon in which atmospheric refraction bends light rays to produce a mirrored image below a real object. It occurs when a hot surface heats the layer of air above it, placing it below a colder & denser layer of air.

Flat-Earthers like to use inferior mirages to “explain” how a distant object can appear partly obstructed if the Earth is flat. In reality, an inferior mirage cannot produce an appearance similar to an object partially obscured by Earth’s curvature.

Refraction: Only a Baseless Technobabble in Flat Earth Community

Flat-Earthers use the phenomenon of refraction as a catch-all “explanation” every time they observe something not appearing in the expected position if the Earth is flat. They use the term “refraction” as mere technobabble without any underlying explanation.

The following are how we can distinguish real-world, scientific refraction from the technobabble used by flat-Earthers:

Distance to the Horizon & the Black Swan Observation

The distance to the horizon depends on the height of the observer and atmospheric refraction. The variability in atmospheric refraction can make the horizon appear in front of a distant object sometimes, and behind it in other times.

Some flat-Earthers raised a case —dubbed the “black swan” case— in which the horizon appears to be behind a distant object which is farther than the distance to the horizon according to calculation, and they use the fact to “disprove” Earth’s curvature. In reality, they did not account for atmospheric refraction, and that the amount of refraction varies.

The Legendary AutoCAD Diagram Involving the Pythagorean Theorem from Flat-Earthers

There is a legendary diagram of spherical Earth from flat-Earthers that was made using AutoCAD involving the Pythagorean theorem, flaunting the fact that it is was created “using AutoCAD 2016 with 15-digit precision”™. The diagram is not wrong, but it does not account for the height of the observer and atmospheric refraction. And therefore, it is unsuitable for determining the obstruction of a distant object due to Earth’s curvature.

The numbers in the diagram will give us a much higher number than it should be. If a distant object is visible, but according to the diagram it should not, then flat-Earthers will incorrectly conclude that the Earth is flat.

The Appearance of Sunset and Atmospheric Refraction

The appearance of sunset depends on atmospheric conditions. Thermal inversion can irregularly refract light coming from the sun and distort the image of the Sun seen from an observer.

Some flat-Earthers observed a specific appearance of sunset and presented the fact as if it “proves” a receding sun in the flat-Earth model. They are wrong. The appearances of the sunsets are the results of atmospheric refraction.

Eight Inches Per Mile Squared

The “eight inches per mile squared” is a rule of thumb to determine the drop height due to the curvature of the Earth. It does not account for the observer’s height and atmospheric refraction. And therefore, the rule is unsuitable to determine the amount of obstruction of a distant object due to Earth’s curvature.

Many Flat-Earthers often use the “8 inches” rule to reach the conclusion similar to “X is visible, but at the distance of Y miles, X should be Z feet below the horizon, so the Earth is flat.” They are wrong. The “8 inches” rule is the wrong tool for the purpose as it does not account for the height of the observer and atmospheric refraction.

Variability of Atmospheric Refraction

In most situations, atmospheric refraction bends light downwards and causes objects to appear higher than they actually are. The strength of atmospheric refraction is not constant. It depends on weather condition and varies on the different seasons, different days, and even different times of the day.

Because of the variability of atmospheric refraction, a distant object that is usually obstructed by Earth’s curvature can sometimes be visible. Some flat-Earthers would cherry-pick moments when the object is visible. They would show such specific moments to everyone and use them to “disprove” Earth’s curvature; happily ignoring the cases where Earth’s curvature partly or entirely obstructs the object.

Ships Disappearing Over the Horizon and the Various “Explanations” Invented by Flat-Earthers

Due to Earth’s curvature, ships traveling over an ocean disappear from the bottom up. This fact is one of the first evidence to confirm the Earth is a sphere, and one of the first facts of which flat-Earthers had to invent various “explanations” for.

Some of the popular “explanations” are: refraction, perspective, zooming reveal distant ships and visibility limitations. None can explain away the fact.

Earth’s Curvature Calculation

The amount of obstruction of a distant object that is caused by Earth’s curvature depends on:

1. The distance of the object.
2. The height of the observer.
3. The height of the object.
4. The magnitude of atmospheric refraction.

Flat-Earthers like to use the visibility of a distant object to prove Earth’s curvature does not exist. Very often, they failed to account for observer’s height and atmospheric refraction, or make other mistakes, like unit conversions errors, distance calculation errors, etc. Once all are considered for, and mistakes are fixed, everything will be consistent with spherical Earth.

Infrared Vision

When we are unable to see a distant object, then it is due to one of these reasons:

• Our eyes lack the sufficient angular resolution to recognize the object.
• Atmospheric condition limits visibility.
• The object is far enough and obscured by Earth’s curvature.

Some flat-Earthers like to show that a previously unseen distant object can be brought into view using infrared vision. They would take that as ‘evidence’ of the non-existence of Earth’s curvature. They are wrong. Infrared vision can reveal hidden objects caused by visibility limitations, but not ones obscured by Earth’s curvature.