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

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Perspective and the Lake Pontchartrain Transmission Line

The Lake Pontchartrain power transmission pylons demonstrate Earth’s curvature. Flat-Earthers invented various excuses to dismiss the observation, including the excuse that it was just a perspective effect.

If it were just a perspective effect, the same parts of the pylons would line up in a straight line, converging into a distant point. In reality, they do not line up in a straight line but are visibly curving downward due to the curvature of the Earth.

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Different Reasons Why a Distant Object is Not Visible

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Different reasons can cause a distant object to be not visible:

  1. The angular resolution limit of the observer.
  2. The visibility limit imposed by the atmosphere.
  3. Obstruction by another object, including by Earth’s curvature.

Flat-Earthers incorrectly presumed “a distant ship is not visible only because of Earth’s curvature.” Incorrectly concluded if we can bring the ship back into view, the curve must not exist. In reality, Earth’s curvature is not the only thing that can cause a distant ship to be not visible; other reasons can also cause it.

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Flight Instruments

Pilots can easily observe that Earth is a sphere, either by visual observation or from the aircraft’s flight instrument.

The level indicator is above the visible horizon. It is the dip of the horizon caused by the fact that Earth is a sphere. Earth’s horizon itself is visibly curving. The horizontal component of the velocity vector is often not the same as the plane’s direction due to the wind & the Coriolis effect from Earth’s rotation.

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Fisheye Lenses and Lens Distortions

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A photographic lens may have a specific distortion characteristic and strength that may differ from another lens. However, despite the differences, a straight line will always appear straight if it crosses the center of the frame.

Flat-Earthers like to dismiss images taken by a fisheye lens. In reality, the attributes of such a lens are well known. We can use the knowledge to determine if a line in the image is straight, even if it was taken using a fisheye lens.

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Zoom, Digital Resolution, and the Visibility of a Distant Object

A camera with a zoom lens has a variable field of view but a fixed output resolution. As a result, its angular resolution depends on the zoom factor. Changing the zoom factor will change the ability of the camera to resolve a distant object.

Flat-Earthers show us zooming in reveals an unseen object & uses it to “disprove” Earth’s curve. In reality, the object was previously unseen due to the angular resolution limit at wide zoom. It is not far enough to be obscured by Earth’s curvature.

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Thompson vs. Garcia, 2019

Thompson vs. Garcia was a lawsuit about a “contest to prove Earth’s curve” set up by the defendant, a flat-Earther. The plaintiff felt he had won the contest & the defendant failed to pay. The plaintiff did provide evidence of Earth’s curvature, but not in the way required by the contest. Thus, the court ruled in favor of the defendant.

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Glass of Water

Earth’s gravity causes water’s surface to curve with the same radius as Earth’s. On glass with a diameter of 10 cm, Earth’s gravity causes the water to form a curvature of 0.00000002 cm, outside other factors like surface tension.

Flat-Earthers like to demand that we prove water curvature in a glass of water. In reality, whatever the container, the curvature of the water surface remains the same as Earth’s radius. On a glass of water, it will be too small to observe.

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Turning Torso Building, Malmö, Sweden

Turning Torso is a skyscraper in Malmö, Sweden, having a distinctive segmented shape & easy to recognize from far. From Copenhagen, Denmark, we can see its lower part is obscured more if we are farther because Earth is a sphere.

The building is close to the strait of Øresund. Copenhagen, Denmark, lies at the other side of the channel, only 15 km (9 miles) away from Malmö. Turning Torso has a distinctive segmented shape that is easy to recognize from far. And more importantly, for our purposes, its segmented form makes it easy to judge its height from far.

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The Wallace Method

In 1870, Alfred Russel Wallace proved Earth’s curvature in the Bedford level experiment. His method was straightforward & effective and can be easily used in a similar experiment.

Wallace’s method fixes the common issues with similar experiments done by flat-Earthers. They usually fail to control, and even disregard, the effect of atmospheric refraction. Using Wallace’s simple method, we can easily prove Earth is a sphere.

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Magnitude of Visible Earth’s Curvature in a Photograph

The magnitude of the curvature that appears in a photograph of Earth’s curvature depends on several factors:

  1. The observer’s altitude.
  2. The camera’s field of view or focal length.
  3. The distortion characteristics of the camera lens.

Flat-Earthers like to dismiss a photo of Earth’s curvature by comparing it to another photo showing a different amount of curvature. In reality, to compare the visible curvature, we need to ensure all the images were taken from the same altitude, same field of view, and account for the lens’ distortion.

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Auguste Piccard, A Scientist Falsely Claimed by Flat-Earthers

In 1931, Auguste Piccard went up in a balloon to an altitude of 15781 m, not far above the cruising altitude of an airliner. At the altitude, Earth’s curvature is still very slight and difficult to see, especially through the small portholes in his chamber.

In a Popular Science interview, Piccard was reported to have said that Earth “seemed a flat disc with an upturned edge.” Flat-Earthers quickly interpreted his statement as if he was telling us Earth is flat. In reality, in another interview, it is clear that he is convinced that Earth is a sphere. In his writings about his expeditions, the word “globe” was also mentioned several times.

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Perspective Compression: Determining If a Curvature Exists

By utilizing perspective compression, it will be easier for us to observe the curvature of an object if the object is curved.

Flat-Earthers like to deliberately choose a vantage point where it is difficult to see Earth’s curvature, and they use it to “prove” the curvature does not exist. In reality, they only make it more difficult to see the curvature. It does not mean the curve is not there.

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Bedford Level Experiment Confirmed the Curvature of the Earth

In 1870, Alfred Russell Wallace took a challenge from a flat-Earther and successfully confirmed the existence of Earth’s curvature in the Bedford level experiment.

Wallace did the experiment on the Bedford Canal. Wallace fixed a black band on the Old Bedford Bridge and observed it with a telescope from the Welney Bridge 6 miles away. Right in the middle, he placed a pole with two discs on it. The telescope, the top disc, and the black band are at the same height above the water.

From the telescope, both discs were seen above the black band, proving the existence of Earth’s curvature.

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Salar de Uyuni and Other Salt Flats

Salt flats are flat expanses of ground covered with salt and other minerals. Salar de Uyuni in Bolivia is the world’s largest salt flat. Salt flats appear visibly flat, but they still follow the curvature of Earth.

Flat-Earthers claim the apparent flatness of Salar de Uyuni and other salt flats “proves” a flat Earth. In reality, while salt flats appear flat, they still follow Earth’s curvature. On a salt flat, the bottom part of a distant object will be obscured by Earth’s curvature, similar to at sea.

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Airplanes Going Over Earth’s Curvature

To an observer on the ground, an airplane at 36000 ft will start going over Earth’s curvature at a distance of 400 km. However, we are also limited by atmospheric visibility. In an extremely best-case scenario, we can only see up to 240 km.

We can easily see ships going over Earth’s curvature, and flat-Earthers like to demand us the same thing for airplanes. In reality, at the typical distance a plane does that, it is far out of visibility range. But if it is very low, like when approaching a runway to land, it is possible to see the plane going over the curvature of the Earth.

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Archimedes and the Surface of any Fluid

In the 3rd century BCE, Archimedes of Syracuse wrote what we now call the Archimedes’ principle in his book “On Floating Bodies” using the spherical Earth model.

Flat-Earthers like to misuse Archimedes’ principle as if it supports their claim that Earth is flat. In particular, they abuse Archimedes’ principle as if it is a competing explanation against gravity. In reality, Archimedes and other Greek scientists at the time already knew Earth is a sphere, and he explicitly mentioned it in his writings.

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Observing Earth’s Curvature From a Flight

At the cruising altitude of a jet airliner, Earth’s curvature is too slight for us to notice casually. But with planning and careful observation, it is not impossible to see the curvature. Continue reading “Observing Earth’s Curvature From a Flight”

Curvature of Water Surface

If Earth is a sphere, why is the surface of water always flat?

Due to Earth’s gravity, water seeks the lowest potential or as close as possible to the center of Earth. It forms a practically spherical surface centered on Earth’s center. The width of the surface in everyday cases is tiny compared to Earth’s radius. Thus, the water surface looks practically flat but not perfectly flat.

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Where’s the Curvature?

A human eye can only visually perceive the curvature of the Earth if we are at a considerable altitude from the surface, which is still beyond the reach of most humans today. A commercial jet airliner is the highest position realistically attainable by most humans today. We can only perceive very slight curvature at such altitude and, even then, only in an ideal condition.

The basis of flat-Earthers’ belief is that the horizon appears flat. They would say if we cannot see the curvature, then there’s no curvature, and therefore, the Earth is flat. In reality, only a few people can travel high enough to see the curve.  The highest we can realistically go is by getting on a commercial jet airliner, which can only go about 11-15 km up, only a fraction of Earth’s radius.

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