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>In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the

>In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius 1, the area equals the triangle excess To all the flat earthers out there, this property can be used to find out earth is not flat, just by drawing a giant triangle on the surface, without leaving the earth. Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. But this approach proves it without the need to refer the sun.

fluoridation 10mo ago
>Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. That wasn't to prove the Earth is round (and it doesn't prove it). Eratosthenes assumed two things when he performed his experiment: 1) the Earth is round, and 2) the Sun is an infinite distance away. By just this experiment he would have been unable to distinguish between this situation and the Earth being flat while the Sun being only a finite distance overhead (and in fact a fair bit closer than it actually is). Eratosthenes and his contemporaries were already convinced of the roundness of the planet, and he simply wanted to measure it. >But this approach proves it without the need to refer the sun. A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. If the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? I.e. continuously moving towards the South because the polar coordinates curve towards your left as you progress.
- roywiggins 10mo ago
  >the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? This is something that was more or less solved a long time ago with surveying instruments. You don't have to move in a straight line, you build triangles out of sight lines.
  - fluoridation 10mo ago
    I can kinda see how that would work, but it presents the challenge that whatever route you plan, it cannot go over water for more than a few kilometers.
    - roywiggins 10mo ago
      I don't think it would be that different than the arc measurements that were actually done, you triangulate a bunch of points to work out distances and angle sufficiently precisely: https://en.wikipedia.org/wiki/Arc_measurement https://en.wikipedia.org/wiki/Arc_measurement
      - fluoridation 10mo ago
        That doesn't help you if you're moving West to East, though. EDIT: Also, that's to measure distance, not direction.
        roywiggins 10mo ago
        You can survey West to East just as well: https://www.noaa.gov/digital-collections/collections/4774/item?page=123 https://www.noaa.gov/digital-collections/collections/4774/it... https://amerisurv.com/2012/10/27/the-longest-line/ https://amerisurv.com/2012/10/27/the-longest-line/ And measurements of, say, very precise equilateral triangles, necessarily imply certain interior angles, which you can compare to the actual angles they make. For instance, on a flat plane, you can fit six equilateral triangles sharing one point to make a hexagon. On a sphere if you make them big enough you'll find that they don't quite fit.
        fluoridation 10mo ago
        >https://www.noaa.gov/digital-collections/collections/4774/item?page=123 https://www.noaa.gov/digital-collections/collections/4774/it... Like I said before, you can't do that over water. It constrains the shape of the triangle you can measure.
- teo_zero 10mo ago
  > A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. Do flat-earther reject the existence of LASER, too?
  - fluoridation 10mo ago
    Flat-earthers don't accept that a flat plane implies infinite line of sight (especially at sea), so who knows.
thaumasiotes 10mo ago
> But this approach proves it without the need to refer the sun. Only if you're happy "proving" your argument to an audience that never had any doubts. You can't use this argument to prove the earth is not flat over the objections of your audience because you can never convincingly show that any given line is straight.
lwansbrough 10mo ago
Once you internalize that flat-Earther-ism isn’t about the Earth being flat you realize that rational arguments are pointless. To expand on that, it’s about community and finding people who share your interests. The movie Behind The Curve explores this idea and it’s quite revealing.
- QuadrupleA 10mo ago
  And the ego boost of it all - being one of the special few who sees "the truth" that others are too brainwashed/dumb/whatever to see. Makes one feel quite important.
  - lordnacho 10mo ago
    Indeed, this might be why religion seems so odd to outsiders. It's implausible, yet that's what stimulates the tribal feelings among the believers.
  - kakacik 10mo ago
    Those are the simple cows to be milked, but numerous 'gurus' in these communities are very well aware of the bullshit they propagate to the weak and gullible, but its just such an easy noncritical prey. You can always just go deeper in paranoia. Makes me think that mr trump switched from being democrat to republican and pushed for magaesque folks who often love him to the death due to very similar principle - just spit out some populist crap that stirs core emotions - the worse the better, make them feel victim, find easy target to blame which can't defend themselves well (immigrants), add some conspiracy (of which he is actually part of as wall street billionaire). Extreme left wouldn't swallow easily that ridiculous mix from nepotic billionaire who managed to bankrupt casinos and avoided military duty (on top of some proper hebephilia with his close friend mr E and who knows what else). But what do I know, just an outside observer, but nobody around the world has umbrella thick enough that this crap doesn't eventually fall on them too.
    - immibis 10mo ago
      I think Trump's just been running a simple popularity-seeking loop for a while. Do a thing; if his people like it, do it more; otherwise do it less. I've heard that even Hitler was like this: that he didn't start out hating Jews, but repeatedly reacted to the fact that he got louder cheers whenever he blamed things on Jews. But I don't know how to verify if this is true.
Sharlin 10mo ago
As they say, you can't reason someone out of something they didn't reason themselves into in the first place.
themafia 10mo ago
> relied on the sun shining directly overhead on wells in different cities. It was just one city actually. The critical piece is that the city's northern latitude was nearly identical to the Earth's angle of axial tilt. Which also means that this shadow phenomenon only occurs during the Summer Solstice. https://www.khanacademy.org/science/shs-physical-science/x0492729e402f0c6b:4th-quarter/x0492729e402f0c6b:evolving-models-of-the-universe/a/eratosthenes-of-cyrene https://www.khanacademy.org/science/shs-physical-science/x04...
taneq 10mo ago
It also means that pi could be equal to 3 if you world is small enough.
schobi 10mo ago
This sounds more like a Matt Parker video idea - get a bunch of people, three theodolites to measure angles accurately, a good location and start measuring angles for line of sight and see how well this determines the earth's radius. Rough estimate - with an excellent 0.5" angular resolution and 35km triangle this could work.
amelius 10mo ago
No, you're talking about a hologram. Everything is flat. https://en.wikipedia.org/wiki/Holographic_principle https://en.wikipedia.org/wiki/Holographic_principle

fluoridation 10mo ago

>Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. That wasn't to prove the Earth is round (and it doesn't prove it). Eratosthenes assumed two things when he performed his experiment: 1) the Earth is round, and 2) the Sun is an infinite distance away. By just this experiment he would have been unable to distinguish between this situation and the Earth being flat while the Sun being only a finite distance overhead (and in fact a fair bit closer than it actually is). Eratosthenes and his contemporaries were already convinced of the roundness of the planet, and he simply wanted to measure it. >But this approach proves it without the need to refer the sun. A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. If the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? I.e. continuously moving towards the South because the polar coordinates curve towards your left as you progress.

roywiggins 10mo ago
>the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? This is something that was more or less solved a long time ago with surveying instruments. You don't have to move in a straight line, you build triangles out of sight lines.
- fluoridation 10mo ago
  I can kinda see how that would work, but it presents the challenge that whatever route you plan, it cannot go over water for more than a few kilometers.
  - roywiggins 10mo ago
    I don't think it would be that different than the arc measurements that were actually done, you triangulate a bunch of points to work out distances and angle sufficiently precisely: https://en.wikipedia.org/wiki/Arc_measurement https://en.wikipedia.org/wiki/Arc_measurement
    - fluoridation 10mo ago
      That doesn't help you if you're moving West to East, though. EDIT: Also, that's to measure distance, not direction.
      - roywiggins 10mo ago
        You can survey West to East just as well: https://www.noaa.gov/digital-collections/collections/4774/item?page=123 https://www.noaa.gov/digital-collections/collections/4774/it... https://amerisurv.com/2012/10/27/the-longest-line/ https://amerisurv.com/2012/10/27/the-longest-line/ And measurements of, say, very precise equilateral triangles, necessarily imply certain interior angles, which you can compare to the actual angles they make. For instance, on a flat plane, you can fit six equilateral triangles sharing one point to make a hexagon. On a sphere if you make them big enough you'll find that they don't quite fit.
        fluoridation 10mo ago
        >https://www.noaa.gov/digital-collections/collections/4774/item?page=123 https://www.noaa.gov/digital-collections/collections/4774/it... Like I said before, you can't do that over water. It constrains the shape of the triangle you can measure.
teo_zero 10mo ago
> A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. Do flat-earther reject the existence of LASER, too?
- fluoridation 10mo ago
  Flat-earthers don't accept that a flat plane implies infinite line of sight (especially at sea), so who knows.

thaumasiotes 10mo ago

> But this approach proves it without the need to refer the sun. Only if you're happy "proving" your argument to an audience that never had any doubts. You can't use this argument to prove the earth is not flat over the objections of your audience because you can never convincingly show that any given line is straight.

lwansbrough 10mo ago

Once you internalize that flat-Earther-ism isn’t about the Earth being flat you realize that rational arguments are pointless. To expand on that, it’s about community and finding people who share your interests. The movie Behind The Curve explores this idea and it’s quite revealing.

QuadrupleA 10mo ago
And the ego boost of it all - being one of the special few who sees "the truth" that others are too brainwashed/dumb/whatever to see. Makes one feel quite important.
- lordnacho 10mo ago
  Indeed, this might be why religion seems so odd to outsiders. It's implausible, yet that's what stimulates the tribal feelings among the believers.
- kakacik 10mo ago
  Those are the simple cows to be milked, but numerous 'gurus' in these communities are very well aware of the bullshit they propagate to the weak and gullible, but its just such an easy noncritical prey. You can always just go deeper in paranoia. Makes me think that mr trump switched from being democrat to republican and pushed for magaesque folks who often love him to the death due to very similar principle - just spit out some populist crap that stirs core emotions - the worse the better, make them feel victim, find easy target to blame which can't defend themselves well (immigrants), add some conspiracy (of which he is actually part of as wall street billionaire). Extreme left wouldn't swallow easily that ridiculous mix from nepotic billionaire who managed to bankrupt casinos and avoided military duty (on top of some proper hebephilia with his close friend mr E and who knows what else). But what do I know, just an outside observer, but nobody around the world has umbrella thick enough that this crap doesn't eventually fall on them too.
  - immibis 10mo ago
    I think Trump's just been running a simple popularity-seeking loop for a while. Do a thing; if his people like it, do it more; otherwise do it less. I've heard that even Hitler was like this: that he didn't start out hating Jews, but repeatedly reacted to the fact that he got louder cheers whenever he blamed things on Jews. But I don't know how to verify if this is true.

Sharlin 10mo ago

As they say, you can't reason someone out of something they didn't reason themselves into in the first place.

themafia 10mo ago

> relied on the sun shining directly overhead on wells in different cities. It was just one city actually. The critical piece is that the city's northern latitude was nearly identical to the Earth's angle of axial tilt. Which also means that this shadow phenomenon only occurs during the Summer Solstice. https://www.khanacademy.org/science/shs-physical-science/x0492729e402f0c6b:4th-quarter/x0492729e402f0c6b:evolving-models-of-the-universe/a/eratosthenes-of-cyrene https://www.khanacademy.org/science/shs-physical-science/x04...

taneq 10mo ago

It also means that pi could be equal to 3 if you world is small enough.

schobi 10mo ago

This sounds more like a Matt Parker video idea - get a bunch of people, three theodolites to measure angles accurately, a good location and start measuring angles for line of sight and see how well this determines the earth's radius. Rough estimate - with an excellent 0.5" angular resolution and 35km triangle this could work.

amelius 10mo ago

No, you're talking about a hologram. Everything is flat. https://en.wikipedia.org/wiki/Holographic_principle https://en.wikipedia.org/wiki/Holographic_principle