Bravo. A very nice article comparing and contrasting standard trig methods and geometric algebra methods.
And as a bonus, it compares both with Dr. Wildberger's Rational Trigonometry. (Side note: I've known a few Australian mathematicians who embrace some very non-standard methods in mathematics, and I think it's great. We need people to shake up our thinking bout Math every once in a while.)
A very nice piece for some relaxing weekend recreational math.
One of Jearl Walker's articles in The Amateur Scientist series in Scientific American discusses a method to measure the size of the earth
with only a foot rule or a stopwatch while observing setting sun. 1979 (05) pp. 172-182
The flat-earth community is actually kind of fascinating. Someone recently paid to fly some flat-earthers down to Antarctica to witness the 24-hour sun which is impossible under most flat-earth models [1]. The trip was extensively documented, and one of the flat-earthers who made the trip actually changed their position. The result, predictably, was that this person was ostracized from the community. The intellectual backflips that the flat-earth community are making to try to debunk this experiment is simultaneously fun and disturbing to watch.
The easiest way to debunk the flat-earth model is simply to observe that that 1) the sun sets below the horizon and 2) this happens at different times in different locations. On a flat earth, if the sun is seen at the horizon in one location it has to be seen at the horizon everywhere.
Not sure "fascinating" is the word I'd use. While there might be some people that are genuinely confused, the flat Earthers that I've seen debate people who went on the Final Experiment are nothing more than pathological liars.
When prominent flat Earthers heard about and were invited to participate in the Final Experiment, none of them were excited that the globers would finally see the Sun isn't up for 24 hours. They attacked the flat Earthers that did agree to go. All of the globers bought new equipment with their own money and sought out experiments to do and record. The flat Earthers were essentially dead weight, none of them actually planed to record a sunset.
Maybe. But I think there's something deeper going on. The flat-earthers-are-nothing-more-than-pathological-liars hypothesis fails to explain why they cling to this particular lie with such vehemence.
I don't mean to get political here, but Donald Trump is a pathological liar. He lies about everything, and he flits effortlessly from lie to lie. He never even offers the pretense of an argument or a justification. He simply proclaims that, say, they're eating the dogs because he read it on the Internet. The flearthers by way of very stark contrast have a laser-focus on this one lie, and they put a lot of effort into coming up with plausible-sounding justifications for it.
Yes, it is true that it is clear that at the end of the day the flearthers don't actually believe their own rhetoric. But it is still, to me, a phenomenon distinct from pathological lying, and which survives despite being so violently at odds with observation. I find that fascinating.
At least of the people who've been on the debates, the motive is quite clearly money. They all have apps, merch, blogs, or Youtube Channels they make money from.
Sure, you could say the same thing about most religious leaders. That doesn't necessarily mean that they don't sincerely believe what they're preaching. Flat-earth doesn't seem to me to be a particularly lucrative scam, the audience is too small. If money was really their primary motive I would think they'd find greener pastures elsewhere.
The original Flat Earth "movement" was an exercise to show that it is impossible to prove anything without some leaps of faith about the nature of the Universe or interpolating between gaps in definite knowledge..
There is always an assumption that can be denied. Turtles all the way down.
Later it grew into to a haven for a certain kind of mentally ill person and grifters that profit from them, illustrating the philosophical Problem of Subjectivity: malice cannot be reliably distinguished from stupidity.
It's a little more nuanced than that. The flat earth movement originated with Samuel Rowbotham who actually conducted an experiment in 1838 whose results (he claimed) demonstrated that the earth was flat [1]. I think it's arguable that he was engaging in the scientific method in good faith.
> The result, predictably, was that this person was ostracized from the community.
This one sentence probably explains why the vast majority of humans don't change how we behave. The fear of losing almost all of the people we love is SO strong.
I wonder how much of a difference refraction of the suns light as it hits the atmosphere makes? It’s definitely the case that we see sunsets happening after the sun has ‘geometrically’ set because of this. I guess one needs to make some argument about the rays from the reflected and non-reflected cases being close enough together that they get sufficiently similar angles of refraction.
Something doesn't sit right with me. I would expect that the semicircle would be reflected nonlinearly. That would imply the reflection is not a true circle. Am I wrong?
I'm also a little skeptical with this approach, as it's not taking into account the curvature of light through the atmosphere. In reality, when the photo was taken, the sun was already fully below the horizon. It just looks like it isn't because the atmosphere is bending the light a little ways around the earth. (As a rule of thumb, the amount of bending is coincidentally nearly equal to the apparent size of the sun's disc, so as soon as the bottom edge appears to touch the horizon, that's the moment it has actually fully dropped below the horizon, and would be totally obscured if the light were travelling in the straight line paths assumed by this article.)
Atmospheric refraction does not matter for this method as it is not based on the time of sunset. Yes, the actual sunset occurs later than the observed sunset, but the relationship between the reflection and the observed position of the Sun remains the same. So, the only difference is you're observing the relationship at a slightly different time than when it actually occurred geometrically.
awesome article. Do you know how to access the other articles in the series that are linked (dead) in the first paragraph? Would be great to read if still available.
I posted the following the other day -- and if you like this post and the maths involved - you will really enjoy Randall Carlson's Sacred Geometry lecture whereby he talks about all the Ancient Sacred Geometry and how it was used to measure the earth, moon, sun, stars and Earth's precession -- and how it all relates down to the measurements from the Human Body - whereby the 12-inch foot comes from, Cubits, etc.
---
>Speaking of "feet" as a measurement - Randall Carlson has an AMAZING video[0] on the source of the 12 inch foot.
>And how its all related to the measurement of the precession of the earth. And yes - its specifically how to measure things in space. And its all from Sacred Geometry.
Bravo. A very nice article comparing and contrasting standard trig methods and geometric algebra methods.
And as a bonus, it compares both with Dr. Wildberger's Rational Trigonometry. (Side note: I've known a few Australian mathematicians who embrace some very non-standard methods in mathematics, and I think it's great. We need people to shake up our thinking bout Math every once in a while.)
A very nice piece for some relaxing weekend recreational math.
One of Jearl Walker's articles in The Amateur Scientist series in Scientific American discusses a method to measure the size of the earth with only a foot rule or a stopwatch while observing setting sun. 1979 (05) pp. 172-182
Here: https://www.jstor.org/stable/24965204
The flat-earth community is actually kind of fascinating. Someone recently paid to fly some flat-earthers down to Antarctica to witness the 24-hour sun which is impossible under most flat-earth models [1]. The trip was extensively documented, and one of the flat-earthers who made the trip actually changed their position. The result, predictably, was that this person was ostracized from the community. The intellectual backflips that the flat-earth community are making to try to debunk this experiment is simultaneously fun and disturbing to watch.
The easiest way to debunk the flat-earth model is simply to observe that that 1) the sun sets below the horizon and 2) this happens at different times in different locations. On a flat earth, if the sun is seen at the horizon in one location it has to be seen at the horizon everywhere.
---
[1] https://www.the-final-experiment.com/
Not sure "fascinating" is the word I'd use. While there might be some people that are genuinely confused, the flat Earthers that I've seen debate people who went on the Final Experiment are nothing more than pathological liars.
When prominent flat Earthers heard about and were invited to participate in the Final Experiment, none of them were excited that the globers would finally see the Sun isn't up for 24 hours. They attacked the flat Earthers that did agree to go. All of the globers bought new equipment with their own money and sought out experiments to do and record. The flat Earthers were essentially dead weight, none of them actually planed to record a sunset.
> nothing more than pathological liars
Maybe. But I think there's something deeper going on. The flat-earthers-are-nothing-more-than-pathological-liars hypothesis fails to explain why they cling to this particular lie with such vehemence.
I don't mean to get political here, but Donald Trump is a pathological liar. He lies about everything, and he flits effortlessly from lie to lie. He never even offers the pretense of an argument or a justification. He simply proclaims that, say, they're eating the dogs because he read it on the Internet. The flearthers by way of very stark contrast have a laser-focus on this one lie, and they put a lot of effort into coming up with plausible-sounding justifications for it.
Yes, it is true that it is clear that at the end of the day the flearthers don't actually believe their own rhetoric. But it is still, to me, a phenomenon distinct from pathological lying, and which survives despite being so violently at odds with observation. I find that fascinating.
At least of the people who've been on the debates, the motive is quite clearly money. They all have apps, merch, blogs, or Youtube Channels they make money from.
Sure, you could say the same thing about most religious leaders. That doesn't necessarily mean that they don't sincerely believe what they're preaching. Flat-earth doesn't seem to me to be a particularly lucrative scam, the audience is too small. If money was really their primary motive I would think they'd find greener pastures elsewhere.
The original Flat Earth "movement" was an exercise to show that it is impossible to prove anything without some leaps of faith about the nature of the Universe or interpolating between gaps in definite knowledge..
There is always an assumption that can be denied. Turtles all the way down.
Later it grew into to a haven for a certain kind of mentally ill person and grifters that profit from them, illustrating the philosophical Problem of Subjectivity: malice cannot be reliably distinguished from stupidity.
It's a little more nuanced than that. The flat earth movement originated with Samuel Rowbotham who actually conducted an experiment in 1838 whose results (he claimed) demonstrated that the earth was flat [1]. I think it's arguable that he was engaging in the scientific method in good faith.
---
[1] https://en.wikipedia.org/wiki/Bedford_Level_experiment
> The result, predictably, was that this person was ostracized from the community.
This one sentence probably explains why the vast majority of humans don't change how we behave. The fear of losing almost all of the people we love is SO strong.
I wonder how much of a difference refraction of the suns light as it hits the atmosphere makes? It’s definitely the case that we see sunsets happening after the sun has ‘geometrically’ set because of this. I guess one needs to make some argument about the rays from the reflected and non-reflected cases being close enough together that they get sufficiently similar angles of refraction.
A list of urls from the beginning of post that don't seem to work anymore:
* Measuring the Earth’s Diameter from a Sunset Photo: https://vanderbei.princeton.edu/tex/sunset/ms.pdf
* The Earth is Not Flat: https://vanderbei.princeton.edu/tex/sunset/34-39.OPN.1108two...
* Sunset: https://vanderbei.princeton.edu/tex/sunset/sunset.pdf
* The Earth is Not Flat: An Analysis of a Sunset Photo: https://vanderbei.princeton.edu/tex/talks/sunset/sunsetTalk_...
Something doesn't sit right with me. I would expect that the semicircle would be reflected nonlinearly. That would imply the reflection is not a true circle. Am I wrong?
I'm also a little skeptical with this approach, as it's not taking into account the curvature of light through the atmosphere. In reality, when the photo was taken, the sun was already fully below the horizon. It just looks like it isn't because the atmosphere is bending the light a little ways around the earth. (As a rule of thumb, the amount of bending is coincidentally nearly equal to the apparent size of the sun's disc, so as soon as the bottom edge appears to touch the horizon, that's the moment it has actually fully dropped below the horizon, and would be totally obscured if the light were travelling in the straight line paths assumed by this article.)
Atmospheric refraction does not matter for this method as it is not based on the time of sunset. Yes, the actual sunset occurs later than the observed sunset, but the relationship between the reflection and the observed position of the Sun remains the same. So, the only difference is you're observing the relationship at a slightly different time than when it actually occurred geometrically.
Links to Vanderbei's work are dead.
Here are fresh links:
https://vanderbei.princeton.edu/tex/sunset/34-39.OPN.1108two...
awesome article. Do you know how to access the other articles in the series that are linked (dead) in the first paragraph? Would be great to read if still available.
https://web.archive.org/web/20160325085811/http://orfe.princ... https://web.archive.org/web/20160508114924/http://orfe.princ... https://web.archive.org/web/20170809073230/http://orfe.princ...
One year of Sun movement captured in a single long exposed photo: https://wesely.org/2019/flughafen-tempelhof-berlin-1-7-2008-...
I wonder how 14 bln years of Sun movement will look if you stand somewhere on the equator?
This is great!
I posted the following the other day -- and if you like this post and the maths involved - you will really enjoy Randall Carlson's Sacred Geometry lecture whereby he talks about all the Ancient Sacred Geometry and how it was used to measure the earth, moon, sun, stars and Earth's precession -- and how it all relates down to the measurements from the Human Body - whereby the 12-inch foot comes from, Cubits, etc.
---
>Speaking of "feet" as a measurement - Randall Carlson has an AMAZING video[0] on the source of the 12 inch foot.
>And how its all related to the measurement of the precession of the earth. And yes - its specifically how to measure things in space. And its all from Sacred Geometry.
[0] https://www.youtube.com/watch?v=R7oyZGW99os