Very Wrong Math
108 comments
·January 10, 2025seanhunter
userbinator
The earth is just much much much bigger than in the diagram so the effect is within the margin of error for the measurement he was taking.
It's actually measurable on a human scale:
https://www.mathscinotes.com/2017/01/effect-of-earths-curvat...
1 5/8" difference over 693', or slightly less than 1 part in 5 thousand --- definitely measurable on a smaller scale with accurate machinists' tools.
pvillano
One can also watch a boat leaving shore descend "under" the horizon with a telescope
sigmoid10
You don't even need a boat or a telescope. Just watch the sun set on the ocean while lying down at the beach just in front of the water. The moment it disappears completely, stand up. You'll see part of it again. If you measure the time it takes disappear completely again and know your own height, you can even get a rough estimate of earth's radius.
munch117
Ah, but would they actually be parallel on a flat earth?
Say the earth is disc-shaped. Then the center of gravity is only directly beneath you if you're standing at the exact center. You get ever-so-slightly not parallel lines, just like on a round earth.
The fun part of a disc-shaped earth comes as you move towards the sides, and gravity, still pointing towards the center, makes you stand at an increasingly acute angle to the surface. The ground beneath you will then appear like one big endless mountainside, with an increasingly steep slope the further away from the center that you get.
tempestn
Depends what causes things to stick to the flat Earth. IIRC flat earthers have various explanations for gravity, including the disc continuously accelerating upward; in that case you'd experience the same force everywhere on it.
munch117
If this mysterious disc-accelerating force also accelerated the people and things on the surface, we'd all be weightless.
I guess it must be a pushing force from below.
So, who's doing the pushing? I'm thinking a big turtle.
lukan
The theological argument I recently heard is, the creator just made up and down. And things move down. But it is not gravity.
f1shy
We can give them points for creativity.
thombat
Standard flat-earther response is to scornfully deny the existence of gravity. It's all density/buoyancy you see... Gravity is a hoax promulgated by the notorious cabalist Newton, in service to his Illuminati/Papal masters, etc, etc.
mp05
Why do we still talk about these people? The more we stand in awe of their calculated ignorance, the more satisfied they are.
I feel like there are better things to do with my time than be as fascinated by it as some people.
Someone
> and gravity, still pointing towards the center, makes you stand at an increasingly acute angle to the surface. The ground beneath you will then appear like one big endless mountainside
That’s why you never hear of people who went to the edge of that dis: they slid down that mountainside, and dropped off :-)
Alternatively, you can postulate that disc to be arbitrarily thick.
That will decrease the deviations. If that’s not enough to make them immeasurable, postulate that the stuff “deeper down” has higher density.
In the limit, just postulate that there’s an enormous black hole millions of light years below the center of the earth.
Flat-earthers probably won’t accept Newton’s theory of gravity, however, so you can make up anything.
phkahler
Even physicists have a hard time with disks and gravity. I can't tell you how many times I've seen them use the shell theorem on galaxies (does not apply). The only dark matter is in their head ;-)
jimmaswell
I'm considering what flat-surfaced shape you could construct with equal gravitational pull at all points. Maybe something where the center is thin as a point, the edges have a lot of depth, and they curve towards the center either convex or concave. Might run some calculus to figure it out.
t_mann
That way you should be able design a disc-shaped earth with constant strength of the gravitational force on the whole surface. But it would still have a center of mass (likely lying outside the shape you're describing, in the void beneath the center point), and the direction of the force should still be pointing towards that center, no? So the problem the GP has described, that you're starting to tilt as you move towards the edge, should remain in principle.
somat
yes, we call it a sphere.
I am just joking with you, I know what you mean, however the fruit was hanging too low not to pick.
rendaw
Doesn't the flat earth extend infinitely in all directions?
jerf
"We joke about LLMs hallucinating but I'm not convinced we are so superior when we are outside our personal "training data"."
In all seriousness one of the things about LLMs that most impress me is how close they get to human-style hallucination of facts. Previous generations of things were often egregiously and obviously wrong. Modern LLMs are much more plausible.
It's also why they are correspondingly more dangerous in a lot of ways, but it really is a legitimate advance in the field.
I observe that when humans fix this problem, we do not fix it by massive hypertrophy of our language centers, which is the rough equivalent of "just make the LLM bigger and hope it becomes accurate". We do other things. I await some AI equivalent of those "other things" with interest; I think that generation of AI will actually be capable of most of the things we are foolishly trying to press hypertrophied language centers into doing today.
ben_w
> We joke about LLMs hallucinating but I'm not convinced we are so superior when we are outside our personal "training data".
Every time I see the phrase "common sense", I expect to see an example of the human failing you describe.
BalinKing
Related Wikipedia article: https://en.wikipedia.org/wiki/String_girdling_Earth#Implicat....
The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. The problem is usually framed like so: If you wrapped a rope around the earth, how much more rope would you need to add so that it would be 1 meter above the ground at all points? The answer is only 2π meters!
travisjungroth
> The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect.
Maybe it’s because I’m a pilot and we never account for altitude when measuring distance, my intuition puts the difference at “effectively zero”. I also have it internalized that the earth’s atmosphere is very thin.
gsf_emergency
This could be why dimensional analysis is one of the few things from physics class that can't be drilled enough..
Without forcefully dumping the geometric "intuition", this would still feel counterintuitive to me!
nayuki
And the text about the airplane problem was added on 2024-11-26: https://en.wikipedia.org/w/index.php?title=String_girdling_E...
shortrounddev2
(2pi * (n + 1)) - (2pi * n)
-> 2pi * (n + 1 - n)
-> 2pi * 1
-> 2pi
If I remember my algebra correctly. Someone else check my work I'm a dropout
freeopinion
For convenience, we set τ=2pi. :-)
x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ
shortrounddev2
How do you pronounce that symbol?
kurthr
The only issue I see with this is that as a classic physics trope, we've approximated the earth as a sphere.
If, instead we approximate it as a fractal... then the distance is infinite, or at least highly dependent on the thickness of the rope!
The error in the original is assuming that the radius is proportional to the height above the earth (Earthradius=0?).
seanhunter
We actually model the earth as a very large spherical cow. This is approximately the same for most purposes but ends up being more convenient.
P.S. Not a physicist, but my child is studying maths and physics at Uni at present, so I have it on good authority that this is still going on. They told me in their first week one of their classes had a worked example where the lecturer used the phrase "Assume the penguin's beak is a cone".
kergonath
> I have it on good authority that this is still going on
Do you mean making simplifying assumptions to make a problem tractable? Of course it’s still going on. It has to be, otherwise you just cannot do anything.
> Assume the penguin's beak is a cone
It is impossible to consider the true shape of a penguin’s beak for several reasons:
- you’d need to go all the way down to the electron clouds of the atoms of the beak, at which point the very concept of shape is shaky
- every penguin has a different beak so even if you describe perfectly one of them, it does not necessarily make your calculation more realistic in general.
There is a spectrum of approximations one can make, but a cone is a sensible shape at a first order. It’s also simple enough that students can actually do it without years of experience and very advanced tools.
What do you think they should do instead?
davrosthedalek
A spherical cow /in vacuum/
Dylan16807
> infinite, or at least highly dependent on the thickness of the rope
The latter. But that's only if it's not somewhat taut. Some tension brings it closer to a circle and makes the actual thickness pretty unimportant.
But I like the idea overall. It means that lifting up the string makes it smoother and it actually needs less length. How's that for being unintuitive?
kurthr
Exactly, if you're only 1cm off the surface you follow every nook and cranny. If you're 10km off the surface only Everest is a blip.
aardvark179
Just because your initial fractal path is infinite does not imply that a line offset from it is also infinite (even for an infinitely thin rope), at least if the offset version is not self intersecting.
null
bruce511
Even if the math of the arc length was correct (and you don't need to be a math professor to figure out it isn't) there's another logic misstep.
Implied in the caption is that the speed is the same at all heights (given that an increase in distance is implied as an increase in time.)
This is again obvious nonsense - speed is a function of thrust versus drag, and it's safe to say that both of those are affected by air density.
It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.
mastermedo
> It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.
Can you expand on this? My brain is not connecting the dots.
db48x
He is talking about orbital mechanics, rather than free space. When you are in an orbit, the shape of the orbit is determined by your speed. At every distance from the center of the object you are orbiting (such as the Earth), there is a speed that makes your orbit a circle. If you are going at any other speed then your orbit will be an ellipse instead. Too fast and your orbit rises higher above the Earth. Too slow and it dips back down closer to it. If you try to “catch up” with an object ahead of you in your orbit by speeding up you will only turn your orbit into an ellipse that gets further away from the Earth, and thus further away from the object you were trying to catch. Instead of catching it you’ll go up and over it. As Niven wrote, “forward is up, up is back, back is down, and down is forward”. It’s rather counterintuitive at first. Playing KSP can help you get a feel for it, especially once you start docking multiple craft together.
Sharlin
It’s even worse than that. By speeding up you end up actually getting further behind your target because in your new higher orbit you actually move slower on average, and as your average orbital radius gets longer, so does the circumference, so you end up on a "detour" trajectory compared to your target!
Whereas if you slow down, you drop to a lower, shorter, higher-speed orbit.
davrosthedalek
Just to point out here what's different between "space" and "not space": "Space" assumes no "height control",i.e. ways to exert force "down or up" along the earth-object direction. That's obviously not true for a plane. If you can exert force in that direction, you can change speed and keep the shape of the trajectory around earth constant.
nyc111
This is called Kepler's Third Law, right? Radius^1.5 :: Period
f1shy
You are literally going way and beyond what the target audience of that post (the original with bad math) was for.
chrismorgan
I think the funny thing about this article is this numeric error (though not so egregious as the one that caused the article!):
> The mean radius of the earth is actually 3,459 miles or over 18 million feet.
That’s off by 500 miles; the correct figure is 3,959 miles. That makes it almost 21 million feet, and yields a ratio of about 1.0013378, even smaller than the quoted 1.0015.
csours
Reminds me of this classic:
https://www.politifact.com/factchecks/2020/mar/06/msnbc/bad-...
“Bloomberg spent $500 million on ads. The U.S. population is 327 million. He could have given each American $1 million and still have money left over.”
n4r9
Don't know if anyone mentioned this yet, but presumably the flight path does not follow a normal vector to gain height, but generally something more diagonal in the direction of travel.
prmph
One question I've always had with this: How does the rotation of the earth affect an airplane's flight time, if any? And how does this change with altitude?
f1shy
Simple answer: Zero. Because the planes move inside the atmosphere, which moves with the earth.
A more nuisance would be that earth rotating generate all sorts of things in the atmosphere, including winds and Coriolis effect on the winds, and you can account for that considering the winds. Btw a flight from Chile to France and back, will have a leg significantly shorter (up to 2 hs in a 13hs flight) and which leg it is, depends on the time of the year.
prmph
Interesting to know about the Coriolis effect.
I get that what really matters is the relative motion, but it still seems to me that there might be a gravitational/inertial effects at play, even if tiny.
Consider this thought experiment: Planes cannot really fly into space, but assume they can. At a certain altitude, it cannot be said the the plane is moving perfectly in step with the gravity of the earth. At infinite altitude, that certainly cannot be the case.
So that tells me there is some deviation due to the inertia of the plane, even at low altitudes. Like I said, the effect might be tiny, but would be interesting to learn more about it nonetheless.
f1shy
I meant coriolis effect on the wind. Not sure if noticeable in a plane.
Derbasti
In general, the air moves with the ground, so the earth's rotation does not affect airplanes.
However, rotation of the earth imparts a coriolis force on the air, which results in jetstream winds. Aircraft routes are optimized to use/avoid jetstreams for shorter travel times.
kergonath
It does not really, at least not directly. What matters is relative velocity compared to the starting and final locations, and relative to the air around the aircraft. It just happens that there are very powerful atmospheric currents that go west to east (those are due to the earth’s rotation, among others phenomena).
So, when flying towards the east, catching these currents can significantly reduce flying time. When flying towards the west, we want to avoid them by flying below or elsewhere.
prmph
Thanks for this explanation; quite interesting.
But it still seems to me that there might be a gravitational/inertial effects at play as well. At a (hypothetical) infinite altitude, it can no longer be said the the plane is moving perfectly in lock-step with the gravity/rotational acceleration of the earth. This implies the inertia of the plane relative to the rotation of the earth still has an effect at lower altitudes.
The effect might be tiny, but would be interesting to learn more about it nonetheless.
BlueTemplar
What gravity/rotational acceleration ?
Something like this does exist under general relativity :
https://en.m.wikipedia.org/wiki/Frame-dragging
However,
> This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation.
lynguist
I would do it like this:
Approximate Earth as a flat line. (The 5000ft path is close enough that it is also represented by the flat line. This is the 5000ft path.)
Then make the 33000ft path which is a slightly looser line on top of this line.
This new path is not 4 times longer. Just a little bit raised, because 33000 ft is “nothing” compared to Earth. (To become 4x longer we would go deep into outer space and back.)
syntex
just 2piR and then extra h change the result very little fraction. How is that counter-intuitive :)
simplicio
Seems intuitively obvious. On a flat Earth the two distances would be the same, and while the Earth isn't flat, its close enough to approximate a flat surface for most purposes, so you'd expect the differences in the two arcs to be ~0
pessimizer
Does "Remember the high you go the further it'll have to travel" really need to be debunked? Did the "design and construction firm" spell "drill" with one "l"?
jodrellblank
Earth in the picture is scaled to roughly 300,000 feet per pixel; Earth's surface and both flying altitudes would be the same pixel if drawn to scale.
(~42M feet diameter shown in ~134 pixels).
I have seen a very similar (incorrect) argument used to justify the idea of a flat earth. A builder on youtube made the argument (with a similar out of scale drawing of the earth) that if he drops a plumb bob and makes a right angle so he has a straight horizontal line and then goes across that line for a bit and drops another plumb bob, the two lines he has dropped are parallel, "proving" that the surface of the earth must be parallel to the horizontal line and therefore flat and not curved. If the earth's surface was actually curved he argued then the two lines he has dropped should tilt slightly inward towards each other. Which of course they do. The earth is just much much much bigger than in the diagram so the effect is within the margin of error for the measurement he was taking.
As a meta point, our intuition often fails us hilariously when we are dealing with stuff that is out of the scale we have commonly seen in our lives. We joke about LLMs hallucinating but I'm not convinced we are so superior when we are outside our personal "training data".