Go east from Seattle(finmoorhouse.com)
finmoorhouse.com
Go east from Seattle
https://finmoorhouse.com/writing/go-east/
162 comments
> Travel exactly due East. Now, ignore what East means.
It's "face east, then travel in a straight line," not "travel due East" (though I agree that the title itself is misleading).
IMO, it's a cool thought experiment because it demonstrates that an instruction to "travel in a straight line" is incompatible with an instruction to "travel due East" (unless you're on the equator!) - as the North Pole example illustrates. Start 3 feet south of the North Pole and face east, then start traveling in a straight line. Obviously after 10 or 100 or 1000 miles of travel you'll be essentially due south of the pole - and in fact, if you take a single step to the left, you will be exactly due south of the pole!
It's "face east, then travel in a straight line," not "travel due East" (though I agree that the title itself is misleading).
IMO, it's a cool thought experiment because it demonstrates that an instruction to "travel in a straight line" is incompatible with an instruction to "travel due East" (unless you're on the equator!) - as the North Pole example illustrates. Start 3 feet south of the North Pole and face east, then start traveling in a straight line. Obviously after 10 or 100 or 1000 miles of travel you'll be essentially due south of the pole - and in fact, if you take a single step to the left, you will be exactly due south of the pole!
> It's "face east, then travel in a straight line," not "travel due East" (though I agree that the title itself is misleading).
It's a fun and interesting problem, but just to be the contrarian:
The solution didn't travel in a straight line any more than the alternate (incorrect solution), both resulted in arcs.
It's a fun and interesting problem, but just to be the contrarian:
The solution didn't travel in a straight line any more than the alternate (incorrect solution), both resulted in arcs.
Any straight line that's tangential to the surface of a sphere will lead off the sphere.
Everyone's replying as if the title and image don't exist
Author here — I agree the image is misleading (and added a caveat to be clearer, though I could have made the arrow smaller).
I don't think the question requires the reader to ignore what ‘east’ means, however. There is unambiguously one direction at Seattle (tangent to the Earth's surface) which is facing due east.
I also didn't mean to take a patronising/arrogant tone to the reader and I'm sorry if that was the impression I gave.
I don't think the question requires the reader to ignore what ‘east’ means, however. There is unambiguously one direction at Seattle (tangent to the Earth's surface) which is facing due east.
I also didn't mean to take a patronising/arrogant tone to the reader and I'm sorry if that was the impression I gave.
I read it after your caveat was added, but I didn't find it intentionally misleading, from the very strong response (which is doing what it accuses the post of doing by claiming it reads 'travel due east') I get the feeling someone may be a bit peeved they didn't 'get it's and instead thought the answer would be some embassy that happens to be directly east of Seattle.
> There is unambiguously one direction at Seattle (tangent to the Earth's surface)
and that tangent is the straight line, at least it is within the R^3 spatial universe I live within.
There's confusion here as the article presumes humans to be bound to an S^2 manifold and to consider a Great Circle to be a "straight line" which it certainly is not unless one lives within an S^2 Flatland.
and that tangent is the straight line, at least it is within the R^3 spatial universe I live within.
There's confusion here as the article presumes humans to be bound to an S^2 manifold and to consider a Great Circle to be a "straight line" which it certainly is not unless one lives within an S^2 Flatland.
My immediate guess was England or maybe a bit lower and France, but then I saw the arrow and thought "DOH! Canada"
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The way I'd explain it is:
You have a perfect paper airplane. It flies in a straight line regardless of wind, rain, hail or snow. And it keeps in the air a really long time.
You get into an air balloon over Seattle and launch your paper airplane directly East.
The plane descends so slowly it can leave North America and go over the ocean and eventually will reach another country and still not touch the ground. What is the first country it will reach?
You have a perfect paper airplane. It flies in a straight line regardless of wind, rain, hail or snow. And it keeps in the air a really long time.
You get into an air balloon over Seattle and launch your paper airplane directly East.
The plane descends so slowly it can leave North America and go over the ocean and eventually will reach another country and still not touch the ground. What is the first country it will reach?
This is so much better. I'd remove even more and forget about descending.
You have a perfect paper airplane. It flies in a straight line regardless of wind and never descends, staying a constant height from the ground as it flies. You get into a hot air balloon over Seattle and launch your paper airplane directly East. After leaving the USA, what is the next country that the plane will fly over?
You have a perfect paper airplane. It flies in a straight line regardless of wind and never descends, staying a constant height from the ground as it flies. You get into a hot air balloon over Seattle and launch your paper airplane directly East. After leaving the USA, what is the next country that the plane will fly over?
No, it says face due east and travel in a straight line. Sure, the wording could have been clearer, but you're seeing misdirection that isn't there.
A straight line would extend into space.
The instructions state the following:
> Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface.
Where does it say to "travel exactly due East"?
> Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface.
Where does it say to "travel exactly due East"?
The instructions are impossible to follow, since there are no straight lines (in all three dimensions) along the Earth's surface.
And if it's supposed to mean "a straight line when projected onto a map", I'd like to know: Which map projection? :)
And if it's supposed to mean "a straight line when projected onto a map", I'd like to know: Which map projection? :)
Imagine you're on a snowmachine 10 feet south of the North Pole pointing due east. I tell you to ride for 10 miles without turning. You don't need to worry about map projections or anything - just don't turn. When you stop, are you still 10 feet south of the North Pole? Clearly not, you'll be about 10 miles south of the pole.
It's exactly the same if you start in Seattle. If you drive for 10 miles without turning, when you stop you will no longer be at Seattle's latitude - you'll have gone south.
It's exactly the same if you start in Seattle. If you drive for 10 miles without turning, when you stop you will no longer be at Seattle's latitude - you'll have gone south.
I know, but that's not what TFA demands. Going 10 miles without turning is not "traveling in a straight line"; it's at most "traveling along a great circle line"!
And even that would require an unusually well aligned steering on a snowmobile :)
And even that would require an unusually well aligned steering on a snowmobile :)
Impossible to follow if you are thinking Euclidean geometry, but a great circle is a straight line in spherical geometry. The earth isn't a sphere, but it is much closer to one than it is a plane.
Certainly not the map projection used to illustrate the question with the giant straight line on it. Not that one
I'm glad I read this comment first. This blog post is terrible and you saved me time.
It really isn't, at least with the author's clarification that the arrow isn't the path. It might sound like it if you misread the instructions as 'travel due east' as the parent comment did.
This inspired similarly annoyed responses last time it was posted too. It genuinely surprises me to see that on HN. The whole thing can be summarized as "You might think of 'start facing East and go in a straight line' as being the same as 'maintain an Eastern heading', but latitude lines are only straight in the Cartesian abstraction we're used to, and not in reality."
I hadn't thought about it before. It's interesting. It's not some kind of cheap or stupid trick intended to embarrass you. The negativity is truly baffling.
I hadn't thought about it before. It's interesting. It's not some kind of cheap or stupid trick intended to embarrass you. The negativity is truly baffling.
I don't think it is negativity. Someone else already called it out, but if you were to face east and then travel in a straight line, you would leave Earth's orbit. Thus, the question inherently is taking Earth into consideration. Their definition of "most obvious" is strange because you would encode a globe using spherical coordinates. A "straight line" East in spherical coordinates could be argued to be keeping the polar angle constant and increasing the azimuthal angle which would land you in France. The crux of the disagreement is what "most obvious" means.
Idk, I think it’s pretty clear based on the description what they mean.
For one, the problem description says the straight line is “along the Earth’s surface” so I don’t think you can reasonably think it might go into space.
And to me it seems reasonable to say a straight line means not turning. I personally imagined getting in a plane and going straight forward, assuming there is no wind to blow me off course.
That is just the common understanding of straight. If somebody asks you to walk in a straight line, do you ask them what coordinate system they’re using? No, you walk forward without turning.
I guess “most obvious” can always be up to interpretation, though.
For one, the problem description says the straight line is “along the Earth’s surface” so I don’t think you can reasonably think it might go into space.
And to me it seems reasonable to say a straight line means not turning. I personally imagined getting in a plane and going straight forward, assuming there is no wind to blow me off course.
That is just the common understanding of straight. If somebody asks you to walk in a straight line, do you ask them what coordinate system they’re using? No, you walk forward without turning.
I guess “most obvious” can always be up to interpretation, though.
I think if you were to ask someone not trained in navigation "to fly straight" between two points on the globe, they are probably not going to come up with the concept of great circles and most likely will end up turning a bunch during their flight "to fly straight". I could be wrong but I think "most obvious" if we are defining it as "the most natural thing most people would do without explicit training" then the article is wrong and this thread would agree with this conclusion.
I don't disagree that it is possible to deduce the great circle if you think about it and have the knowledge to coordinate systems. But why even ask the question this way when you can get to the core of the question by explicitly asking "Along the great circle that reaches its max latitude at Seattle, what's the next country it touches going eastward from Seattle?" This would yield many more correct answers and why people think its mostly wordplay (because it is).
I don't disagree that it is possible to deduce the great circle if you think about it and have the knowledge to coordinate systems. But why even ask the question this way when you can get to the core of the question by explicitly asking "Along the great circle that reaches its max latitude at Seattle, what's the next country it touches going eastward from Seattle?" This would yield many more correct answers and why people think its mostly wordplay (because it is).
The problem just says to fly straight, not straight between two points. Though this thread is proof that a lot of people interpreted it differently, so all I can say is that I personally think it makes sense.
Even so, my guess after a very brief thought about it was actually Canada, so this understanding didn’t help me any :)
I had never heard of this concept of “great circles” before, so it personally wouldn’t have made it clearer to me without some research.
Given my ignorance on this topic, I found the fact that Australia was the first country you’d hit to be really quite remarkable.
Even so, my guess after a very brief thought about it was actually Canada, so this understanding didn’t help me any :)
I had never heard of this concept of “great circles” before, so it personally wouldn’t have made it clearer to me without some research.
Given my ignorance on this topic, I found the fact that Australia was the first country you’d hit to be really quite remarkable.
> the straight line is “along the Earth’s surface”
Well, you've already run into a problem, because any line that follows the Earth's surface can't possibly be straight.
Well, you've already run into a problem, because any line that follows the Earth's surface can't possibly be straight.
Yeah as soon as I saw the "answer" I was like "yeah sure blah blah blah, you can make a trick question, but what IS the country that you'll actually hit if you walk East from Seattle". I was guessing England, but the right answer is France.
> what IS the country that you'll actually hit if you walk East from Seattle". I was guessing England, but the right answer is France.
Actually it's Canada, if "walk East" means "walk eastward along a line of constant latitude".
Actually it's Canada, if "walk East" means "walk eastward along a line of constant latitude".
Well they called it "Go east" so maybe it feels like a trick that they didn't go east? Instead they worded their actual question in a subtly different way and then said "gotcha". It's a cool observation and would have been cool without the pedantry, too.
Did they change the title of the blog post? When I open it, it says “A path from Seattle”. The “Go east” part is only in the Hacker News post.
The Substack version says "Go east from Seattle": https://finmoorhouse.substack.com/p/go-east-from-seattle
Yes, I changed the title to the linked page to ‘A path from Seattle’ from ‘Go east from Seattle’ because, from reading the comments here, I agree the original title was too misleading. Just updated the title of the Substack post also.
It's a trick question regardless of if the intent was to educate the reader or to make them feel stupid. You have to spot that the question both isn't asking what the title suggests it's asking and that the image isn't actually showing the direction you'd go. People tend not to like trick questions.
As they state at the bottom, they are referring specifically to geodesics when they mean straight line. They then rule out the due east curve, despite stating you must travel due east. They claim this is because to do so would make the line no longer straight, but this claim is utterly nonsensical in the context of three dimensional geometries.
This is a puzzle born of a misunderstanding of three dimensional Euclidean geometry.
This is a puzzle born of a misunderstanding of three dimensional Euclidean geometry.
HN is full of people who like to be the "well actually" guy, and they're all getting well actually'd by this article. Of course they're annoyed. It must be just awful.
If you ask a question to which the answer is "no, actually...", don't be upset when you get a lot of "well, actually actually" retorts, especially on this site :)
And don't bring a geodesic to a semantics fight!
And don't bring a geodesic to a semantics fight!
After this, let's have a calm discussion about whether an airplane can take off from a treadmill that exactly matches the plane's wheel's speed in the reverse direction.
I enjoyed many weeks of debate on a different website about that.
> start facing East and go in a straight line
This is where the trick is hidden: To do this, you'd generally require a spaceship (or a very impressive ladder) :)
This is where the trick is hidden: To do this, you'd generally require a spaceship (or a very impressive ladder) :)
So it's hardly "go east". You're going momentarily east, then southeast.
It is interesting what a "straight line" really means, though, in different contexts. I would have to say this is a poor puzzle, though, filed under cheap word tricks & ambiguities.
It is interesting what a "straight line" really means, though, in different contexts. I would have to say this is a poor puzzle, though, filed under cheap word tricks & ambiguities.
The mental bias towards the Cartesian 2d map projections being what we're so familiar with is strong, and I too instinctively think of "a straight line" as being a straight line on a map. But if you think of the problem at a different scale — say you're standing 50 feet south of the north pole — I feel like you would consider the definition of "face east, then walk forward in a straight line" far less ambiguous.
I'd also point out that it's a puzzle, not a standardized test question. Wordplay, tricks and ambiguities are par for the course.
I'd also point out that it's a puzzle, not a standardized test question. Wordplay, tricks and ambiguities are par for the course.
It's not a bias for 2d projection. If someone tells me to "go east in a straight line", I think about how practically I'd go about it. In this case, I'd take a compass, point myself to go east, and keep adjusting my bearing so that I keep facing east. Bonjour, France.
Nobody familiar with navigation would say "go east in a straight line" because it's a paradox. That's why they say "maintain an easterly heading", or, for shorter distances (e.g. car directions), "head straight east", as in "exactly east", and not "east in a perfectly straight line", which would be less relevant for shorter distances.
It's also why the question is (paraphrased) "start facing east and go in a straight line", and not "go east in a straight line".
It's also why the question is (paraphrased) "start facing east and go in a straight line", and not "go east in a straight line".
Of course it's not a perfectly straight line, because the earth is a curved object. Perfectly straight puts you in outer space.
Now that is, in fact, a silly gotcha. It's a third abstraction, irrelevant to the topic of two dimensional navigation. It is abstracted away for both cartesian and global concepts, and nobody is realistically confused by its absence.
The article quite clearly says “in a straight line along the Earth’s surface.”
Emphasis mine.
Emphasis mine.
Perhaps the HN headline doesn't match the article, but the instructions say:
> Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface.
where there isn't any wording saying "go east in a straight line".
> Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface.
where there isn't any wording saying "go east in a straight line".
The most reasonable interpretation of this is to follow the latitudenal geodesic along its eastern path. You cannot claim that one geodesic is more “straight” than another in 3d Euclidean geometry, that is nonsense. But that is what the author does.
Edit: Ok, the latitudinal geodesic only exists at the equator, so the question is fundamentally impossible, with how the author defines a straight line.
Edit: Ok, the latitudinal geodesic only exists at the equator, so the question is fundamentally impossible, with how the author defines a straight line.
> the latitudenal geodesic
There is no such thing. A curve of constant latitude on Earth, except for the equator, is not a geodesic.
> You cannot claim that one geodesic is more “straight” than another in 3d Euclidean geometry
In terms of 3D Euclidean geometry, neither a curve of constant latitude on Earth's surface nor a great circle on Earth's surface is a straight line/geodesic. Both are curved.
If you restrict to the 2D surface of the Earth, a great circle is a geodesic but a curve of constant latitude, except for the equator, is not.
There is no such thing. A curve of constant latitude on Earth, except for the equator, is not a geodesic.
> You cannot claim that one geodesic is more “straight” than another in 3d Euclidean geometry
In terms of 3D Euclidean geometry, neither a curve of constant latitude on Earth's surface nor a great circle on Earth's surface is a straight line/geodesic. Both are curved.
If you restrict to the 2D surface of the Earth, a great circle is a geodesic but a curve of constant latitude, except for the equator, is not.
You're right, so if the puzzle restricts one (arbitrarily) to geodesics as the only valid straight-lined curve (?), it becomes nonsense because the only easterly geodesic is at the equator, and seattle is not at the equator.
> the only easterly geodesic
Wrong. There is a perfectly good great circle passing through Seattle and pointing due east at the point where it passes through Seattle. The author showed it to you in the article. That geodesic does not always point due east, but nothing in the article said it had to. The article only said you have to face due east at the start, not that you need to continue facing that way.
Wrong. There is a perfectly good great circle passing through Seattle and pointing due east at the point where it passes through Seattle. The author showed it to you in the article. That geodesic does not always point due east, but nothing in the article said it had to. The article only said you have to face due east at the start, not that you need to continue facing that way.
Except that the original instruction was "straight line", not "geodesic", so does it matter all that much which kind of non-straight-line one follows?
> the original instruction was "straight line", not "geodesic"
If you're working within a 2-sphere, such as the Earth's surface, or indeed any non-Euclidean geometry, they mean the same thing. More precisely, there are no "straight lines" in the exact sense you mean in a non-Euclidean geometry, but there are geodesics that satisfy all of the geometric properties of "straight lines" within that non-Euclidean geometry.
If you're working within a 2-sphere, such as the Earth's surface, or indeed any non-Euclidean geometry, they mean the same thing. More precisely, there are no "straight lines" in the exact sense you mean in a non-Euclidean geometry, but there are geodesics that satisfy all of the geometric properties of "straight lines" within that non-Euclidean geometry.
Surely you mean Paris, Ontario, Canada? :)
Erm, is it bias if it's the standard for navigation?
Historically, maintaining constant latitude (colloquial "Go East") was straightforward.
By contrast, maintaining a Great Circle Route other than the equator was effectively impossible until the 1800s. It requires the ability to track longitude which requires the invention of the marine chronometer.
Historically, maintaining constant latitude (colloquial "Go East") was straightforward.
By contrast, maintaining a Great Circle Route other than the equator was effectively impossible until the 1800s. It requires the ability to track longitude which requires the invention of the marine chronometer.
> maintaining a Great Circle Route other than the equator was effectively impossible until the 1800s.
It was at sea because the sea is constantly forcing your ship to change heading (wind, currents, etc.), and you have to compensate for that, and before the required navigation techniques were developed, there was no way to know what heading you should be on at any given time to maintain a great circle route.
But the article is asking you to imagine that you don't have to worry about that--which, if you're just walking across flat land, you don't. You can just "put one foot in front of the other", doing what comes naturally, and you will follow a great circle--not a curve of constant latitude.
It was at sea because the sea is constantly forcing your ship to change heading (wind, currents, etc.), and you have to compensate for that, and before the required navigation techniques were developed, there was no way to know what heading you should be on at any given time to maintain a great circle route.
But the article is asking you to imagine that you don't have to worry about that--which, if you're just walking across flat land, you don't. You can just "put one foot in front of the other", doing what comes naturally, and you will follow a great circle--not a curve of constant latitude.
Did navigators before that actually follow rhumb lines? I'd have imagined they would at least approximate a great circle line somewhat by using a compass and dead reckoning, at least on very long crossings (where the difference between a rhumb line and a great circle line becomes significant).
A straight line along a constant latitude is a perfectly valid interpretation of the words. That interpretation was reinforced by the image. You're right that warning was given that it was a "puzzle". Still I'm with what it seems the GP was saying and felt the statements of correctness with certainty were intellectually dishonest. Rather than enjoying a clever thoughtfulness, perspective, and new knowledge, I appreciated the description of how they got that answer while feeling a bit abused. If the good and invalid interpretations had been recognized and the bad dismissed then the chosen "selected" I may have had smaller objections but recognized the right to choose, especially if there was something in the statement of the puzzle teasing that interpretation. In lieu of that it is an unsatisfying a game of guess the teacher's answer for me.
> A straight line along a constant latitude is a perfectly valid interpretation of the words.
It's a perfectly understandable interpretation, and I wouldn't belittle anyone who read it that way, but it can't be correct. A line of constant latitude cannot be straight unless it's on the equator. (Of course technically no line that remains on the Earth's surface is truly straight, but the puzzle's definition of "straight" meaning "turning neither left nor right" does seem like the best interpretation overall.)
It's a perfectly understandable interpretation, and I wouldn't belittle anyone who read it that way, but it can't be correct. A line of constant latitude cannot be straight unless it's on the equator. (Of course technically no line that remains on the Earth's surface is truly straight, but the puzzle's definition of "straight" meaning "turning neither left nor right" does seem like the best interpretation overall.)
> it can't be correct
You are making a good point and I feel like I understand the basis of the original article's definition of straight as well as its basis for that definition. However, it does not seem to fully define the frame. If it had disambiguated the underlying terminology and assumptions, I'd maybe think some of the definitions unintuitive but ceded their right to define.
The way I would regard my interpretation to be valid is to take the sphere and slice it at the current line of latitude. That seems like a historic if not reasonable interpretation of "exactly due East". Then for further visualization, consider a cylinder that is perpendicular to and centered on the edge of the flat surface. If you cut the cylinder at the starting point and unrolled it, the resulting path to the other end would be a 2D ray. Traveling the center of that cylinder along the ray seems like a way to go "exactly due East". As a bonus, the tradition of following a compass would take you on the same path. Given the ambiguity and what I think are reasonable interpretations of some of the terms, I believe that this is a reasonably reachable result that can be called valid and/or correct.
The challenge here is the underdefinition of and opportunity to attach different semantics to the words in use.
You are making a good point and I feel like I understand the basis of the original article's definition of straight as well as its basis for that definition. However, it does not seem to fully define the frame. If it had disambiguated the underlying terminology and assumptions, I'd maybe think some of the definitions unintuitive but ceded their right to define.
The way I would regard my interpretation to be valid is to take the sphere and slice it at the current line of latitude. That seems like a historic if not reasonable interpretation of "exactly due East". Then for further visualization, consider a cylinder that is perpendicular to and centered on the edge of the flat surface. If you cut the cylinder at the starting point and unrolled it, the resulting path to the other end would be a 2D ray. Traveling the center of that cylinder along the ray seems like a way to go "exactly due East". As a bonus, the tradition of following a compass would take you on the same path. Given the ambiguity and what I think are reasonable interpretations of some of the terms, I believe that this is a reasonably reachable result that can be called valid and/or correct.
The challenge here is the underdefinition of and opportunity to attach different semantics to the words in use.
> The way I would regard my interpretation to be valid
Involves getting rid of the 2-sphere and substituting a cylinder. Which changes the geometry. Yes, a circle cut in the way you describe out of a cylinder is a geodesic on the cylinder. But that doesn't mean it's a geodesic on a 2-sphere. It's not.
> That seems like a historic if not reasonable interpretation of "exactly due East".
It's a reasonable interpretation of "exactly due East", yes, but it's not a reasonable interpretation of the directions the article actually gave, which are:
"Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface."
So you start facing "exactly due East", but then you travel "in a straight line along the Earth's surface". The most natural interpretation of "in a straight line" is "without doing anything to change direction", i.e., just "put one foot in front of the other" continuously. And if you do that, you will follow a great circle, not a curve of constant latitude. To follow a curve of constant latitude, you would need to be making a conscious effort to turn left as you went. That's not traveling in a straight line.
Involves getting rid of the 2-sphere and substituting a cylinder. Which changes the geometry. Yes, a circle cut in the way you describe out of a cylinder is a geodesic on the cylinder. But that doesn't mean it's a geodesic on a 2-sphere. It's not.
> That seems like a historic if not reasonable interpretation of "exactly due East".
It's a reasonable interpretation of "exactly due East", yes, but it's not a reasonable interpretation of the directions the article actually gave, which are:
"Imagine you begin a journey in Seattle WA, facing exactly due east. Then start traveling forward, in a straight line along the Earth’s surface."
So you start facing "exactly due East", but then you travel "in a straight line along the Earth's surface". The most natural interpretation of "in a straight line" is "without doing anything to change direction", i.e., just "put one foot in front of the other" continuously. And if you do that, you will follow a great circle, not a curve of constant latitude. To follow a curve of constant latitude, you would need to be making a conscious effort to turn left as you went. That's not traveling in a straight line.
The author admits that:
> "by ‘straight line’ here I really mean ‘geodesic’"
In other words, the puzzle hinges on one's interpretation of "straight line". I am merely asserting that other valid interpretations exist even if you are correct and geodesic is "the most natural interpretation".
Incidentally, the article appears revised and seems far more intellectually honest now (thank you Fin, that's a lot more fun!)
Consider the tomato. Classified as vegetable by the culinary discipline based on its use in the kitchen, it is classified as a fruit botanically based on the characteristics of the plant[0]. Neither is correct over the other, they are both frames for understanding the world and a classification can be correct (or not) according to the definitions of each.
In the same way, I described a means of acquiring a line that I believe most would accept as meeting our ideas of "straight" that was consistent with my interpretation. It made how I failed to understand the author's intention for the puzzle exceedingly clear which can be beneficial for anyone debugging their thoughts and/or social patterns. Do you tend to get into conflicts instead of expanding your perspective? Mine doesn't seem to be the best interpretation and it certainly wasn't that intended by the author but it seems a valid interpretation, especially given the brevity of the problem statement. I find this is generally a forum where we discuss the meaning we intended and benefit from the varieties of perspectives available. Of course we are unified on almost nothing.
[0] https://www.eufic.org/en/healthy-living/article/is-a-tomato-...
> "by ‘straight line’ here I really mean ‘geodesic’"
In other words, the puzzle hinges on one's interpretation of "straight line". I am merely asserting that other valid interpretations exist even if you are correct and geodesic is "the most natural interpretation".
Incidentally, the article appears revised and seems far more intellectually honest now (thank you Fin, that's a lot more fun!)
Consider the tomato. Classified as vegetable by the culinary discipline based on its use in the kitchen, it is classified as a fruit botanically based on the characteristics of the plant[0]. Neither is correct over the other, they are both frames for understanding the world and a classification can be correct (or not) according to the definitions of each.
In the same way, I described a means of acquiring a line that I believe most would accept as meeting our ideas of "straight" that was consistent with my interpretation. It made how I failed to understand the author's intention for the puzzle exceedingly clear which can be beneficial for anyone debugging their thoughts and/or social patterns. Do you tend to get into conflicts instead of expanding your perspective? Mine doesn't seem to be the best interpretation and it certainly wasn't that intended by the author but it seems a valid interpretation, especially given the brevity of the problem statement. I find this is generally a forum where we discuss the meaning we intended and benefit from the varieties of perspectives available. Of course we are unified on almost nothing.
[0] https://www.eufic.org/en/healthy-living/article/is-a-tomato-...
> I described a means of acquiring a line that I believe most would accept as meeting our ideas of "straight" that was consistent with my interpretation.
Which, as I said, involves throwing away the 2-sphere and substituting a cylinder. Which does not strike me as a valid interpretation of anything. There is no cylinder anywhere in the problem. Saying "I will only count a Euclidean straight line going off into space as a straight line" would be one thing--still not really applicable to the problem, but the Earth is embedded in a Euclidean 3-space so at least you're dealing with something that's actually there. But saying "I will not count a great circle on a 2-sphere as straight, but I will count as straight a circle drawn around an imaginary cylinder that I just made up" is something else.
Which, as I said, involves throwing away the 2-sphere and substituting a cylinder. Which does not strike me as a valid interpretation of anything. There is no cylinder anywhere in the problem. Saying "I will only count a Euclidean straight line going off into space as a straight line" would be one thing--still not really applicable to the problem, but the Earth is embedded in a Euclidean 3-space so at least you're dealing with something that's actually there. But saying "I will not count a great circle on a 2-sphere as straight, but I will count as straight a circle drawn around an imaginary cylinder that I just made up" is something else.
That would be something else if I had written it.
I accept that one can mean a great circle on a 2 sphere when they write "straight".
I accept that one can mean a great circle on a 2 sphere when they write "straight".
This is not how three dimensional Euclidean geometry works!
Please see https://en.m.wikipedia.org/wiki/Geodesic
Geodesics are “straight” along a curved surface, regardless of if they follow the circumference.
Edit: The above is an incorrect description of geodesics, which must follow the circumference.
Perhaps a more accurate description of the problem would be to say that it is impossible to have an easterly vector field except at the equator, and Seattle is not at the equator.
Please see https://en.m.wikipedia.org/wiki/Geodesic
Geodesics are “straight” along a curved surface, regardless of if they follow the circumference.
Edit: The above is an incorrect description of geodesics, which must follow the circumference.
Perhaps a more accurate description of the problem would be to say that it is impossible to have an easterly vector field except at the equator, and Seattle is not at the equator.
The GP is correct. Only great circles on a 2-sphere are geodesics. Curves of constant latitude, except for the equator, are not.
I'm not sure why you referenced the Wikipedia article you did, since it supports what the GP says and contradicts what you say.
I'm not sure why you referenced the Wikipedia article you did, since it supports what the GP says and contradicts what you say.
In Euclidean geometry, one generally does not return to the point of departure in finite time when following a straight line!
So is a geodesic a straight line?
So is a geodesic a straight line?
No interpretation can be correct, since there are no straight lines on the surface of the earth.
The headline is poorly worded. The actual puzzle is unambiguous (begin facing east, then travel forward in a straight line along the surface).
But there are no straight lines on the surface of a sphere (or geoid)!
Yes, there are--great circles are geodesics of the 2-sphere. They are not geodesics of the surrounding 3-dimensional Euclidean space in which our Earth is embedded, but that's irrelevant since the problem is not asking us to travel off of the Earth's surface.
But geodesics are not straight lines, they're curves!
On a sphere, you have to pick one: Stick to the surface or follow a straight line.
On a sphere, you have to pick one: Stick to the surface or follow a straight line.
> geodesics are not straight lines, they're curves!
If you are working in the 3D Euclidean space in which the Earth is embedded, yes.
But not if you're working within the 2D non-Euclidean space of the Earth's surface alone. Within that space great circles, geodesics, satisfy all of the geometric properties of "straight lines".
In other words, the geometric properties that define "straight lines" do not actually pick out the precise concept that Euclid thought they did. They pick out his intended straight lines in Euclidean space, but they also pick out great circles on 2-spheres, and hyperbolas on a hyperboloidal 2-surface. This discovery was made in the 1800s by several different people, and it led, among other things, to the theory of General Relativity in physics, so it's not something that can just be put aside.
If you are working in the 3D Euclidean space in which the Earth is embedded, yes.
But not if you're working within the 2D non-Euclidean space of the Earth's surface alone. Within that space great circles, geodesics, satisfy all of the geometric properties of "straight lines".
In other words, the geometric properties that define "straight lines" do not actually pick out the precise concept that Euclid thought they did. They pick out his intended straight lines in Euclidean space, but they also pick out great circles on 2-spheres, and hyperbolas on a hyperboloidal 2-surface. This discovery was made in the 1800s by several different people, and it led, among other things, to the theory of General Relativity in physics, so it's not something that can just be put aside.
But I live in Euclidean space, not on a 2-sphere!
> I live in Euclidean space, not on a 2-sphere!
You live in both. In your daily life, you're restricted to the Earth's surface, and if someone asks you to walk in a straight line, you will walk in a straight line (geodesic/great circle) along the Earth's surface. You're not going to walk off into space, and you're not going to constantly be turning to maintain constant latitude. Over short distances the difference is negligible, but the article is not restricting you to short distances.
You live in both. In your daily life, you're restricted to the Earth's surface, and if someone asks you to walk in a straight line, you will walk in a straight line (geodesic/great circle) along the Earth's surface. You're not going to walk off into space, and you're not going to constantly be turning to maintain constant latitude. Over short distances the difference is negligible, but the article is not restricting you to short distances.
If someone asks me to walk in a straight line unassisted by navigational aids, I'd probably end up walking in a medium-sized circle of (optimistically) a few kilometers' diameter.
For everything beyond that, I'd need a compass or GPS and map, and then... I'd ask them to clarify if they meant a rhumb line, great circle line, or something else :)
For everything beyond that, I'd need a compass or GPS and map, and then... I'd ask them to clarify if they meant a rhumb line, great circle line, or something else :)
> If someone asks me to walk in a straight line unassisted by navigational aids, I'd probably end up walking in a medium-sized circle of (optimistically) a few kilometers' diameter.
Why? If you just put one foot in front of the other, you will naturally walk along a geodesic--a great circle.
Why? If you just put one foot in front of the other, you will naturally walk along a geodesic--a great circle.
No, it’ll really be a not-so-great circle :)
https://www.sciencedaily.com/releases/2009/08/090820123927.h...
https://www.sciencedaily.com/releases/2009/08/090820123927.h...
Yeah; the "trick" of the question is "a straight line along a curved surface is actually a curved line", but even this isn't taking into account the Y-axis curvature of the earth; and thus the assumed-to-be "straight" line you're walking isn't even straight on the globe, its more like an arc along the planet's surface.
You can't walk in a universally straight line on the planet. You can only walk in a straight line relative to something. In their example, when they say "walk in a straight line" they really mean something more-like "walk in a radially straight line, relative to the center of the earth, assuming the earth is a perfect sphere, etc"
My assumption when reading the text of the question, which I think is most peoples' very reasonable assumption, is that you are traveling straight relative to the eastward direction; meaning, you're re-adjusting to continue to travel east, because that is a reasonable extension of the pretense "you leave Seattle traveling east".
Its a bad question that is purposefully bad to trick people.
You can't walk in a universally straight line on the planet. You can only walk in a straight line relative to something. In their example, when they say "walk in a straight line" they really mean something more-like "walk in a radially straight line, relative to the center of the earth, assuming the earth is a perfect sphere, etc"
My assumption when reading the text of the question, which I think is most peoples' very reasonable assumption, is that you are traveling straight relative to the eastward direction; meaning, you're re-adjusting to continue to travel east, because that is a reasonable extension of the pretense "you leave Seattle traveling east".
Its a bad question that is purposefully bad to trick people.
Yeah it feels like a logical/semantic confusion, where the author gets to "well actually" the reader.
To me "go east" means something more like "travel with a velocity vector that is always aligned with east on a compass", not "travel with a velocity vector that is fixed with east from your initial position"
To me "go east" means something more like "travel with a velocity vector that is always aligned with east on a compass", not "travel with a velocity vector that is fixed with east from your initial position"
In my opinion all maps of single airplane routes should be shown with a two-point equidistant projection. This clearly shows the straight line between the start and end as the shortest direction, and minimizes distortion along that line. Additionally, all points on the map have accurate distances from either point, and deviations from the geodesic (such as for administrative or weather related reasons) are easily seen.
https://www.jasondavies.com/maps/two-point-equidistant/
https://www.jasondavies.com/maps/two-point-equidistant/
Tiniest clarification to the author, you should include the speed at which this person is travelling. Long duration of travel could result in techtonic shifts, so the traveller might even strike Seattle again. Maximum pedantry
Depending on the speed of travel, countries and Seattle may not even exist. A cataclysmic event may occur and "east" may mean something very different (ask our friend Uranus).
Semi related to this exercise, I really forgot how far north most of Northern Europe is. Not even talking Scandinavia... mentally in my head I didn't have France and Germany on the same-ish latitude as WA.
All of England is north of the north-most point in the lower 48? Huh.
All of England is north of the north-most point in the lower 48? Huh.
Over half of Canada's population lives farther south than Seattle, which is also crazy to think about.
I had to check that for myself, and wow, you weren't kidding.
2015 population of Canada south of Seattle (~47.6°N): ~22 million (https://imgur.com/a/azeTbwz)
2015 Canada population: ~35 million
Sources:
- NASA SEDAC Population Estimator (https://sedac.ciesin.columbia.edu/mapping/popest/gpw-v4/)
- Statistics Canada (https://www150.statcan.gc.ca/n1/daily-quotidien/150929/dq150...)
2015 population of Canada south of Seattle (~47.6°N): ~22 million (https://imgur.com/a/azeTbwz)
2015 Canada population: ~35 million
Sources:
- NASA SEDAC Population Estimator (https://sedac.ciesin.columbia.edu/mapping/popest/gpw-v4/)
- Statistics Canada (https://www150.statcan.gc.ca/n1/daily-quotidien/150929/dq150...)
Its the moderating effect of the Mediterranean! Its wild to see the temperature differences of Wisconsin, Montana, Dakota(s) compared to France & Spain. All that water helps insulate Europe even those its at a northern latitude.
It's actually primarily from the AMOC which is bringing warm water up to the Northern Atlantic from the gulf [0].
Fun fact: a possible consequence of climate change is the near term collapse of the AMOC which would cause drastic cooling in Europe (it also has more severe long-term consequences like leading to an anoxic ocean that emits hydrogen-sulfide instead of oxygen, such events have happened multiple times in Earth's past and played a part in major extinction events [1]).
0. https://en.wikipedia.org/wiki/Atlantic_meridional_overturnin...
1. https://en.wikipedia.org/wiki/Anoxic_event
Fun fact: a possible consequence of climate change is the near term collapse of the AMOC which would cause drastic cooling in Europe (it also has more severe long-term consequences like leading to an anoxic ocean that emits hydrogen-sulfide instead of oxygen, such events have happened multiple times in Earth's past and played a part in major extinction events [1]).
0. https://en.wikipedia.org/wiki/Atlantic_meridional_overturnin...
1. https://en.wikipedia.org/wiki/Anoxic_event
There's also the oceanic currents carrying warm water from the gulf. Europe is in a very lucky position.
Currently in Spain and originally from Seattle. Was thinking this very thing earlier today.
Looking at a map Seattle is roughly the same latitude as Zurich and Budapest.
It's great teeing off after dinner in Scotland.
Yeah, my mental map of Europe is definitely severely warped.
For the longest time, I used to consider myself safe from the detrimental effects of Vitamin D deficiency, which I'd read can be felt as far south as London in Winter.
"Good thing I live in Berlin, which is surely south of London" :)
For the longest time, I used to consider myself safe from the detrimental effects of Vitamin D deficiency, which I'd read can be felt as far south as London in Winter.
"Good thing I live in Berlin, which is surely south of London" :)
More wordplay than anything else. “Go east and then continue straight” is different than “Go east and continue going east”. Results from each are more predictable than misunderstanding the prompt.
Even if the prompt had been "Imagine a traveling from Seattle along a Great Circle path that initially heads east" to take out the "trick" part of the puzzle, I think I probably would have had a hard time visualizing the path without a physical globe in front of me.
Yes, if it had just mentioned that it meant a great circle, I would have felt like it wasn't such a gotcha question.
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> So maybe half a point for “France” if that’s what went through your head.
I was thinking France, but I assumed the answer was non-obvious, so I guessed St. Pierre and Miquelon.
But you'd have to start somewhere just south of Tacoma and just miss the tip of Nova Scotia on your way to the northern tip of Miquelon.
I was thinking France, but I assumed the answer was non-obvious, so I guessed St. Pierre and Miquelon.
But you'd have to start somewhere just south of Tacoma and just miss the tip of Nova Scotia on your way to the northern tip of Miquelon.
Um, before that you'd pass through a good chunk of Ontario and Quebec.
Right, but it said "You will travel across North America, and onto the Atlantic Ocean. Eventually, you will hit another country."
So I was thinking the next country you would hit after hitting the Atlantic Ocean if you moved along a line of latitude.
So I was thinking the next country you would hit after hitting the Atlantic Ocean if you moved along a line of latitude.
> it said "You will travel across North America, and onto the Atlantic Ocean. Eventually, you will hit another country."
Yes, but it didn't say that country would be the first one you would hit. Of course the intended answer of the article was that you would travel along a great circle and would not hit Canada prior to hitting the Atlantic Ocean. But when the article talked about France, it was assuming that you were thinking of traveling along a curve of constant latitude instead. And if you do that, as I said, you will hit Canada before you hit the Atlantic Ocean. And if the author of the article is going to be pedantic enough to say the reader should have thought of great circles, I'm going to turn that right back on him and say he should have thought of Canada when talking about a curve of constant latitude, and should have recognized that his wording is perfectly consistent with that answer.
Yes, but it didn't say that country would be the first one you would hit. Of course the intended answer of the article was that you would travel along a great circle and would not hit Canada prior to hitting the Atlantic Ocean. But when the article talked about France, it was assuming that you were thinking of traveling along a curve of constant latitude instead. And if you do that, as I said, you will hit Canada before you hit the Atlantic Ocean. And if the author of the article is going to be pedantic enough to say the reader should have thought of great circles, I'm going to turn that right back on him and say he should have thought of Canada when talking about a curve of constant latitude, and should have recognized that his wording is perfectly consistent with that answer.
I got really pissed when the headline didn't match the puzzle specs. Still kinda cool to think about at least.
In this case it seems particularly egregious because the entire trick of the puzzle is that "face East and walk in a straight line" and "Go east from Seattle" aren't the same thing.
Yes, and the difference is key: one attempts to travel the same distance with your left foot as with your right foot, and the other chases the direction the stars come up over the horizon.
In a truly "straight" line you'd go into space soon...
Correct answer.
BTW, this is all about which coordinate system for a "straight line", is being used.
BTW, this is all about which coordinate system for a "straight line", is being used.
https://www.landfallnavigation.com/nga-nautical-chart-17-gre...
For small scale marine charts, you can get ones that show the great circle effect.
For small scale marine charts, you can get ones that show the great circle effect.
If you go south from downtown Detroit, what’s the first (non-USA) country you enter?
bonus question: this was unknown to the writer of what song?
Hmm. There’s a reference in Don’t Stop Believin’, but I don’t think that’s the song you’re referring to.
No, that’s what I meant - the “South Detroit” in the song doesn’t exist because that’s Canada.
"Don't Stop Believin'"
Another neat aspect of a curved path but a straight line - Finding the longest straight line you could sail without hitting land https://www.technologyreview.com/2018/04/30/143150/computer-... ( https://news.ycombinator.com/item?id=16965650 | 145 comments)
The most expensive trip you can take while staying within WA! Perth is in Western Australia, or WA.
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I knew what this was gonna be and enjoyed the puzzle within those constraints. My first thought was Australia but then I decided on "somewhere in Africa, probably South Africa" which it looks like was a close guess.
Wonder how much question's success rate would increase if the question was, "what is the first continent you hit?"
Author here.
Reading these comments I agree that the title is far too misleading (and unnecessarily so). Changed to “A path from Seattle”.
Reading these comments I agree that the title is far too misleading (and unnecessarily so). Changed to “A path from Seattle”.
The title is asking for something different from the problem statement.
> in a straight line along the Earth’s surface.
This already seems suspect, given that the earth's surface is curved. A true straight line from the starting point would take us off into space.
So this feels like it's all just quibbling about different curved lines and which is closest to some definition of straight.
This already seems suspect, given that the earth's surface is curved. A true straight line from the starting point would take us off into space.
So this feels like it's all just quibbling about different curved lines and which is closest to some definition of straight.
From the article:
> To be pedantic, by ‘straight line’ here I really mean ‘geodesic’ — the
> generalisation of ‘straight line’ for curved surfaces and space as well
> as ‘flat’, Euclidian spaces.
> A geodesic is just the shortest path between two points in a surface. If
> you’re placed on a surface and continually walk “forward”, by definition
> you will trace out a geodesic.
> Some people replying to my original tweet pointed out that a straight
> line from the original point and direction would take you into space."Straight line of travel means you do not turn left or right. Different from heading continuously east. In the former, walking straight on the curved surface creates a curved path. In the latter, the compass needle will shift a bit as you travel, so you will be turning a bit to stay on the due easterly course.
To your point, this a straight line relative to the traveller while staying on the surface of the globe (which is the premise).
To your point, this a straight line relative to the traveller while staying on the surface of the globe (which is the premise).
> A true straight line from the starting point would take us off into space.
How do you know? "Straight" relative to what? The expanding universe? Would continuous adjustments need to be made to account for the displacement due to that expansion? Does this then assume some 'outer' universe / reference container?
How do you know? "Straight" relative to what? The expanding universe? Would continuous adjustments need to be made to account for the displacement due to that expansion? Does this then assume some 'outer' universe / reference container?
Obviously he meant a geodesic though spacetime, assuming constant speed. Of course your speed does become relevant, but as long as it's positive, "space" seems broadly correct.
At least if we ignore the first 10km or so of your journey. A true straight line (geodesic) from the starting point would take us straight into a tree, hill or building. At that point the journey would end since humans can't phase through solid objects. You never reach another country.
At least if we ignore the first 10km or so of your journey. A true straight line (geodesic) from the starting point would take us straight into a tree, hill or building. At that point the journey would end since humans can't phase through solid objects. You never reach another country.
A true straight line from the starting point would take us into the Earth, because space is curved in its proximity.
This is not correct, not even technically correct. Look up the term "geodesic".
We need a word like "geodesic" precisely because "a straight line on a sphere" is an ambiguous concept.
Man, talking about missing the point even pedantically. Good job on impressing me with that
Is not pedantry the aim of this article?
At least the article gets it right though
The article doesn’t though. East is a path parallel to the equator on a spheroid. East of Seattle is a circle around the globe at a particular latitude. Facing the thing 1mm east of Seattle is a different heading than facing the thing 5000km east of Seattle.
As has already been said as soon as you “go”, if you aren’t turning some infinitesimal amount you aren’t going east. It’s only a surprising result when misleading people by arguing that east is a particular heading.
As has already been said as soon as you “go”, if you aren’t turning some infinitesimal amount you aren’t going east. It’s only a surprising result when misleading people by arguing that east is a particular heading.
But it just said in a straight line to the east. It gave you a cardinal direction to start. From there forward, you go in a straight line.
It gets tricky if you suggest that you can only travel along a great circle in the direction of the rotation along that axis. Since the axis is actually tilted, this is the result you get from traveling along that great circle.
It gets tricky if you suggest that you can only travel along a great circle in the direction of the rotation along that axis. Since the axis is actually tilted, this is the result you get from traveling along that great circle.
The issue people have is with "straight". It's ambiguous. And the simplest, most intuitive interpretation of "straight" in the context of someone directing you to travel East, is to continue travelling East.
It's like handing someone a compass pointing East, then saying "walk in a straight line", and expecting them to slowly travel in a wandering arc. It's insane.
It's like handing someone a compass pointing East, then saying "walk in a straight line", and expecting them to slowly travel in a wandering arc. It's insane.
Maybe so, but that's not how geometry works. If geometry worked by my interpretation, then I would have skated through that class with an even higher grade than I got by following the rules.
Also, the GGGGP that I replied to started by trying to walking a tangential line to a sphere resulting in a space walk. That's like THC induced levels of strained thinking of the subject.
Also, the GGGGP that I replied to started by trying to walking a tangential line to a sphere resulting in a space walk. That's like THC induced levels of strained thinking of the subject.
The puzzle doesn’t say “go east”. Only the headline does.
Really? The title of the puzzle isn't part of the puzzle?
Then the first line of the puzzle should be "completely ignore the title of this puzzle because it's somehow not part of the puzzle. It's unrelated. It will completely mislead you"...
Then the first line of the puzzle should be "completely ignore the title of this puzzle because it's somehow not part of the puzzle. It's unrelated. It will completely mislead you"...
If you have a problem with the title, then take it up with the submitter. The title of the TFA is "A path from Seattle". The article just says to walk in a straight line while facing due east. It does not say to travel East. I know I sometimes have an issue of reading over things too quickly to actually comprehend properly. Maybe this happened to you too?
They changed the title after all the complaints. Also the HN title hasn't been updated yet.
https://news.ycombinator.com/item?id=40567623
https://news.ycombinator.com/item?id=40567623
I guess that's the risk of being pedantic on the interweb. The thing being complained about can change, but those pedantic comments will be written in ink and look foolish in hindsight
[deleted]
I feel sorry for the first cop who gives you a field sobriety test.
Lots of the comments are salty about the question's formulation, but it seems really straightforward to me within a couple sentences. I guessed Cuba, which was a too aggressive with the curve, but not too bad conceptually.
I was not aggressive enough with the curve and thought it would hit the coast of west Africa somewhere around Senegal.
[deleted]
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(rot13-encoded, in case you want the fun of thinking about this yourself)
(rot13-encoded, in case you want the fun of thinking about this yourself)
Did you get it, idiot? No? Let me explain to you why you're wrong.
First, you completely missed that the Great Circle I told you not to worry about doesn't actually follow along the white arrow I used to illustrate the problem. Clearly, if you completely ignore all the information that was given to you, and what East is, the answer is obviously Australia. Because if you were at the North Pole, a line going East would be a tight circle.
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