5 ms·

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.

by arh68 2y ago

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.

re 2y ago
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.
- bialpio 2y ago
  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.
  - w-j-w 2y ago
    [dead]
  - happytoexplain 2y ago
    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".
    - davidw 2y ago
      Of course it's not a perfectly straight line, because the earth is a curved object. Perfectly straight puts you in outer space.
      - mynameisvlad 2y ago
        The article quite clearly says “in a straight line along the Earth’s surface.” Emphasis mine.
      - happytoexplain 2y ago
        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.
  - mc10 2y ago
    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".
    - tacitusarc 2y ago
      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.
      - pdonis 2y ago
        > 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.
        lxgr 2y ago
        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?
        pdonis 2y ago
        > 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.
  - lxgr 2y ago
    Surely you mean Paris, Ontario, Canada? :)
- bsder 2y ago
  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.
  - lxgr 2y ago
    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).
  - pdonis 2y ago
    > 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.
- erikerikson 2y ago
  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.
  - PhasmaFelis 2y ago
    > 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.)
    - erikerikson 2y ago
      > 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.
wrs 2y ago
The headline is poorly worded. The actual puzzle is unambiguous (begin facing east, then travel forward in a straight line along the surface).
- lxgr 2y ago
  But there are no straight lines on the surface of a sphere (or geoid)!
  - pdonis 2y ago
    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.
    - lxgr 2y ago
      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.
      - pdonis 2y ago
        > 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.
        2y ago
NegativeLatency 2y ago
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"
015a 2y ago
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.

by arh68 2y ago

re 2y ago

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.

bialpio 2y ago
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.
- w-j-w 2y ago
  [dead]
- happytoexplain 2y ago
  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".
  - davidw 2y ago
    Of course it's not a perfectly straight line, because the earth is a curved object. Perfectly straight puts you in outer space.
    - mynameisvlad 2y ago
      The article quite clearly says “in a straight line along the Earth’s surface.” Emphasis mine.
    - happytoexplain 2y ago
      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.
- mc10 2y ago
  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".
  - tacitusarc 2y ago
    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.
    - pdonis 2y ago
      > 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.
      - lxgr 2y ago
        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?
        pdonis 2y ago
        > 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.
- lxgr 2y ago
  Surely you mean Paris, Ontario, Canada? :)
bsder 2y ago
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.
- lxgr 2y ago
  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).
- pdonis 2y ago
  > 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.
erikerikson 2y ago
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.
- PhasmaFelis 2y ago
  > 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.)
  - erikerikson 2y ago
    > 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.

wrs 2y ago

The headline is poorly worded. The actual puzzle is unambiguous (begin facing east, then travel forward in a straight line along the surface).

lxgr 2y ago
But there are no straight lines on the surface of a sphere (or geoid)!
- pdonis 2y ago
  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.
  - lxgr 2y ago
    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.
    - pdonis 2y ago
      > 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.
      - 2y ago

NegativeLatency 2y ago

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"

015a 2y ago

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.