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1361 clicks; posted to STEM » on 31 May 2022 at 8:50 AM (9 weeks ago)   |   Favorite    |   share:

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There's a logical flaw in that article.  AND I HAVE TO TALK ABOUT IT!!!

5.  And that any two lines that are parallel to each other will always remain equidistant and never intersect.

This isn't necessarily so, and it's the fifth postulate's fault. To understand why, just look at the lines of longitude on a globe.

Every line of longitude you can draw makes a complete circle around the Earth, crossing the equator and making a 90° angle wherever it does. Since the equator is a straight line, and all the lines of longitude are straight lines, this tells us that-at least at the equator-the lines of longitude are parallel. If Euclid's fifth postulate were true, then any two lines of longitude could never intersect.

If the angle of intersection between longitude and latitude keeps changing, then you're not dealing with an actual straight line.  Merely an approximately straight line.  Or, as it's more commonly known, a curve.

Thank you for coming to my ted talk.

SpockYouOut: There's a logical flaw in that article.  AND I HAVE TO TALK ABOUT IT!!!

5.  And that any two lines that are parallel to each other will always remain equidistant and never intersect.

This isn't necessarily so, and it's the fifth postulate's fault. To understand why, just look at the lines of longitude on a globe.

Every line of longitude you can draw makes a complete circle around the Earth, crossing the equator and making a 90° angle wherever it does. Since the equator is a straight line, and all the lines of longitude are straight lines, this tells us that-at least at the equator-the lines of longitude are parallel. If Euclid's fifth postulate were true, then any two lines of longitude could never intersect.

If the angle of intersection between longitude and latitude keeps changing, then you're not dealing with an actual straight line.  Merely an approximately straight line.  Or, as it's more commonly known, a curve.

Thank you for coming to my ted talk.

Your example of why two parallel lines never intersect in a flat universe is based on a sphere. Solid logic right there.

SpockYouOut: If the angle of intersection between longitude and latitude keeps changing, then you're not dealing with an actual straight line.  Merely an approximately straight line.  Or, as it's more commonly known, a curve.

Except no.

What you describe is true of Euclidean geometry, but not of non-Euclidean geometries. Which is more or less the whole point of the article. To make sense of it you also have to understand that the meaning of basic concepts like "line" and "point" is more abstract than the intuitive sense we get from living in a (more or less) Euclidean space.

In a spherical geometry, such as the two-dimensional surface of a sphere, lines of longitude are "lines". So is the equator, and indeed, any other "great circle" on the sphere. A great circle is the largest circle you can draw around the sphere, and has the critical properties that

- you can draw one and only one great circle through two points on the sphere; and
- following the great circle drawn between two points is the shortest distance between them

That's what makes a great circle the mathematical equivalent in spherical space of a conventional straight line in flat (Euclidean) space.

So from our external POV great circles curve in three dimensions, sure, but to somebody living on the surface and confined to two dimensions, a great circle is a straight line. HOWEVER a line of latitude [except for the special case of the equator] is not a straight line; it's a curve. That's why different lines of latitude intersect lines of longitude (straight lines) at different angles: because the longitudes are "straight", but the latitudes are "curved".

Anyway, all of this can be made mathematically rigorous, and extended to three or more dimensions - the above is just a sketch to try and give an idea. The key thing is that you have to let go of your intuitive sense of what a "line" is and how it behaves in a Euclidean space to understand what is going on in curved spaces.

If the universe is flat, why haven't cosmic cats batted everything off the edge yet?

Or is that why galaxies keep disappearing over what we wrongly imagine to be our cosmic horizon?

SpockYouOut: There's a logical flaw in that article.  AND I HAVE TO TALK ABOUT IT!!!

5.  And that any two lines that are parallel to each other will always remain equidistant and never intersect.

This isn't necessarily so, and it's the fifth postulate's fault. To understand why, just look at the lines of longitude on a globe.

Every line of longitude you can draw makes a complete circle around the Earth, crossing the equator and making a 90° angle wherever it does. Since the equator is a straight line, and all the lines of longitude are straight lines, this tells us that-at least at the equator-the lines of longitude are parallel. If Euclid's fifth postulate were true, then any two lines of longitude could never intersect.

If the angle of intersection between longitude and latitude keeps changing, then you're not dealing with an actual straight line.  Merely an approximately straight line.  Or, as it's more commonly known, a curve.

Thank you for coming to my ted talk.

Congratulations, you just flunked noneuclidean geometry.

HugeMistake: The opposite case is hyperbolic space. It's harder to visualize, but one example is like the surface of a saddle: it curves down at the sides and up at the ends.

I write excellent book on topic. Index copied from old Vladivostok telephone directory.

I still subscribe to Homer's theory of a doughnut shaped universe.

flat earthers everywhere: see!!!

The universe is not flat subby even though it is "flat."

What we have here is equivocation: a word with more than one meaning. The universe is not flat as as the word is used in everyday speech which is the type of flat that flat earth morons are talking about.  The evidence supports that the universe has a flat spacetime as astronomers understand the word when they are discussing astrophysics.

Next up: Earth's atmosphere is mostly metal. That is true in astronomy jargon and utterly false if you are a chemist.

TheMysteriousStranger: Next up: Earth's atmosphere is mostly metal. That is true in astronomy jargon and utterly false if you are a chemist.

It's also mostly false if you write for Revolver.

SpockYouOut: To understand why, just look at the lines of longitude on a globe.

Those are not lines, they are planes.

TheMysteriousStranger: Next up: Earth's atmosphere is mostly metal.

But every so often it opts to listen to soft rock.

Up isn't real.

Nimbull: [Fark user image 600x600]

"Time is a flat tire at the Circle K"

TheMysteriousStranger: SpockYouOut: There's a logical flaw in that article.  AND I HAVE TO TALK ABOUT IT!!!

5.  And that any two lines that are parallel to each other will always remain equidistant and never intersect.

This isn't necessarily so, and it's the fifth postulate's fault. To understand why, just look at the lines of longitude on a globe.

Every line of longitude you can draw makes a complete circle around the Earth, crossing the equator and making a 90° angle wherever it does. Since the equator is a straight line, and all the lines of longitude are straight lines, this tells us that-at least at the equator-the lines of longitude are parallel. If Euclid's fifth postulate were true, then any two lines of longitude could never intersect.

If the angle of intersection between longitude and latitude keeps changing, then you're not dealing with an actual straight line.  Merely an approximately straight line.  Or, as it's more commonly known, a curve.

Thank you for coming to my ted talk.

Congratulations, you just flunked noneuclidean geometry.

In cartography, lines are called arcs and then there's nautical miles.....

// you aren't wrong, adding to you

I believe it prefers being called a member of the Itty Bitty Omega Committee. You don't necessarily want to have a Universe that has a large, positive curvature, no matter how fun they may be to play in, as they will eventually squeeze the life right out of you. And the eldritch horrors that are less than flat? They'll tear you apart.

FTA: If you had come along before the 1800s, it likely never would have occurred to you that the Universe itself could even have a shape.

Holy shiat this is so wrong, there is no reason to trust anything in the article. WT actual F?

HugeMistake: But this saddle is infinite; it doesn't have edges. And the interesting thing about this space is that given a "straight line", you can draw infinitely many "parallel" lines, i.e. lines that never intersect with that line - in contradiction to Euclid's infamous fifth postulate.

Important to note: the reason it doesn't follow the 5th postulate isn't that you can make infinitely many non-intersecting parallel lines, which you can surely do in Euclidian geometry, it's that they don't remain equidistant from each other.

Bennie Crabtree: FTA: If you had come along before the 1800s, it likely never would have occurred to you that the Universe itself could even have a shape.

Holy shiat this is so wrong, there is no reason to trust anything in the article. WT actual F?

Dammit Jim, he's an astronomer not a historian of science.

/Once had a quiz in a history of science class which the prof quoted Stephen Hawking's "A Brief History of Time" and asked what was wrong with it.

Bennie Crabtree: FTA: If you had come along before the 1800s, it likely never would have occurred to you that the Universe itself could even have a shape.

Holy shiat this is so wrong, there is no reason to trust anything in the article. WT actual F?

Gimme a yell, a yell
an existential yell
and when we yell
we yell like hell
and this is what we yell
Ala-bam
Ala-bam

Alabam-diego
San Diego
Aristarchus
Kiss my carcass
Rah Rah
[team name]

Ivo Shandor: [wompampsupport.azureedge.net image 657x1080]

I'm glad I don't need to illustrate every lame thought about prepubescent schoolgirls, it sounds like hell.

Ivo Shandor: [wompampsupport.azureedge.net image 657x1080]

She'd also be wet and crawling with things.  Ew.

freidog: I still subscribe to Homer's theory of a doughnut shaped universe.

Does that mean the universe was born in May... 'cause it's a torus?

/ducks, runs

CheatCommando: HugeMistake: The opposite case is hyperbolic space. It's harder to visualize, but one example is like the surface of a saddle: it curves down at the sides and up at the ends.

I write excellent book on topic. Index copied from old Vladivostok telephone directory.

This I know from nothing.

madgonad: SpockYouOut: To understand why, just look at the lines of longitude on a globe.

Those are not lines, they are planes.

Lines of longitude are actually contrails.

HugeMistake: madgonad: SpockYouOut: To understand why, just look at the lines of longitude on a globe.

Those are not lines, they are planes.

Lines of longitude are actually contrails.

Ha! Someday I'll learn to check my spellchecker.

Those are not lines, they are planes.

Lines of longitude are actually contrails.

Ha! Someday I'll learn to check my spellchecker.

Wait, it's spelled right. I guess 4 hours of sleep isn't good for the brain in general.

LrdPhoenix: HugeMistake: But this saddle is infinite; it doesn't have edges. And the interesting thing about this space is that given a "straight line", you can draw infinitely many "parallel" lines, i.e. lines that never intersect with that line - in contradiction to Euclid's infamous fifth postulate.

Important to note: the reason it doesn't follow the 5th postulate isn't that you can make infinitely many non-intersecting parallel lines, which you can surely do in Euclidian geometry, it's that they don't remain equidistant from each other.

Um, not really.

The statements about an infinite number of parallel lines through a given point and lines not remaining equidistant are formally equivalent. So I don't know what it means to say that one of them is important and the other is not?

Here's the fifth postulate as usually stated, with a key part emboldened.

If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

This is equivalent to the statement known as Playfair's Axiom (although it was known in antiquity):

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

Now the attentive reader will have noticed the phrase "at most" in the above statement of Playfair's axiom. It is possible to prove, by combining the 5th postulate/Playfair's Axiom with the other four that there must exist at least one parallel line; so you end up with the result of "one and only one" in Euclidean geometry. This is not stated in Playfair's axiom because it would be redundant.

By "equivalent" here we mean that you can assume Euclid's 5th postulate and then you can derive Playfair's Axiom; or you can assume Playfair's Axiom and then you can derive Euclid's 5th. Either way you get the same (Euclidean) geometry. They are equally good as 5th axioms.

Spherical geometry violates Playfair's Axiom: there are no lines that are parallel to each other. And so does hyperbolic geometry: there are infinitely many lines parallel to a given line through a point not on it. Consequently, that's why it also violates Euclid's 5th.

Now, you can also quote Euclid's 5th as follows:

There exists a pair of straight lines that are at constant distance from each other.

Although perhaps not immediately obvious, this is also equivalent to Euclid's 5th and Playfair's Axiom. It is easy to see that on the sphere there are no such lines: all lines intersect (remember that a "line" here is a great circle). In hyperbolic space it's not so easy to visualize, but it turns out that this postulate is also violated there; so we might say "there exists no pair of straight lines that are at constant distance from each other". Or as you originally phrased it, "they don't remain equidistant from each other."

Hence: completely equivalent.

HugeMistake: LrdPhoenix: HugeMistake: But this saddle is infinite; it doesn't have edges. And the interesting thing about this space is that given a "straight line", you can draw infinitely many "parallel" lines, i.e. lines that never intersect with that line - in contradiction to Euclid's infamous fifth postulate.

Important to note: the reason it doesn't follow the 5th postulate isn't that you can make infinitely many non-intersecting parallel lines, which you can surely do in Euclidian geometry, it's that they don't remain equidistant from each other.

Um, not really.

That's not what you said though.  You said:

"And the interesting thing about this space is that given a "straight line", you can draw infinitely many "parallel" lines, i.e. lines that never intersect with that line - in contradiction to Euclid's infamous fifth postulate."

Which makes it sound like the ability to construct infinitely many lines which are parallel to some other line violates flat geometry, which it obviously doesn't.  In Euclidian geometry, given any line you can easily construct infinitely many parallel lines by first making a line which is perpendicular to that line, and then making as many more lines as you wish that are perpendicular to the new one.

You never specified a given point that the parallel lines must pass through.

xanadian: TheMysteriousStranger: Next up: Earth's atmosphere is mostly metal.

But every so often it opts to listen to soft rock.

Not true. It has both kinds: Country AND Western!

LrdPhoenix: That's not what you said though.  You said:

"And the interesting thing about this space is that given a "straight line", you can draw infinitely many "parallel" lines, i.e. lines that never intersect with that line - in contradiction to Euclid's infamous fifth postulate."

No, that's what you made it seem like I said by cutting out the previous part of my explanation. Anybody with an 8th grade education knows the Euclidean properties of parallel lines; even you yourself described it as "obvious". And I think it should be clear from both my previous posts that I have a better than 8th grade understanding of both Euclidean and non-Euclidean geometry, no?

In other words, you're trying to look clever by being pedantic about a quote taken out of context - and it's not working.

TheMysteriousStranger: Next up: Earth's atmosphere is mostly metal.

HugeMistake: LrdPhoenix: That's not what you said though.  You said:

"And the interesting thing about this space is that given a "straight line", you can draw infinitely many "parallel" lines, i.e. lines that never intersect with that line - in contradiction to Euclid's infamous fifth postulate."

No, that's what you made it seem like I said by cutting out the previous part of my explanation. Anybody with an 8th grade education knows the Euclidean properties of parallel lines; even you yourself described it as "obvious". And I think it should be clear from both my previous posts that I have a better than 8th grade understanding of both Euclidean and non-Euclidean geometry, no?

In other words, you're trying to look clever by being pedantic about a quote taken out of context - and it's not working.

It's not taken out of context, everything before what I initially quoted deals with spherical geometry and great circles and has nothing to do with parallel lines in neither planar nor hyperbolic geometry forced through a specific point.

By the way, your great circle definition is flawed.  Specifically: "you can draw one and only one great circle through two points on the sphere;"  This is only true so long as the two points do not directly oppose each other.  All lines of longitude are great circles which pass through the same two points at the poles.

Let me show you how lines work on my Timecube

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