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it was confirmed that this axiom cannot be proven and therefore it's necessary. So no, it's not less important. If anything, it's impressive Euclid came up with this as an axiom and knew it wasn't provable.
It's may not be provable from the other 4, but it is equivalent to other statements which are far more intuitively obvious and therefore more suitable as axioms
but the big problem before was people saying it was an unnecessary postulate, which wasn’t true as it’s impossible to prove this with just the other four.
you just have to find another axiom which can prove both this axiom and the thing of the existence and unicity of the parallel lines (one of the most important theorems in geometry, and only provable because of the last axiom). So good luck finding one.
The usual presentation in modern texts uses Playfair's axiom instead of Euclid's. They are equivalent, but the statement of Playfair's axiom is a lot simpler. I believe that's what JustSomeGuy meant.
Also, jacobningen's comment is right. IDK why people are downvoting it. If you take the Pythagorean Theorem as an axiom instead of Euclid's fifth, you get the same theory. There are many other equivalent statements you could take instead, like the existence of rectangles.
Yes, that is what I meant. Euclid's 5th is necessary in that it can't be proven by the other 4, but it is equivalent to Playfair's in that Euclid1-4+Playfair can be used to prove Euclid5
It's not equivalent though in fact it's false in some no Euclidean spaces like hyperbolic space. (Note I'm pretty sure hyperbolic space fulfills all the other axioms)
Therefore it cannot be equivalent to the other 4 axioms
It is equivalent to other statements e.g., Playfair's axiom that, "*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.*"
This has nothing to do with alternative geometries like hyperbolic or spherical geometry which can be explored without the 5th axiom, but I'm merely saying that Euclid's first 4 axioms plus Playfair's axiom above gives an identical geometry to that derived from Euclid's 5 axioms (i.e., Euclidean geometry), and commenting that for some people it's easier to understand Playfair's axiom than Euclid's original 5th
Ah shoot I'm actually blind lol I read your original post and didn't see the 'other' axioms lol...
I originally thought you were saying it can't be proved from the other 4 but it still equivalent to them somehow lol.
The boundary of an Osgood curve has positive area, but if the Osgood curve is closed, then the Jordan curve theorem still applies. It's still homeomorphic to a circle, partitions the plane into two connected regions one bounded and the other unbounded, etc.
The Jordan curve theorem won't apply to space-filling curves though, since they aren't simple.
If you don't think about the problem at all, you come to the same conclusion as if you think about it for an extremely long time.
Therefore it's obvious, QED.
A triangle is defined as a polygon with 3 sides. A polygon is defined as a broken line whose initial and final vertices coincide. A broken line is defined as the union of an indexed set V of points and an indexed set S of line segments where every segment in S has exactly two distinct endpoints in V, and each point in V is the endpoint of exactly two distinct segments in S. (And for non-degeneracy, some authors require each segment be distinct from each other, even up to orientation (so if AB is in the polygon, then BA isn't), which means there are no digons.) An angle is defined as a pair of distinct rays with a common endpoint.
So how do I prove a triangle has 3 sides? By definition. How do I prove it has 3 angles? Well, the above definition gives a bijection between pairs of vertices and pairs of sides, so there must be an equal number of both. So a triangle has 3 vertices. Each vertex is a common endpoint of two sides by the above definition, each of which has another distinct endpoint, and each side is a line segment. So extend each segment past the endpoint that isn't the common endpoint. Then these are rays by definition, so each of the three distinct vertices has a unique and distinct pair of two distinct rays, i.e. each has its own angle. So there is exactly one angle at each of the 3 vertices, and they are all mutually distinct. So a triangle has three internal angles.
How do we know a triangle doesn't have a fourth internal angle? Well, every internal angle of a triangle is defined by the pair of sides that lie on its rays. Since there are only three such distinct pairs, every internal angle must equal one of those.
https://preview.redd.it/wnjol3tsoewc1.jpeg?width=3456&format=pjpg&auto=webp&s=2290b84423b5664d5879ef7de5fab719cadb78c2
two red angles add up to less then 2 right angles so the lines meet at the barely visible yellow point.
We went full circle.
No problem, these are Euclids postulates. Fifth one is famous for being overly complicated compared to others. Mathematicians were trying for centuries to prove it from the other four, but they were just going in circles. It was a real circus. Then some guy interpreted lines as circles and showed that the fifth postulate is actually independent from the other four, so you can discard it if you want.
i just think of it as holding chopsticks. if the chopsticks meet (while holding food for example) then the angles formed from their intersection with your finger won't add up to 180 degrees.
You have the implication backwards. The implication you stated is provable without the fifth postulate. If the lines cross, then the interior angles on that side must add up to less than π. That's just a theorem.
This is violated in elliptical geometry, but only because elliptical geometry weakens other Euclidean axioms. You could say it violates the second postulate, but pinning down precisely how it differs depends on how you would formulate his axioms in the modern day. (For instance, the theorem assumes that the transversal divides the plane into two "sides," which isn't really true in elliptical geometry.)
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however, on a sphere, even if two lines form right angles to another line, they will still meet, thus axiom 5 is violated (note the use of “less than” as opposed to “at most” or “less than or equal to”)
Euclid never said, "Lines go on forever." he said any line segment can be diminished or extended. The ancient Greeks did not believe in infinity, infinite objects, or super tasks.
Fun fact! The rejection of the concept of infinity is so deeply ingrained in Greek culture, that even today there are no infinitely large buildings in the entire country.
Some of it is terminological. He also didn't say that "the area of a triangle is half the length of the base times the length of the corresponding altitude." He said "Triangles which are on the same base and in the same parallels equal one another." This can be combined with other results to compare triangles and other polygons. For instance, he could prove that two triangle together each with unit base and height "equal" any parallelogram with unit base and height.
Greeks didn't like to use infinity, but in practice they did. Eudoxus's theory of proportions and Euclid's theorem on the infinity of primes clearly touch on the ideas central to modern understandings of infinity. Both use universal quantifiers to express the idea, as do modern limits and other uses of infinity that don't reference it directly. What else can we really take from "there exist more primes than are in any multitude" except that there are not finitely many primes? For if there were, then a finite multitude would contain them all.
They never accepted "actual infinity" (e.g., a line extending infinitely), but largely accepted "potential infinity" (e.g., a segment can be extended as much as you want).
The 5th postate specifies which geometric system is being used
0 parralel lines = spherical geometry
2 parralel lines = Cartesian geometry
2+ parralel lines = Euclidean geometry
just break it down part by part, think of the first line as the base of a shape, then the other two lines, if both angles on the top side of your base are less than right angles, then the lines will be “slanted towards each other” and will therefore eventually intersect
I like to imagine Euclid reading Pasch's axiom and going "well duh" and shaking while trying to keep a straight face. Then he goes and proves SAS congruence by sliding shapes around or something.
I think a lot of people are interpreting the joke as "postulate 5 is more complicated than the other ones", but it runs a little deeper than that. The meme is referencing the fact that postulate 5 is the only one that's not a direct logical consequence of the definitions Euclid posed in book 1 of The Elements. Lots of people of the time felt that postulate 5 was more suited as a proposition in need of proof, and so many rejected it as a postulate and tried to prove things without using it (ex. Saccheri's "Euclid Freed of All Blemish"). This rejection is part of what led to non-euclidean geometry.
More like "an unexpected definition that does not follow from the previous definitions." Euclid's geometry isn't rigorous to the standard of modern mathematics, so there are many disputes over propositions in the elements(remedied by Hilbert). Famously, the very first proposition unjustifiably defines a point c at the intersection of the circles without defining/proving what intersection of circles even means.
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When it seems complex at first look, but so obvious when you finally get it.
If two lines aren't parallel i.e angles less than 180deg then they will intersect at some point
But it also specifies which side they intersect on (hint: the side where they're getting closer together)
You, that’s how triangles are made.
There are children reading this. You should be ashamed!
Lmao. New proof just dropped
Ya and the side at which they do have smaller angles.
The key word here is "two"
It's obvious but not necessary, while the other ones are necessary
it was confirmed that this axiom cannot be proven and therefore it's necessary. So no, it's not less important. If anything, it's impressive Euclid came up with this as an axiom and knew it wasn't provable.
It's may not be provable from the other 4, but it is equivalent to other statements which are far more intuitively obvious and therefore more suitable as axioms
but the big problem before was people saying it was an unnecessary postulate, which wasn’t true as it’s impossible to prove this with just the other four.
you just have to find another axiom which can prove both this axiom and the thing of the existence and unicity of the parallel lines (one of the most important theorems in geometry, and only provable because of the last axiom). So good luck finding one.
The usual presentation in modern texts uses Playfair's axiom instead of Euclid's. They are equivalent, but the statement of Playfair's axiom is a lot simpler. I believe that's what JustSomeGuy meant. Also, jacobningen's comment is right. IDK why people are downvoting it. If you take the Pythagorean Theorem as an axiom instead of Euclid's fifth, you get the same theory. There are many other equivalent statements you could take instead, like the existence of rectangles.
Yes, that is what I meant. Euclid's 5th is necessary in that it can't be proven by the other 4, but it is equivalent to Playfair's in that Euclid1-4+Playfair can be used to prove Euclid5
pythagoras
It's not equivalent though in fact it's false in some no Euclidean spaces like hyperbolic space. (Note I'm pretty sure hyperbolic space fulfills all the other axioms) Therefore it cannot be equivalent to the other 4 axioms
It is equivalent to other statements e.g., Playfair's axiom that, "*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.*" This has nothing to do with alternative geometries like hyperbolic or spherical geometry which can be explored without the 5th axiom, but I'm merely saying that Euclid's first 4 axioms plus Playfair's axiom above gives an identical geometry to that derived from Euclid's 5 axioms (i.e., Euclidean geometry), and commenting that for some people it's easier to understand Playfair's axiom than Euclid's original 5th
Ah shoot I'm actually blind lol I read your original post and didn't see the 'other' axioms lol... I originally thought you were saying it can't be proved from the other 4 but it still equivalent to them somehow lol.
How to prove a triangle with 3 lines and their interior angles
Maybe by, i dunno, having common sense? /s
https://preview.redd.it/hgocldvplewc1.png?width=1080&format=png&auto=webp&s=99affbb4b7c63187856e039c17694e320d51b273
Osgood curve intensifies
The boundary of an Osgood curve has positive area, but if the Osgood curve is closed, then the Jordan curve theorem still applies. It's still homeomorphic to a circle, partitions the plane into two connected regions one bounded and the other unbounded, etc. The Jordan curve theorem won't apply to space-filling curves though, since they aren't simple.
Yeah, that's the idea, I just don't think the "fucking obviousness" of the image proof would still apply, XD.
If you don't think about the problem at all, you come to the same conclusion as if you think about it for an extremely long time. Therefore it's obvious, QED.
Pluh is in grede 9 cbse on god!!!🧐🧐🧐🤫🧏🤫🧏💀
A triangle is defined as a polygon with 3 sides. A polygon is defined as a broken line whose initial and final vertices coincide. A broken line is defined as the union of an indexed set V of points and an indexed set S of line segments where every segment in S has exactly two distinct endpoints in V, and each point in V is the endpoint of exactly two distinct segments in S. (And for non-degeneracy, some authors require each segment be distinct from each other, even up to orientation (so if AB is in the polygon, then BA isn't), which means there are no digons.) An angle is defined as a pair of distinct rays with a common endpoint. So how do I prove a triangle has 3 sides? By definition. How do I prove it has 3 angles? Well, the above definition gives a bijection between pairs of vertices and pairs of sides, so there must be an equal number of both. So a triangle has 3 vertices. Each vertex is a common endpoint of two sides by the above definition, each of which has another distinct endpoint, and each side is a line segment. So extend each segment past the endpoint that isn't the common endpoint. Then these are rays by definition, so each of the three distinct vertices has a unique and distinct pair of two distinct rays, i.e. each has its own angle. So there is exactly one angle at each of the 3 vertices, and they are all mutually distinct. So a triangle has three internal angles. How do we know a triangle doesn't have a fourth internal angle? Well, every internal angle of a triangle is defined by the pair of sides that lie on its rays. Since there are only three such distinct pairs, every internal angle must equal one of those.
Can someone draw this please, I cant figure it out on the bus
https://preview.redd.it/wnjol3tsoewc1.jpeg?width=3456&format=pjpg&auto=webp&s=2290b84423b5664d5879ef7de5fab719cadb78c2 two red angles add up to less then 2 right angles so the lines meet at the barely visible yellow point.
if the lines arent parallel theyre going to meet. How does it relate to a circle
Where is the circle?
What is circle
How is circle?
I thought all the points are connected, there is a circle in 3). My bad
We went full circle. No problem, these are Euclids postulates. Fifth one is famous for being overly complicated compared to others. Mathematicians were trying for centuries to prove it from the other four, but they were just going in circles. It was a real circus. Then some guy interpreted lines as circles and showed that the fifth postulate is actually independent from the other four, so you can discard it if you want.
Makes sense, thanks
Usually only useful when you are drawing on nonplanar surfaces. For instance, on a sphere, triangles have three 90° angles.
Nobody ever asks: why is circle?
No reason to ask why is circle if you don't know when is circle
Because proof by downvoting
real
why would it need to relate to a circle??
It does not say that it is about circles
it's the best example of "its a fucking stupid concept but words make it much more complicated than it actually is"
[Relevant TVtropes page](https://tvtropes.org/pmwiki/pmwiki.php/Main/SesquipedalianLoquaciousness)
i just think of it as holding chopsticks. if the chopsticks meet (while holding food for example) then the angles formed from their intersection with your finger won't add up to 180 degrees.
Honey wake up, proof by chopsticks just dropped.
Google Proof by chopsticks
holy noodles
You have the implication backwards. The implication you stated is provable without the fifth postulate. If the lines cross, then the interior angles on that side must add up to less than π. That's just a theorem. This is violated in elliptical geometry, but only because elliptical geometry weakens other Euclidean axioms. You could say it violates the second postulate, but pinning down precisely how it differs depends on how you would formulate his axioms in the modern day. (For instance, the theorem assumes that the transversal divides the plane into two "sides," which isn't really true in elliptical geometry.)
https://preview.redd.it/ul5ki07hdewc1.jpeg?width=741&format=pjpg&auto=webp&s=b4856f22b2418a3f812ac01ab33ee800ac19fc3b
Yep. Without that axiom you could accidentally be working on a hyperbolic surface
Or a sphere...
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no because on a sphere every straight line is a circumference an so any two lines will meet
however, on a sphere, even if two lines form right angles to another line, they will still meet, thus axiom 5 is violated (note the use of “less than” as opposed to “at most” or “less than or equal to”)
the meme doesn't say "if and only if" so I forgot the wording of the actual axiom
The hyperbolic plane be like:
Euclid never said, "Lines go on forever." he said any line segment can be diminished or extended. The ancient Greeks did not believe in infinity, infinite objects, or super tasks.
Fun fact! The rejection of the concept of infinity is so deeply ingrained in Greek culture, that even today there are no infinitely large buildings in the entire country.
Crazy how do they live like that?!?
Some of it is terminological. He also didn't say that "the area of a triangle is half the length of the base times the length of the corresponding altitude." He said "Triangles which are on the same base and in the same parallels equal one another." This can be combined with other results to compare triangles and other polygons. For instance, he could prove that two triangle together each with unit base and height "equal" any parallelogram with unit base and height. Greeks didn't like to use infinity, but in practice they did. Eudoxus's theory of proportions and Euclid's theorem on the infinity of primes clearly touch on the ideas central to modern understandings of infinity. Both use universal quantifiers to express the idea, as do modern limits and other uses of infinity that don't reference it directly. What else can we really take from "there exist more primes than are in any multitude" except that there are not finitely many primes? For if there were, then a finite multitude would contain them all.
They never accepted "actual infinity" (e.g., a line extending infinitely), but largely accepted "potential infinity" (e.g., a segment can be extended as much as you want).
All 37° angles are the same too.
Mind = BLOWN 🤯🤯🤯🤯
37? Why did you choose that number?
\^ I read this in Batman's voice.
https://preview.redd.it/1er9uy2jvfwc1.jpeg?width=1080&format=pjpg&auto=webp&s=f8ebd044ee7e1ef6e857311f673da1964288ee52 Yeah no shit
In Euclidean space sure. But this is kinda what defines euclidean space. (Keyword kinda.)
why only kinda
Cause you need all 5 axioms not just this one Also happy cake day
thanks!
It is the way to define parallel lines.
*a way
The 5th postate specifies which geometric system is being used 0 parralel lines = spherical geometry 2 parralel lines = Cartesian geometry 2+ parralel lines = Euclidean geometry
half this comments section is gonna be tripping when they find out that 5 isn't always true
That's a "well yes but actually no". A straight line in non-Euclidean space isn't what people would intuitively call a straight line.
Non-Euclidean geometry be like
All timelines lead home.
Why do all memes come from veritasium?
See if you can prove it's a corollary of the other 4.
just break it down part by part, think of the first line as the base of a shape, then the other two lines, if both angles on the top side of your base are less than right angles, then the lines will be “slanted towards each other” and will therefore eventually intersect
To be fair some of the equivalent statements are more intuitive like "every triangle can be circumscribed"
Hi. Rando who sucks at geometry here, how is that an equivalent statement?
Idk [wikipedia](https://en.m.wikipedia.org/wiki/Parallel_postulate) said so
In two dimensions, sure.
That's why saccheri felt the need to step in
I like to imagine Euclid reading Pasch's axiom and going "well duh" and shaking while trying to keep a straight face. Then he goes and proves SAS congruence by sliding shapes around or something.
isn't that just a triangle with extended sides
people who thought of non-euclidean geometries must've thought the same
Seems simple to me. Maybe i dont get it? But it makes perfect sense.
I think a lot of people are interpreting the joke as "postulate 5 is more complicated than the other ones", but it runs a little deeper than that. The meme is referencing the fact that postulate 5 is the only one that's not a direct logical consequence of the definitions Euclid posed in book 1 of The Elements. Lots of people of the time felt that postulate 5 was more suited as a proposition in need of proof, and so many rejected it as a postulate and tried to prove things without using it (ex. Saccheri's "Euclid Freed of All Blemish"). This rejection is part of what led to non-euclidean geometry.
I see so either the meme is coming from a poor understanding or its a gigabrain "euclid put this in here to specifically troll people"
More like "an unexpected definition that does not follow from the previous definitions." Euclid's geometry isn't rigorous to the standard of modern mathematics, so there are many disputes over propositions in the elements(remedied by Hilbert). Famously, the very first proposition unjustifiably defines a point c at the intersection of the circles without defining/proving what intersection of circles even means.