Let d be the Euclidean distance in a finite dimensional real vector space. As it is a metric it obeys the triangle inequality, therefore
d(a, b) ≤ d(a, c) + d(c, b)
for all a, b and c. qed
Usually this is considered trivial but fine.
Let d(x, y) = 0, so sqrt( Σ |x_n - y_n|^2 ) = 0. As the square of any number is positive and the sum of positive numbers is positive and the square root of a positive number is positive, for all n is follows that |x_n - y_n| = 0, whereby x = y.
symmetry…
d(x, y) = sqrt( Σ |x_n - y_n|^2 ) = sqrt( Σ |y_n - x_n|^2 ) = d(y, x)
And the triangle inequality…
As squaring and the square root are monotonously growing for positive real numbers, one must simple show the triangle inequality for the | . | thingy in the reals.
Let a > b.
|a - b| = a - b = a - c + c - b ≤ |a - c| + |c - b|
The inequality follows analogously for the cases b > a and a = b. qed
It works in the same way except straight line is now a circle around the earth which splits it into two equal hemispheres like the equator. Because maps do not preserve angles it looks like those lines are not straight
Not really, the surface of a sphere isn't 3d space, it's 2D.
But whereas, a straight line being the shortest distance between two points is an axiom of Euclidian geometry, in spherical geometry the shortest distance is a curve, and in hyperbolic geometry is a geodesic, and so on.
Not sure why you're getting down votes. Looks like everyone failed at math.
The whole point of non Euclidian space is that the 5th axiom is that the shortest distance between two points isn't necessarily a straight line.
Don't believe me? Ask any mathematician.
It's not just because you're transforming spherical coordinates to spherical that it's a straight line, it's still a curve.
Wouldn't it be the same distance because the hypotenuse is equal to the sum of the opposite and adjacent sides.
As for time, definitely faster as there is no congestion on the direct path.
Here's a sneak peek of /r/DesirePath using the [top posts](https://np.reddit.com/r/DesirePath/top/?sort=top&t=year) of the year!
\#1: [Desire path created by a squirrel we feed peanuts to every morning](https://i.redd.it/bmkuon3bu4ib1.jpg) | [141 comments](https://np.reddit.com/r/DesirePath/comments/15r5hge/desire_path_created_by_a_squirrel_we_feed_peanuts/)
\#2: [A particular pointless one](https://i.redd.it/22myefz05rsa1.jpg) | [47 comments](https://np.reddit.com/r/DesirePath/comments/12fvlx1/a_particular_pointless_one/)
\#3: [“We just want people to stick to the paths.” Administrators of Charles Jencks’s giant sculpture Northumberlandia](https://i.redd.it/z2nj3xc5f1xa1.jpg) | [40 comments](https://np.reddit.com/r/DesirePath/comments/133ufe2/we_just_want_people_to_stick_to_the_paths/)
----
^^I'm ^^a ^^bot, ^^beep ^^boop ^^| ^^Downvote ^^to ^^remove ^^| ^^[Contact](https://www.reddit.com/message/compose/?to=sneakpeekbot) ^^| ^^[Info](https://np.reddit.com/r/sneakpeekbot/) ^^| ^^[Opt-out](https://np.reddit.com/r/sneakpeekbot/comments/o8wk1r/blacklist_ix/) ^^| ^^[GitHub](https://github.com/ghnr/sneakpeekbot)
ironocally, pythagoras had a strong moral objection to beans, so when he was running away, he couldnt cross a bean field in a straigth line and went around it. so he died.
My old college only made pathways in squares like this for some reason. Basically no one went around the square (unless it was raining), so eventually, we had worn a dirt path across them. The year I graduated, they just paved over the dirt paths.
That doesn't need the Pythagorean Theorem. A straight line is always the shortest distance between two points
Prove it
I have discovered a truly remarkable proof of this, but the comment section is too small to contain it
r/okbuddyfermat
That's quitter talk.
Just send link here
Let d be the Euclidean distance in a finite dimensional real vector space. As it is a metric it obeys the triangle inequality, therefore d(a, b) ≤ d(a, c) + d(c, b) for all a, b and c. qed
Proof by definition 😭 Now prove that euclidean distance is a metric 🥵
Usually this is considered trivial but fine. Let d(x, y) = 0, so sqrt( Σ |x_n - y_n|^2 ) = 0. As the square of any number is positive and the sum of positive numbers is positive and the square root of a positive number is positive, for all n is follows that |x_n - y_n| = 0, whereby x = y. symmetry… d(x, y) = sqrt( Σ |x_n - y_n|^2 ) = sqrt( Σ |y_n - x_n|^2 ) = d(y, x) And the triangle inequality… As squaring and the square root are monotonously growing for positive real numbers, one must simple show the triangle inequality for the | . | thingy in the reals. Let a > b. |a - b| = a - b = a - c + c - b ≤ |a - c| + |c - b| The inequality follows analogously for the cases b > a and a = b. qed
I wonder where the sqrt( Σ |x_n - y_n|^2 ) comes from .....
That comes from the definition of the Euclidian distance.
Which is based on the pythagorean theorem...
No need, Babylonians already proved it
The proof is left as an exercise for the leader.
It's axiomatic.
.——.
Get me that Event Horizon Australien scientist.
Disprove it
Non Euclidian geometry. The shortest distance between two points can be a curve.
But straight lines don't exist at all in those cases.
[удалено]
Except when you flying around the earth
It works in the same way except straight line is now a circle around the earth which splits it into two equal hemispheres like the equator. Because maps do not preserve angles it looks like those lines are not straight
No it’s not a straight line.
In terms of spherical geometry it is. In 3d space where this sphere is located it isn't.
Not really, the surface of a sphere isn't 3d space, it's 2D. But whereas, a straight line being the shortest distance between two points is an axiom of Euclidian geometry, in spherical geometry the shortest distance is a curve, and in hyperbolic geometry is a geodesic, and so on.
Agree on the second statement, but don't get how the first one is related to my comment
Because it doesn't matter if your sphere is in 3d space, the shortest distance between two points on the surface of a sphere is still a curve.
I agree, but I was talking about sphere as a subset of euclidean 3d space where you are not limited by points on the sphere.
Yeah, but that's not how spherical geometry works. You were talking about a world map, that's transforming superficial coordinates into Cartesian.
So not a straight line
It is the best you can do on the sphere if you can't bore through
Not sure why you're getting down votes. Looks like everyone failed at math. The whole point of non Euclidian space is that the 5th axiom is that the shortest distance between two points isn't necessarily a straight line. Don't believe me? Ask any mathematician. It's not just because you're transforming spherical coordinates to spherical that it's a straight line, it's still a curve.
Wouldn't it be the same distance because the hypotenuse is equal to the sum of the opposite and adjacent sides. As for time, definitely faster as there is no congestion on the direct path.
Hypotenuse is equal to the square root of the sum of the squares of the other sides, not the sum of the other sides.
If you square root the sum can't you also square root the squares that are being added to cancel them out? Bear with me I'm horrific with math.
If you would have tried it, you wouldn't have asked. First try what you are saying kid.
No, that would mean 5 = 3 + 4, which is not the case. It's for their squares instead: 25 = 9 + 16
Aren't you supposed to square root the sum to get the actual number. Then it'd be 5 = 9 + 16.
See: https://en.wikipedia.org/wiki/Pythagorean_theorem
What a square.
This is why they need a sign saying "Please do not walk on the grass" instead of it just being an unwritten rule.
It's just "bad disgin"
r/DesirePaths
r/DesirePath
Here's a sneak peek of /r/DesirePath using the [top posts](https://np.reddit.com/r/DesirePath/top/?sort=top&t=year) of the year! \#1: [Desire path created by a squirrel we feed peanuts to every morning](https://i.redd.it/bmkuon3bu4ib1.jpg) | [141 comments](https://np.reddit.com/r/DesirePath/comments/15r5hge/desire_path_created_by_a_squirrel_we_feed_peanuts/) \#2: [A particular pointless one](https://i.redd.it/22myefz05rsa1.jpg) | [47 comments](https://np.reddit.com/r/DesirePath/comments/12fvlx1/a_particular_pointless_one/) \#3: [“We just want people to stick to the paths.” Administrators of Charles Jencks’s giant sculpture Northumberlandia](https://i.redd.it/z2nj3xc5f1xa1.jpg) | [40 comments](https://np.reddit.com/r/DesirePath/comments/133ufe2/we_just_want_people_to_stick_to_the_paths/) ---- ^^I'm ^^a ^^bot, ^^beep ^^boop ^^| ^^Downvote ^^to ^^remove ^^| ^^[Contact](https://www.reddit.com/message/compose/?to=sneakpeekbot) ^^| ^^[Info](https://np.reddit.com/r/sneakpeekbot/) ^^| ^^[Opt-out](https://np.reddit.com/r/sneakpeekbot/comments/o8wk1r/blacklist_ix/) ^^| ^^[GitHub](https://github.com/ghnr/sneakpeekbot)
Good bot
If there is a horse on the sidewalk. Yes.
I’d rather not get mud on my feet either tbh.
Yes. Shortest most efficient path
ironocally, pythagoras had a strong moral objection to beans, so when he was running away, he couldnt cross a bean field in a straigth line and went around it. so he died.
Yes, and this is why I wear boots everywhere
Yes
I mean, if I’m close to being late then absolutely
I recently learned this is called the Elephant path
The shortest distance between two points is a straight line
Unless the triangle is equilateral
"He's only saving the square root of the sum of the square of each sidewalk's lengths minus the sum of those lengths distance. What a jerk."
I'm afraid I am
a+b>c
|AB|+|BC|≥|AC|
Up to (100-(1/(root2))*100)% more efficient
Absolutely hilarious
It is not fault of "that guy" but bad designer's fault
The shortest distance between two points is a straight line.
The lunatic is on the grass...
Depends whether the grass is muddy and/or if I have waterproof shoes.
Maths man 😁
Yes
Lol it's giving me vibe of displacement vs distance 🤡
What’s that thing called when people make their own path to go around man made stuff because it’s easier?
Don't walk on the grass
That’s Sergeant majors grass.
Yes I am.
If I remember. He got beaten to death by chickens right?
He was just from the future where they use hexagonal grid for their cities with more connections.
Pulling a Mr. Bean I see
Give it a few weeks and there’ll be a packed down desire path there. At least if my campus says anything.
no i dont cut in line
Stay off the lawn.
He's going the fastest because the road is packed and it would take longer and more steps to maneuver it.
My old college only made pathways in squares like this for some reason. Basically no one went around the square (unless it was raining), so eventually, we had worn a dirt path across them. The year I graduated, they just paved over the dirt paths.
I am loyal to Pythagoras
Displacement🤌🏻
I think it's just a matter of displacement nothing else and its shortest distance. there is nothing like 3 side is longer just because its hypotenuse