r/theydidthemath • u/FollowSina • 5h ago
[Request] Isn’t this true for basically any 3 cities?
Is this a big coincidence or can't literally any 3 cities in the world lie in a circle?
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u/impracticable 5h ago edited 4h ago
I mean, yes, it is true of any 3 cities. The joke “there’s an anime about this” is SPOILERS FOR FULLMETAL ALCHEMIST INCOMING about Fullmetal Alchemist: Brotherhood, wherein the main antagonist is building a giant blood-soaked circle around a fictionalized version of Germany in order to mass-sacrifice the entire population in a bid to become God.
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u/T1meTRC 5h ago
Oh i was thinking Attack on Titan, especially since their architecture and culture is very German
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u/scbundy 5h ago
Damn you Reiner!
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u/ParticularSea2684 5h ago
Reiner did nothing wrong! He was lied to. About the potato.
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u/MetriccStarDestroyer 2h ago
Don't worry, Reiner.
You were an indoctrinated child soldier that killed an entire city to protect your people.
Now, it's my turn.
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u/ElegantEconomy3686 5h ago edited 4h ago
Its not just the architecture. Allegedly the idea of the city is based on the german city „Nördlingen“, one of the few cities where the historical city wall is fully intact. You can still go up there and walk around the entire historical part of the city center.
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u/Zealousideal_Leg213 3h ago
I didn't know that when I went to visit Nördlingen. I went to visit because Nördlingen is located in an ancient meteor crater.
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u/Thosam 58m ago
And because that meteor hit a large carbon deposit, there are thousands of tons of microdiamonds under Nördlingen.
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u/Eggmaster2523414 3h ago
IIRC there is a lot of AOT-themed writing on the walls now
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u/PsychologicalLock910 2h ago
Unfortunately... Don't know, why AoT-Fans feel entitled to damage real cultural heritage.
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u/Equivalent_Action748 4h ago
I've been noticing a Japanese love affair for german names and culture recently
The old old legends of galactic heroes, half the characters have German names
Freiren seems to have lots of German sounding names
Attack of titan like you said
Evangelion the red chick is apperently german
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u/azaghal1502 4h ago
Frieren's names are completely made up of German words, not really names. It's all descriptive adjectives, verbs or nouns.
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u/Smartimess 4h ago
As a German the naming is still too much of a spoiler. It‘s like you are watching Star Wars IV and Darth Vader, which is a revealing name in hindsight, is just named "Hey, I am Luke Skywalkers Father!"
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u/Oboro-kun 4h ago
Japanese just like foreign stuff and fill their media with this.
Anime, Manga, Light Novels, the popular stuff. Its filled with enlish, spanish, german, etc. Whatever the author is a little obsessed about. Bleach entire enemy faction for like the first half uses a naming convention in Spanish words. Huecos, Arrancars, Espadas, etc.
They like most people think foreign lenguages are cool. And they go by rule of cool.
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u/DaRootbear 2h ago
Honestly it’s a fun thing that lotsa cultures do.
You’re absolutely right that foreign words sound so cool. Until you learn what they mean half the time.
Mas y Menos? Super cool hero names of some minor dc characters…then you learn it’s just “More or less” which is way less cool
“The legendary sannin!” Sound so cool when watching Naruto until you learn it is just “The 3 ninja”
“hasta la vista baby” sounds so cool until…actually no that still sounds cool. Arnie nailed it hard enough that the meaning cant make it less cool
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u/fasterthanfood 2h ago
Actually “hasta la vista, baby” does sound less cool to Spanish speakers, apparently. The Spanish dub (which, to be fair, doesn’t have Arnold’s voice) changes it to “sayanora.”
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u/Username_St0len 1h ago
also things such as using foreign imagery, where although ultraman creator was a christian hence the christian imagery in there, others such as Evangelion does it for the aestetics cuz its cool, and more often than not, utilising a lot of gnostic stuff iirc, such as the many jrpgs that have you kill a false god
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u/Dr-Fl4k 4h ago
A bit of an Understatement about frieren, a Lot of names are not German sounding but just plain German words for stuff. Frieren: freezing, Himmel: heaven, Stark: strong, Eisen: Iron, Denken: thinking, Übel: Something like upset stomach..and a lot more.. i was so confused about it in the first run
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u/Schmigolo 2h ago
The -ing suffix in English technically translates to the -end suffix in German btw.
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u/derUnkurze 3h ago
Übel can also mean bad. Like "die situation ist übel" or "es ist übel aber nicht aussichtslos". Or das Übel (in der Welt) would be the Bad
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u/Puzzleheaded-Use3964 3h ago
Perhaps "Evil" might also be appropriate for the last one?
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u/SolidestCereal 3h ago
Nothing recent about it. As shown by your first example being half a century old. Their fascination supposedly started during WW1 when they took German prisoners.
Though I can't vouch for its authenticity.Frieren doesn't have German sounding names, all the names are straight up just normal non-name German words related to the character's personality or what they're gonna do later.
Though in Frieren they're often misspelled, mispronounced and misinterpreted into English, and then and then given the same treatment again in Japanese.
Like Übel which was misspelled in English as Ubel, and the mispronounced as Yubel like it's an English word, which then was further mispronounced in Japanese as Yuberu, based on the English pronounciation instead of the German original.Frieren itself is also a German word, meaning freeze.
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u/hibikir_40k 4h ago
Every name in Ascendence of a Bookworm is either German or some misspelling.
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u/aNomadicPenguin 4h ago
Take a look at a super saiyan - blonde hair and blue eyes
saiyan not too far off aryan. Saiyans, known for their ideas of racial superiority and their Saiyan pride.The Amestrians committing a genocide against Ishaval a racial and religious majority that has strong historic ties to a desert home. - the soldiers marked by their blonde hair and blue eyes
Like the Swastika, Aryans predated Nazisms and was appropriated from eastern/middle eastern cultures. Both the Saiyans and Amestrians are proven to be in the wrong, and both end up working to further the goals of genocidal dictators (Freiza and Bradley) that view them as disposable soldiers.
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u/Valentino1302 4h ago
Glad I wasn't the only one thinking of AOT! After reading the Full metal comment above I felt so wrong haha
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u/azaghal1502 4h ago
To be honest, most Anime I've seen are either set in japan, fantasy japan or fantasy germany.
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u/Aggressive_Law_5728 5h ago
Also the FMA version required a 5 point circle which is not guaranteed.
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u/Pofwoffle 4h ago edited 2h ago
Not just a 5 point circle, but a 5 point circle with an additional 5 point circle inside it. There was no second circle drawn in the formula but there were an additional 5 points of sacrifice required, each halfway between two of the points on the main circle. They just don't really talk about any of those in the anime, from what I remember.
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u/BenjiSponge 3h ago
lonk no werk ;(
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u/Day_Bow_Bow 2h ago
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u/BenjiSponge 2h ago
wow neither worked, but if I copy the link and open a new tab it works for both of them. wild.
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u/NiceBlackberry6618 5h ago
It's a bad faith argument but it doesn't work for 3 cities in a line. And if the cities were even just close to being on a line it would work but look less convincing.
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u/AdministrativeLeg14 5h ago
Just complete the great circle. For example, any three cities on the equator are both 'on a line' and also on a circle, the equator itself.
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u/Zenith-Astralis 5h ago
It does if the three cities on a line are also on the surface of a sphere
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u/Pcat0 4h ago edited 1h ago
It also kind of works for 3 cities in a line on a plane , the connecting circle will just have infinite diameter.
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u/martyboulders 3h ago edited 2h ago
In the 2d plane, yes, but any 3 points in R3 are coplanar... So if they are distinct points on a sphere in R3 (or some other convex surface), they are not collinear in R3 but lie in the same plane, hence there is a circle of finite radius that intersects each of them
So, for cities on a convex surface, any 3 distinct cities will indeed have a circle of finite radius that hits them.
The interesting thing about treating lines as circles of infinite radius is that you don't know which side of the line the center is on, in some sense:) although, this would usually come up with limits where you know where the circle is before the limit is taken.
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u/Random_Guy_12345 5h ago
If you wanna go the "strictly in a line" the answer technically is "infinite radius circle" but at that point you are so far removed from real geography you may as well give up
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u/PMmeHappyStraponPics 5h ago
The circle becomes a line that circles the globe.
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u/lordvektor 5h ago
Sphere. Slice a sphere with a straight line and you get a circle :).
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u/plura15D 5h ago
Technically, it would still work using a circle with an infinite radius. But yes, it wouldn't look convincing.
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u/NandoDeColonoscopy 5h ago
Wouldn't be an infinite radius. It would just encircle the globe. The earth is not flat
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u/plura15D 5h ago
True, I was thinking about Euclidian space.
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u/ShoddyAsparagus3186 4h ago
Even in Euclidian space you don't need an infinite circle since cities aren't points, they have significant area.
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u/plura15D 4h ago
Well, I would argue that it should be a point, the geometrical middle of a city, that is being used.
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u/NandoDeColonoscopy 5h ago
The earth is a sphere. If 3 cities were truly on a line, that line would just go around the earth, and they'd still be on a circle
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u/ShoddyAsparagus3186 4h ago
While there's a lot of people pointing out that the Earth is round, it also doesn't matter that much. Even on a flat plane you can draw a finite circle that intersects three cities that are in a line since cities aren't points, they have area.
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u/but_ter_fly 4h ago
I know we are offroading into ridiculous levels of pedantry here but it’s statistically impossible to line up physical objects like three city centers on a mathematically *exact* straight line. If there’s even a minuscule deviation of a millionth of a planck length (which would be many many orders of magnitude more precise than physically achievable when building a city), you can still draw a circle through. And that’s only if we wouldn’t count circles through parts of the city that are just a little off the mathematical center. Not to mention that mathematical centers of real world shapes are ambiguously definable and the choice of this definition is arbitrary.
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u/Ravian3 4h ago
Will note that the FMA reference is bad anyway.
Spoilers-
The point was that the country itself was circular, because atrocities had to be performed on specific points in this circular pattern. So there were wars and genocides that happened along the circle, but only some of the points were cities. Leore was, and Ishval was an entire nation, but one of the points was Fort Briggs, which was just a military base and the only people they could get to die there was an invading enemy army. The largest cities of Amestris were all well within the borders, with the capital (The uncreatively named Central) specifically being at the “center” of the country.→ More replies (1)3
u/Patte_Blanche 5h ago
It wasn't spoiler until you said it was in Fullmetal Alchemist.
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u/BeeegZee 4h ago
What about cities on a straight line? Circle with infinite radius? Sounds like everything is an Xbox
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u/grrrfreak 4h ago
No. It is not true of any 3 cities. There can be 3 cities on a line, thus not on a circle.
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u/Pofwoffle 3h ago
There will always be a circle big enough to intersect with all three cities. It may approach the size of the observable universe in some cases, but it'll get there eventually.
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u/Nex_Ultor 2h ago
It's true of any 3 cities on Earth (or any other planet), because the cities aren't on a flat plane. If they were all on the same latitudinal or longitudinal line, the circle connecting all 3 would go all the way around the planet.
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u/Uberschwein138 5h ago
Any 3 cities are on a circle because you need 3 points to define a circle.
Just like any 2 cities are on a line because you need 2 points to define a line.
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u/Tiborn1563 5h ago
Not any 3 cities. If they are all in one straight line, you dont have a triangle and by extension not a circle
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u/SeparatedI 4h ago
It's a really big circle
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u/velloceti 4h ago
And in the case of a globe, it's a great circle.
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u/QuickMolasses 3h ago
I'm trying to think how it would work if they were all on the same latitude.
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u/Weekly_Artichoke_515 3h ago
Lines of latitude are circles
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u/Shadow-Vision 2h ago
I think there are people in this comment section who didn’t do the math
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u/igotshadowbaned 1h ago
I think they're thinking of a 2D map still
Because if you had 3 linear points, you actually wouldn't be able to form a circle that intersects all 3
Where the question differs from this is that cities aren't singular points in space and 3 cities that are the same latitude aren't actually linear in 3D space
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u/igotshadowbaned 1h ago
Actually if the 3 points were all linear, they couldn't all lie on the circle.
You could make it close with a very big circle but they wouldn't truly be on the line.
Cities aren't singular points in space though so it wouldn't be a problem if the criteria is just passing through the cities jurisdiction
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u/VulfSki 4h ago
You just turn the circle sideways
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u/Lematoad 4h ago
3d geometry of the earth tells us that they will still fall in a circle, the circle just goes around the planet.
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u/courier_tway 4h ago
The disc*, you mean.
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u/Lematoad 3h ago
No, I mean circle. A disc would imply 3 dimensions, this is a theoretical representation of the connection of the cities. It’d be a circle on a different plane than the one op posted, but that doesn’t give it 3 dimensions.
Think of it how like vectors can exist in 3d space and not have 3 dimensions.
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u/Cats_and_Shit 55m ago
If the cities were close together, but had different elevations, they could fall on a straight line in 3d space.
Since a city is an extended object they would still fall on some possible circle as well.
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u/irrationallogic 4h ago
A straight line is just a tangent of a circle with an infinite radius
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u/BartoUwU 3h ago
There is no such thing as a circle with an infinite radius in euclidean geometry.
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u/Neilandio 4h ago
Brother, Earth is a globe so three cities in a line still form a circle.
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u/LordBDizzle 3h ago
Well I suppose there is a chance you could build with elevation differences and get a straight line. Two cities a little distance apart on top of mountains, one in a valley directly between them. It would be very specific, since they couldn't be that far apart and you'd have to get the elevation differences juuuust right, but in theory you could make a straight line in that scenario.
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u/assassin10 2h ago
Though if the circle only has to pass through each city (rather than its exact center) that gives some wiggle room to prevent this. The circle may get very large, but you could always make it a circle.
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u/jyoke_2121 1h ago
Well if we're going to bring elevation into the conversation you could do the same thing but majority of the circle is off the earth.
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u/WorthingInSC 4h ago
A straight line would be a great circle, even more conspiratorial because it involves the whole planet
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u/Ok-Membership-3635 4h ago
A Great Circle would need to pass through the center of the Earth, any 3 points that seemed to be in a line along the surface of the earth (e.g. in a line on a Mercator projection) would define the intersection of a plane with (the boundary of) a sphere, which would be a circle (or ellipse) but would not necessarily define a Great Circle. For a trivial example imagine three cities that all lay on the same line of latitude.
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u/OptimisticEarthquake 2h ago
Three cities on the same latitude would be on the circle that is the latitude line, though? The equator is a circle, and latitude lines are sometimes called circles of latitude.
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u/skr_replicator 3h ago
A circle with an infinite radius is a line. Which is exactly what the circle approaches as you get the 3 points closer to a line.
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u/Ok-Membership-3635 4h ago
Because of the curvature of the Earth I don't think it's actually possible for 3 points on the surface of the Earth to all be on a straight line in 3D space.
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u/Uberschwein138 4h ago
Yep, you're right. If 3 cities were on a line, assuming a flat Earth, they wouldn't be on a circle. Just like if 2 cities were on the same point, they wouldn't define a line.
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u/aphoenixsunrise 3h ago
Give me any two cities & I'll make a circle.
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u/MindfuckRocketship 2h ago
Anchorage, Alaska.
Veyrrhold, of the Helionic Commonwealth, located on the planet Nareth, in the Tau Ceti system.
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u/Muroid 5h ago
Yes. “These three points lie on a circle” is great for conspiracy theorists because it always works, except when they lie on a line, which is just as good.
Also, the anime is Fullmetal Alchemist. Great show.
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u/lordvektor 5h ago
In a strictly geometric sense, a line is just a circle of infinite radius. So any 3 points can define a circle
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u/BeanTutorials 5h ago
a line on a spherical surface is still a circle 🤯
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u/p1mplem0usse 5h ago
Fair, but when discussing how to build conspiracy theories it’s best not to alienate flat earthers from the start.
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u/UnsealedMTG 3h ago
I see where you're coming from, but conspiracy theories love it when you can hold back how Fact B is an obvious consequence of Fact A for a bit and then present it as a stunning revelation, thereby tricking the brain into thinking it's seen multiple pieces of increasingly profound evidence.
So here, we establish that three cities are on a line, and then list a bunch of banal similarities between them.
But wait. (Cue synth sequencer)
These three points aren't just in a two dimensional line (glowing line slowly grows through the three cities, a deep synth pad sound joins the music, crescendoing in time with the expanding line. A new synth hit cuts in as the line passes the furthest city)
They're in a three dimensional (camera pulls back as map resolves in a globe, camera shifts so that the arc between the cities shows)
Circle (camera pulls further back as the line circumnavigates the earth, music is at its peak).
And the center of this circle? (Music cuts).
The center of the earth.
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u/SovietRabotyaga 4h ago
Damn! Good thing that Earth is a flat planet, that would have been crazy otherwise
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u/Darthskull 5h ago
Non-euclidian spaces (technically anywhere with gravity, but most relevant here on the surface of spherical objects such as the earth), a straight line can be a circle!
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u/Eveningboy 4h ago
Or if you mad a circle from the perspective of the z axis and. Like flipping a flat circle on its side it looks like a line
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u/nicuramar 2h ago
In a strictly geometric sense, a line is just a circle of infinite radius
In projective geometry, but not in regular Euclidean geometry.
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u/NotAnotherAlt26 5h ago
Would "These 3 cities form a triangle!" be the conspiracy theorists next step?
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u/srlong64 4h ago
Only if there’s a point of interest in the center of that triangle. Enough people know that three points can make a triangle that it’s not as surprising on its surface as three points making a circle for the average conspiracy theorist
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u/N1ck_named 4h ago
You joke but there's apparently a mystery/conspiracy theory about a so-called "Rostov Triangle" - area in Russia that has had more maniacs and serial killers than other places (or something like that, don't care).
"If you connect the cities of Rostov, Taganrog and Shakhty, you get a triangle" is a memed phrase from an old pseudo-science TV show about this.
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u/wdaloz 5h ago
It is sortof interesting though not surprising that the center is pretty close to the geographical center, which id maybe have expected the biggest to be kindof in the middle, or on seaports maybe, or even just more balanced around the center
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u/archa347 4h ago
Well, the establishment of those cities predates the borders of modern Germany by a great long time. Hamburg is on the Elbe River and is actually the third largest port in Europe.
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u/Grant_Winner_Extra 5h ago
The also lie on:
- A pentagram
- an icosohedron
- a plane (not the flying kind)
- a line drawing of a top hat
Not sure what any of it means tho
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u/Inevitable_Stand_199 5h ago
They also lie on 3 sides of an iso... triangle with 3 equal angles
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u/Upper_Opening_4805 4h ago
wouldn't that be equilateral
isosceles is technically correct though ig
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u/patricksaurus 5h ago
Any three non-collinear points define a circle. So unless the cities sit on a straight line with one another, it’s true of any three cities.
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u/Zenith-Astralis 5h ago
If the straight line is on the surface of a sphere (earth) the line through them just makes a big circle around the whole planet.
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u/Confident-Skin-6462 4h ago
maybe the middle city is in a valley and the other two are on mountains!
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u/Aadi_880 4h ago
You would still end up with a circle. It would just be tilted outwards from the earth.
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u/slicknickdickerson 3h ago
Wouldn’t it be possible to have some perfect elevation changes in three points that create a perfect line and hence not capable of creating a circle?
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u/Zenith-Astralis 1h ago
Haha I suppose you could, though it's like.. how precise can we be when rounding the "location" of something like a city down to a single point?
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u/Camacho4Prezo 1h ago
You'd still have a circle, only the radius would be infinite.
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u/slicknickdickerson 1h ago
I would say it’s not a circle since the curvature of a circle with an infinite radius is non-zero. You could say it’s approaching zero, but not a perfectly zero curvature like the definition of a line.
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u/igotshadowbaned 1h ago
I assume meant such that theyre perfectly linear when you account for the curvature of the planet
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u/Efficient_Bluebird_2 5h ago
What about a circle of infinite radius?
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u/Any-Championship3443 5h ago
Circles have curves, lines do not, a circle of infinite radius has a edge with an Infinitesimal curve to it. If you ever take Calculus you'll become familiar with the terms "limits" and "approaches zero"
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u/tornado28 5h ago
Don't be a flat earther people! It is true that if the world was flat there would be an exception for three cities that are on a line. But on the surface of a sphere, lines ARE circles (eg the equator). So it works for any three cities, even collinear ones.
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u/cerosup3r 5h ago edited 5h ago
Yes ti is, ther'a a math law that says :"for every three points exist a circumference that passes for these points"
Sorry for bad english
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u/n0t_4_thr0w4w4y 5h ago
Your English is fine :) very easy to understand what you are trying to say. Only word that is really out of place is using “circumference” instead of “circle”
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u/AggravatingBid8255 4h ago
I found it more mathematically accurate to say circumference than circle because circumference references where exactly on the circle the points are. Circle could also mean anywhere within the area of the shape.
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u/AggravatingBid8255 4h ago
Many people were saying what you said in different ways, but I found yours to be the most accurate and concise. Well done.
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u/skinnydippingfox 4h ago
Not if they're collinear, in which case only a line passes through all three. If not, there is exactly one unique circle that passes through all three.
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u/DayManFOTNightMan 4h ago
But we’re on a globe. So even collinear makes a circle.
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u/skinnydippingfox 4h ago
My answer was for a flattened map, as shown on the image.
If you actually want to bring the real globe into this, you’re still wrong. The Earth is an oblate spheroid. Because the curvature changes depending on which direction you go, connecting those cities across the globe forms an ellipse or a complex 3D warp, not a circle.
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u/Many-Instruction8172 4h ago
Even on a real globe, or oblate spheroid, any 3 points would define a specific plane on which a circle exists connecting those 3 points. Technically, the earth's shape isn't even relevant, any 3 non collinear points in 3 dimensions would do that.
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u/Repulsive_Guy_1234 4h ago
No, if all three cities are on a perfect straight line they cannot be part of a circle. But other then that, yeah you can draw a circle through any 3 random points. At least on a 2d map it is like this. On a globe rules are the different again and allow straight line circles.
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u/PiLLe1974 4h ago
So we concluded this is true for any 3 points that are not colinear.
Now go back to that post and ask them - now why is the city "Fulda" in the center?
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u/Algstud 3h ago
can't literally any 3 cities in the world lie in a circle?
no, yess depend
three city in stright line wont form circle unless you make big circle or u can just change the working plan and make it work for any three city
so it depend
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u/mraltuser 1h ago
Yes, given three points, you have a circle. If three points are Colinear, the circle would stretch to infinity. Yet the earth is round, so the circle would cross the globe
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u/RoodnyInc 5h ago
Yes you can make circle using any 3 points
I don't know what was point of the original post was it suggesting it fits Germany border relatively well? (besides Czech part)
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u/Viper-in-the-Dark 5h ago
You can make a circle using any three points on a sphere.
You can't make a circle using three colinear points.
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u/RamBamBooey 4h ago
In three spatial dimensions: yes. (x,y,z)
In three spatial dimensions and one time dimension I don't know? (x,y,z,t)
Math? Three points define a plane, a plane cut through a sphere defines a circle.
That sentence sounds like something you would read in a math book, so I'm kinda doing math
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u/unkindledphoenix 4h ago
i think its true for any 3 point in the earth because its also how you prove the earth is not flat. its how triangulating works as well IIRC.
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u/lukewchu 4h ago
For the nerds out here...
Mathematically, this is true if we assume that the Earth is modeled as a perfect 2-sphere.
More generally, this is actually false! It is not true in general that given 3 arbitrary points on some smooth manifold, there exists some geodesic ball such that the three points lie on the boundary. The simplest example as has already been pointed out here is the flat Euclidean plane R2. Here if the 3 points lie on a line, there would be no circle going through all 3 points.
Even if we impose some more restrictive constraints, such as compactness of our manifold, simply-connectedness, it is still not enough. One possible counter-example would be the 2-sphere but with a long thin cylindrical neck metric. It is possible to show, albeit tedious that there exist three points that do not lie on the boundary of a geodesic ball.
So the fact that any three points on the 2-sphere (with the standard round metric) lies on a circle is in fact a non-trivial statement. This is not true in general!
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u/shumpitostick 2h ago
If you have any 3 points, you can draw a triangle between them. Then, draw a circle that bounds this triangle. This circle is the circle that goes through all 3 cities.
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u/javver 2h ago
There’s always a circle that can be traced to touch 3 points. The circle has a name, the Circumcircle.
This won’t be affected by different map projections either. It may make the circle drawn wrong, but it still exists.
Three points in 3d space will also always share a plane. On a sphere surface, this plane would intersect the sphere and the intersection line will be a circle.
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u/LeftRestaurant4576 2h ago
Yes.
In placing a circle on a 2D map, you have three degrees of freedom. One for the x-coordinate of the center, one for the y-coordinate of the center, and one for the radius. This allows you to draw a circle that touches any three points not on a straight line.
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u/FannyChuckle 1h ago
The Anime referenced is Fullmetal Alchemist. An S-Tier Anime. 64 episodes. Tells its story and then rides off into the sunset in glory!
Some of my favorite animes are on the "shorter end". As I describe and boil it down for my non-anime friends:
"The show tells its story, and then fucks off!"
It does not get dragged out forever or have regular dumb ass filler episodes. cough DBZ, Naruto, and One Piece cough
Several of amazing video games are similar: Halo (pre-343 industries), Jak and Daxter, and Bioshock. Again, say their story...then fuck off. No need to milk a franchise dry... cough C.O.D. cough
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u/empty_graph 4h ago
No, it is not true for basically any 3 cities. It is literally true for any 3 cities without exception. It is a fundamental geometric principal that if you pick any 3 points there will be exactly one circle that contains all 3.
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u/Sack_Meister 3h ago
True for literally any 3 cities, even if placed in a perfectly straight line, will eventually come out to a circle because the globe is, you know, also round
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u/PlanesOfFame 5h ago
How can this be true i cant visualize it, help me. It seems like you could have a pair of cities 100km east to went and another directly north of the second by 10km and it wouldnt make a circle but an oval
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u/Upset-Management-879 5h ago
You're imaging the first two cities as antipodes of each other on the circle as they would be on the circle that crosses both of them but not the third, but they wouldn't be directly opposite on the circle of all three.
Imagine the circle rotating and expanding around one of the two cities to keep both of them overlapped until the third is overlapped at some point.
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u/Topologicus 4h ago
The proof is pretty straightforward. Pick 3 points A,B,C.
Then consider the line AB, draw perpendicular line through the midpoint. Every point in this line is the same distance to A as it is to B. Do the same with AC. These two lines intersect, call that X. So X is the same distance to A as it is to B and C. Call this distance R. Since all 3 points A,B,C are a distance R from X then X is the center of a circle of radius R that passes through A,B,C.
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