>In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius 1, the area equals the triangle excess
To all the flat earthers out there, this property can be used to find out earth is not flat, just by drawing a giant triangle on the surface, without leaving the earth. Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. But this approach proves it without the need to refer the sun.
Once you internalize that flat-Earther-ism isn’t about the Earth being flat you realize that rational arguments are pointless.
To expand on that, it’s about community and finding people who share your interests. The movie Behind The Curve explores this idea and it’s quite revealing.
And the ego boost of it all - being one of the special few who sees "the truth" that others are too brainwashed/dumb/whatever to see. Makes one feel quite important.
Those are the simple cows to be milked, but numerous 'gurus' in these communities are very well aware of the bullshit they propagate to the weak and gullible, but its just such an easy noncritical prey. You can always just go deeper in paranoia.
Makes me think that mr trump switched from being democrat to republican and pushed for magaesque folks who often love him to the death due to very similar principle - just spit out some populist crap that stirs core emotions - the worse the better, make them feel victim, find easy target to blame which can't defend themselves well (immigrants), add some conspiracy (of which he is actually part of as wall street billionaire).
Extreme left wouldn't swallow easily that ridiculous mix from nepotic billionaire who managed to bankrupt casinos and avoided military duty (on top of some proper hebephilia with his close friend mr E and who knows what else).
But what do I know, just an outside observer, but nobody around the world has umbrella thick enough that this crap doesn't eventually fall on them too.
I think Trump's just been running a simple popularity-seeking loop for a while. Do a thing; if his people like it, do it more; otherwise do it less.
I've heard that even Hitler was like this: that he didn't start out hating Jews, but repeatedly reacted to the fact that he got louder cheers whenever he blamed things on Jews. But I don't know how to verify if this is true.
What could be expected to be the "shared interests" of a community of people organized around supposedly believing something that they aren't actually about believing?
It's since being replaced by similar isms like climate change hoax-ism. Very similar way of arguing, dealing with contradicting evidence and seeing a conspiracy whenever a large body of scientists has a consensus.
Unfortunately, the climate change deniers in all their forms have made it much further by having support in politics and having a real impact on people's lives. In contrast to flat earthers.
Just the mere fact that my post here could be interpreted as political (which it really isn't) is evidence of this.
It's more about discrediting conspiracy theories to shift the Overton window so the real ones with the flavor of 'the government is spying on you' also seems crazy to most people.
>Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities.
That wasn't to prove the Earth is round (and it doesn't prove it). Eratosthenes assumed two things when he performed his experiment: 1) the Earth is round, and 2) the Sun is an infinite distance away. By just this experiment he would have been unable to distinguish between this situation and the Earth being flat while the Sun being only a finite distance overhead (and in fact a fair bit closer than it actually is). Eratosthenes and his contemporaries were already convinced of the roundness of the planet, and he simply wanted to measure it.
>But this approach proves it without the need to refer the sun.
A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. If the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? I.e. continuously moving towards the South because the polar coordinates curve towards your left as you progress.
>the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there?
This is something that was more or less solved a long time ago with surveying instruments. You don't have to move in a straight line, you build triangles out of sight lines.
I can kinda see how that would work, but it presents the challenge that whatever route you plan, it cannot go over water for more than a few kilometers.
I don't think it would be that different than the arc measurements that were actually done, you triangulate a bunch of points to work out distances and angle sufficiently precisely:
And measurements of, say, very precise equilateral triangles, necessarily imply certain interior angles, which you can compare to the actual angles they make. For instance, on a flat plane, you can fit six equilateral triangles sharing one point to make a hexagon. On a sphere if you make them big enough you'll find that they don't quite fit.
I'm not sure how that matters for this purpose, they used these surveys to measure the shape of the Earth (specifically, the circumference, and later the flattening- ie, plenty precise enough to measure the curvature more or less directly).
Is the idea that land is spherical while the oceans are flat? Do this survey on each continent, including Antarctica, again just constructing big triangles and measuring the deviation from flatness. Eventually your model of the flat earth looks like a bunch of pieces of an eggshell separated by flat water that just happens to assemble into an egg- or rather an ellipsoid- if rearranged, up to and including the expected flattening at the poles.
With sufficient elbow grease you could extend a survey from the far north of the Americas down to the south, measuring the curvature as you went, eventually finding that you must be almost upside-down compared to where you'd started.
> A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths.
Do flat-earther reject the existence of LASER, too?
> relied on the sun shining directly overhead on wells in different cities.
It was just one city actually. The critical piece is that the city's northern latitude was nearly identical to the Earth's angle of axial tilt. Which also means that this shadow phenomenon only occurs during the Summer Solstice.
This sounds more like a Matt Parker video idea - get a bunch of people, three theodolites to measure angles accurately, a good location and start measuring angles for line of sight and see how well this determines the earth's radius.
Rough estimate - with an excellent 0.5" angular resolution and 35km triangle this could work.
> But this approach proves it without the need to refer the sun.
Only if you're happy "proving" your argument to an audience that never had any doubts. You can't use this argument to prove the earth is not flat over the objections of your audience because you can never convincingly show that any given line is straight.
> Note also that the triangle has infinite perimeter but finite area.
How common is this property in geometry? I know that fractals like the Koch Snowflake also have infinite perimeter over finite area, but I don't know what else does.
Any function that infinitely slowly converges to a finite number will have this property. Discretely, think of 1/2 + 1/4 + 1/8 and so on. The sequence goes on forever but adds up to 1.
> I have no idea why you think the geometric series has anything to do with this -
IgorPartola is perfectly right to mention geometric series, you can easily use a geometric progression to construct a shape with infinite perimeter and finite area, e.g. by gluing together rectangles with height one and width decreasing in geometric progression. With a bit more thought you can also construct a smooth shape having this property.
> your glued together rectangles will have perimeter 2*(1+2)
No. You should think through that perimeter calculation one more time, preferably while drawing a picture.
Here's a hint: the perimeter of a rectangle is no less than its height; you can glue so that the perimeter of each rectangle contributes at least 1 to the perimeter of the union.
I think you're both right. But there are two ways to do what you said and you didn't specify which one.
First, a rectangle of height 1 and width 1/2. The perimeter is 1 * 2 + 1/2 * 2, two sides of height 1 and two sides of width 1/2.
You "glue" the second rectangle. As one may understand this, you glue them by putting them one beside the other standing up, i.e. you glue them along one of the heights. Sorry for the crude ascii art:
Now you have a single rectangle, height 1, and width 1/2 + 1/4. The perimeter is 1 * 2 + (1/2+1/4) * 2. The "added perimeter" in this step is just 1/4 * 2 = 1/2.
Go on doing that and for a rectangle of width 1/n, you only add 2 * 1/n to the perimeter. In the end you get a single rectangle with height 1 and width 2. The perimeter is 2 * 1 + 2 * 2.
---
Now, maybe, you may want to specify that you glue the rectangles along their widths, not their heights.
That way, the resulting shape when you add the second rectangle is not a rectangle but an irregular shape with 6 sides. Sorry for the crude ascii art again:
The added perimeter now is exactly 2 * 1 on each step. Now the final perimeter is infinite but the area is not.
But you didn't specify this option over the other one. And, honestly, if we talk about putting rectangles in a sequence, I think it's more common to think of the rectangles as standing up side by side with their heights together as in the first option. For the second option I would describe the rectangles as having a fixed width of 1 and decreasing heights.
Worth noting that the hyperbolic triangle in the article contains "points at infinity" which are not actually a part of the hyperbolic plane, so this is really an "improper triangle" as the article notes. One could construct a similar improper triangle in the Euclidean plane that consisted of two parallel lines meeting at infinity. Such a triangle would still have 180 degrees of internal angle but it's area and perimeter would be infinite.
However, by the fith axiom of euclid, the lines in your example cannot be parallel AND converge (not even in infinity). Thus, it's more an open rectangle.
Either they are overlapping which violates the definition of a triangle, or they don't and the parallel lines always maintain the same distance X to each other and consequently maintain distance X at infinity (let's say X=1, bc you can just scale it).
Girard's Theorem - Spherical Geometry - Deriving The Formula For The Area Of A Spherical Triangle: https://youtu.be/Y8VgvoEx7HY
T = r^2 (alpha + beta + gamma - pi)
The discussion is about triangles in hyperbolic space. In hyperbolic space, if you keep extending a triangle's lines out by moving the intersection farther away, you'll tend toward a triangle with a constant area (pi in the article because the curve was chosen for that, you can have any arbitrary finite value you want by varying the curvature) even though the perimeter keeps going up.
If that sounds like so much technobabble, that's because this article assumed what I think is a very specific level of knowledge about hyperbolic space, as it doesn't explain what it is, yet this is one of the very first things you'll ever learn about it. So it has a rather small target audience of people who know what hyperbolic space is but didn't know that fact about triangles. If you'd like to catch up with what hyperbolic space is, YouTube has a lot of good videos about it: https://www.youtube.com/results?search_query=hyperbolic+spac... And as is often the case with geometry, videos can be a legitimate benefit that is well taken advantage of and not just a "my attention span has been destroyed by TikTok" accomodation.
Including CodeParade's explanations, which are notable in that he made a video game (Hyperbolica) in which you can even walk around in it if you want, with an option for doing it in VR (though that is perhaps the weirdest VR experience I had... I didn't get motion sick per se, but my brain still objected in a very unique manner and I couldn't do it for very long). It's been out and on Steam for a while now, so you can run through the series where he is talking about the game he is in the process of creating at the time and go straight to trying it out, if you want.
> So it has a rather small target audience of people who know what hyperbolic space is but didn't know that fact about triangles.
Accidentally, I’m in that small set: I have a hand-wavy understanding of hyperbolic spaces (the high school I went to was named after Lobachevsky!), but I haven’t studied the geometry and didn’t know the formulae for area.
As far as I understand, the closer the points are to the line, the more distant they get to the rest of the plane. That's why he says that "this is an improper triangle", as the point of intersections of the hyperbolic lines are theoretically at an infinite distance from the "origin", and thus that the lines connecting those points have an infinite length.
It's a bit analogous to the way train tracks shrink toward the horizon and make an angle with each other where they appear to meet it, even though they don't actually meet in the plane. These hyperbolic lines won't actually ever meet in the hyperbolic plane either but they approach the same point on the horizon.
That edge is basically an artifact of the model, you can equally model the hyperbolic plane space as a disk and then the boundary is a circle, or on an actual hyperboloid in 3D and it extends out forever.
The disk model of hyberolic geometry is made to map hyperbolic 2 space (which is infinite in area) into the finite interior of the disk. In order to capture this, the normal euclidean notion of distance is distorted by a function which allows "distances" to go to infinity as a curve approaches the boundary of the disk.
Let's go to to the normal infinite plane for a moment.
You can use a map that is inside a circle with r=1. The objects get deformed, but points have a 1 to 1 correspondence. Lines that pass though 0 look straight, but other lines are curved.
Measuring a distance is hard, you have to use some weird rules.
If you draw a segment of length 0.001 segment in the circular map, it has almost the same length in the real infinite map.
If you draw a segment of length 0.001 segment near the border of the circular map, it's a huge thing in the infinite map.
Moreover, a line that pass thorough 0 has apparent length 2 in the map, but represent an infinite length in the plane
Note that the border of the circle is outside the plane.
---
The reverse happen if you have a map of the Earth. You can draw on the map with a pencil a long segment near the pole, but it represents a small curved segment in the Earth.
---
Back to your question ,,,
It's on the hyperbolic plane, not in the usual euclidean plane. So the map is only the top half, and the horizontal line = axis x is outside, it's the border.
Length are weird, and a 0.001 segment draw with a pencil on the map far away from the x axis is small in the actual hyperbolic plane, but a 0.001 segment draw with a pencil on the map near the x axis is very long in the actual hyperbolic plane.
The circles "touch" the x axis. In spite they look short when you draw them with a pencil, they part that is close to the x axis has a huge length in the hyperbolic plane.
It must be that the figure with the half circles is just a representation of the hyperbolic space into 2D. Such projections are not faithful; you cannot take measurements in the projection and take them literally.
We can make an analogy to cartography: you can't trust areas and distances on distorted projections like Mercator.
Look, even the angles don't look to be zero in that diagram. We have to imagine that we zoom in on an infinitesimal zone around each corner to see the almost zero angle; i.e. the circle tangent lines actually go almost parallel. So to speak.
Thus the angles are locally correct, since they are measurable on arbitrarily small scales and can easily be imagined to be even when glancing at the entire figure. But distances between the points aren't localizable; they have to follow a measure which somehow correctly spans the abstract hyperbolic space that they represent.
How about this (almost certainly incorrect) imagining: pretend that the real line shown, on which the three points lie, is actually a horizon line, which lies in a vast distance (out at infinity). Just like the horizon when you do drawings with two-point perspective. Imagine the three points are vanishing points on the horizon. Vanishing points are not actually points; they just directions into infinity.
if, in a two-point perspective, you draw a curve whose endpoints are tangent to two vanishing point traces, that curve is infinitely long.
For instance if you draw an intersection between two infinite roads, where the curb has a round corner, you will get some kind of smiley curve joining two vanishing points. That curve is understood to be infinitely long.
Hyperbolic geometry! Note how lines on the chart don't appear straight. That's because this is just a projection of an infinite hyperbolic space. The rules of this projection move points at infinity to the real line, and straight lines to circles. That means each of the points on the mentioned triangle is infinitely far away in some direction.
It's one of those things where it's technically correct but the headline is misleading. When you say "a triangle" without any qualification as the headline does, people are going to interpret that as a good old fashioned triangle. Using the term without clarification that you mean spherical geometry is kind of underhanded writing, imo.
I think it's just a normal ages-old pattern for writing headlines that pique people's curiosity. It's super common in popular math in particular, because math is always about generalizing. There's a fine line between that and actual clickbait meant to actively mislead.
>In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius 1, the area equals the triangle excess
To all the flat earthers out there, this property can be used to find out earth is not flat, just by drawing a giant triangle on the surface, without leaving the earth. Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. But this approach proves it without the need to refer the sun.
Once you internalize that flat-Earther-ism isn’t about the Earth being flat you realize that rational arguments are pointless.
To expand on that, it’s about community and finding people who share your interests. The movie Behind The Curve explores this idea and it’s quite revealing.
A bunch of flat-earthers went to Antarctica to see if the midnight sun was real. Turns out it was.
Jeran from Behind The Curve was one of the ones to flip, and since then, he's been making videos on how the earth is actually round.
He has a lot of thoughts on what it actually takes to convince other flat-earthers. I found it somewhat interesting: https://www.youtube.com/watch?v=1grMf17PeEk
And the ego boost of it all - being one of the special few who sees "the truth" that others are too brainwashed/dumb/whatever to see. Makes one feel quite important.
Indeed, this might be why religion seems so odd to outsiders.
It's implausible, yet that's what stimulates the tribal feelings among the believers.
Those are the simple cows to be milked, but numerous 'gurus' in these communities are very well aware of the bullshit they propagate to the weak and gullible, but its just such an easy noncritical prey. You can always just go deeper in paranoia.
Makes me think that mr trump switched from being democrat to republican and pushed for magaesque folks who often love him to the death due to very similar principle - just spit out some populist crap that stirs core emotions - the worse the better, make them feel victim, find easy target to blame which can't defend themselves well (immigrants), add some conspiracy (of which he is actually part of as wall street billionaire).
Extreme left wouldn't swallow easily that ridiculous mix from nepotic billionaire who managed to bankrupt casinos and avoided military duty (on top of some proper hebephilia with his close friend mr E and who knows what else).
But what do I know, just an outside observer, but nobody around the world has umbrella thick enough that this crap doesn't eventually fall on them too.
I think Trump's just been running a simple popularity-seeking loop for a while. Do a thing; if his people like it, do it more; otherwise do it less.
I've heard that even Hitler was like this: that he didn't start out hating Jews, but repeatedly reacted to the fact that he got louder cheers whenever he blamed things on Jews. But I don't know how to verify if this is true.
The feedback response post is true, but specifically about Jews is not true. He hated Jews long before he rose anywhere near power
Hitler was enthralled by Henry Ford, and copied what he learned about anti-semitism.
https://www.thehistoryreader.com/historical-figures/hitlers-...
What could be expected to be the "shared interests" of a community of people organized around supposedly believing something that they aren't actually about believing?
It's since being replaced by similar isms like climate change hoax-ism. Very similar way of arguing, dealing with contradicting evidence and seeing a conspiracy whenever a large body of scientists has a consensus.
Unfortunately, the climate change deniers in all their forms have made it much further by having support in politics and having a real impact on people's lives. In contrast to flat earthers.
Just the mere fact that my post here could be interpreted as political (which it really isn't) is evidence of this.
It's more about discrediting conspiracy theories to shift the Overton window so the real ones with the flavor of 'the government is spying on you' also seems crazy to most people.
>Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities.
That wasn't to prove the Earth is round (and it doesn't prove it). Eratosthenes assumed two things when he performed his experiment: 1) the Earth is round, and 2) the Sun is an infinite distance away. By just this experiment he would have been unable to distinguish between this situation and the Earth being flat while the Sun being only a finite distance overhead (and in fact a fair bit closer than it actually is). Eratosthenes and his contemporaries were already convinced of the roundness of the planet, and he simply wanted to measure it.
>But this approach proves it without the need to refer the sun.
A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. If the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? I.e. continuously moving towards the South because the polar coordinates curve towards your left as you progress.
>the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there?
This is something that was more or less solved a long time ago with surveying instruments. You don't have to move in a straight line, you build triangles out of sight lines.
I can kinda see how that would work, but it presents the challenge that whatever route you plan, it cannot go over water for more than a few kilometers.
I don't think it would be that different than the arc measurements that were actually done, you triangulate a bunch of points to work out distances and angle sufficiently precisely:
https://en.wikipedia.org/wiki/Arc_measurement
That doesn't help you if you're moving West to East, though.
EDIT: Also, that's to measure distance, not direction.
You can survey West to East just as well:
https://www.noaa.gov/digital-collections/collections/4774/it...
https://amerisurv.com/2012/10/27/the-longest-line/
And measurements of, say, very precise equilateral triangles, necessarily imply certain interior angles, which you can compare to the actual angles they make. For instance, on a flat plane, you can fit six equilateral triangles sharing one point to make a hexagon. On a sphere if you make them big enough you'll find that they don't quite fit.
>https://www.noaa.gov/digital-collections/collections/4774/it...
Like I said before, you can't do that over water. It constrains the shape of the triangle you can measure.
I'm not sure how that matters for this purpose, they used these surveys to measure the shape of the Earth (specifically, the circumference, and later the flattening- ie, plenty precise enough to measure the curvature more or less directly).
Is the idea that land is spherical while the oceans are flat? Do this survey on each continent, including Antarctica, again just constructing big triangles and measuring the deviation from flatness. Eventually your model of the flat earth looks like a bunch of pieces of an eggshell separated by flat water that just happens to assemble into an egg- or rather an ellipsoid- if rearranged, up to and including the expected flattening at the poles.
With sufficient elbow grease you could extend a survey from the far north of the Americas down to the south, measuring the curvature as you went, eventually finding that you must be almost upside-down compared to where you'd started.
> A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths.
Do flat-earther reject the existence of LASER, too?
Flat-earthers don't accept that a flat plane implies infinite line of sight (especially at sea), so who knows.
> relied on the sun shining directly overhead on wells in different cities.
It was just one city actually. The critical piece is that the city's northern latitude was nearly identical to the Earth's angle of axial tilt. Which also means that this shadow phenomenon only occurs during the Summer Solstice.
https://www.khanacademy.org/science/shs-physical-science/x04...
This sounds more like a Matt Parker video idea - get a bunch of people, three theodolites to measure angles accurately, a good location and start measuring angles for line of sight and see how well this determines the earth's radius.
Rough estimate - with an excellent 0.5" angular resolution and 35km triangle this could work.
As they say, you can't reason someone out of something they didn't reason themselves into in the first place.
> But this approach proves it without the need to refer the sun.
Only if you're happy "proving" your argument to an audience that never had any doubts. You can't use this argument to prove the earth is not flat over the objections of your audience because you can never convincingly show that any given line is straight.
It also means that pi could be equal to 3 if you world is small enough.
No, you're talking about a hologram. Everything is flat.
https://en.wikipedia.org/wiki/Holographic_principle
> Note also that the triangle has infinite perimeter but finite area.
How common is this property in geometry? I know that fractals like the Koch Snowflake also have infinite perimeter over finite area, but I don't know what else does.
Any function that infinitely slowly converges to a finite number will have this property. Discretely, think of 1/2 + 1/4 + 1/8 and so on. The sequence goes on forever but adds up to 1.
A continuous function with that property is f(x) := 2^-x (when summed over the non-negative part of the x-axis). Another example is g(x) := 1/x^2.
I have no idea why you think the geometric series has anything to do with this - this is related to continuous but nowhere differentiable functions: https://en.wikipedia.org/wiki/Weierstrass_function
> I have no idea why you think the geometric series has anything to do with this -
IgorPartola is perfectly right to mention geometric series, you can easily use a geometric progression to construct a shape with infinite perimeter and finite area, e.g. by gluing together rectangles with height one and width decreasing in geometric progression. With a bit more thought you can also construct a smooth shape having this property.
> together rectangles with height one and width decreasing in geometric progression
The geometric series sums to 2 - your glued together rectangles will have perimeter 2*(1+2) and area 2*1.
> your glued together rectangles will have perimeter 2*(1+2)
No. You should think through that perimeter calculation one more time, preferably while drawing a picture.
Here's a hint: the perimeter of a rectangle is no less than its height; you can glue so that the perimeter of each rectangle contributes at least 1 to the perimeter of the union.
I think you're both right. But there are two ways to do what you said and you didn't specify which one.
First, a rectangle of height 1 and width 1/2. The perimeter is 1 * 2 + 1/2 * 2, two sides of height 1 and two sides of width 1/2.
You "glue" the second rectangle. As one may understand this, you glue them by putting them one beside the other standing up, i.e. you glue them along one of the heights. Sorry for the crude ascii art:
Now you have a single rectangle, height 1, and width 1/2 + 1/4. The perimeter is 1 * 2 + (1/2+1/4) * 2. The "added perimeter" in this step is just 1/4 * 2 = 1/2.Go on doing that and for a rectangle of width 1/n, you only add 2 * 1/n to the perimeter. In the end you get a single rectangle with height 1 and width 2. The perimeter is 2 * 1 + 2 * 2.
---
Now, maybe, you may want to specify that you glue the rectangles along their widths, not their heights.
That way, the resulting shape when you add the second rectangle is not a rectangle but an irregular shape with 6 sides. Sorry for the crude ascii art again:
The added perimeter now is exactly 2 * 1 on each step. Now the final perimeter is infinite but the area is not.But you didn't specify this option over the other one. And, honestly, if we talk about putting rectangles in a sequence, I think it's more common to think of the rectangles as standing up side by side with their heights together as in the first option. For the second option I would describe the rectangles as having a fixed width of 1 and decreasing heights.
Gabriels Horn for another example.
Doesnt seem that uncommon.
Gabriel's horn is the same phenomenon one dimension up: finite surface area but infinite volume.
You mixed it up. The horn has infinite surface area but finite volume.
You're right. Thanks.
Spherical geometers: the trolls of the math world
Ah! I just realized that there is an infinity of different triangles passing through those three points: two poles and any other point. Wild!
Worth noting that the hyperbolic triangle in the article contains "points at infinity" which are not actually a part of the hyperbolic plane, so this is really an "improper triangle" as the article notes. One could construct a similar improper triangle in the Euclidean plane that consisted of two parallel lines meeting at infinity. Such a triangle would still have 180 degrees of internal angle but it's area and perimeter would be infinite.
However, by the fith axiom of euclid, the lines in your example cannot be parallel AND converge (not even in infinity). Thus, it's more an open rectangle.
Either they are overlapping which violates the definition of a triangle, or they don't and the parallel lines always maintain the same distance X to each other and consequently maintain distance X at infinity (let's say X=1, bc you can just scale it).
Triangle Man hates Person Man
Girard's Theorem - Spherical Geometry - Deriving The Formula For The Area Of A Spherical Triangle: https://youtu.be/Y8VgvoEx7HY T = r^2 (alpha + beta + gamma - pi)
> infinite perimeter
I don’t follow, how/why?
The discussion is about triangles in hyperbolic space. In hyperbolic space, if you keep extending a triangle's lines out by moving the intersection farther away, you'll tend toward a triangle with a constant area (pi in the article because the curve was chosen for that, you can have any arbitrary finite value you want by varying the curvature) even though the perimeter keeps going up.
If that sounds like so much technobabble, that's because this article assumed what I think is a very specific level of knowledge about hyperbolic space, as it doesn't explain what it is, yet this is one of the very first things you'll ever learn about it. So it has a rather small target audience of people who know what hyperbolic space is but didn't know that fact about triangles. If you'd like to catch up with what hyperbolic space is, YouTube has a lot of good videos about it: https://www.youtube.com/results?search_query=hyperbolic+spac... And as is often the case with geometry, videos can be a legitimate benefit that is well taken advantage of and not just a "my attention span has been destroyed by TikTok" accomodation.
Including CodeParade's explanations, which are notable in that he made a video game (Hyperbolica) in which you can even walk around in it if you want, with an option for doing it in VR (though that is perhaps the weirdest VR experience I had... I didn't get motion sick per se, but my brain still objected in a very unique manner and I couldn't do it for very long). It's been out and on Steam for a while now, so you can run through the series where he is talking about the game he is in the process of creating at the time and go straight to trying it out, if you want.
> So it has a rather small target audience of people who know what hyperbolic space is but didn't know that fact about triangles.
Accidentally, I’m in that small set: I have a hand-wavy understanding of hyperbolic spaces (the high school I went to was named after Lobachevsky!), but I haven’t studied the geometry and didn’t know the formulae for area.
As far as I understand, the closer the points are to the line, the more distant they get to the rest of the plane. That's why he says that "this is an improper triangle", as the point of intersections of the hyperbolic lines are theoretically at an infinite distance from the "origin", and thus that the lines connecting those points have an infinite length.
It's a bit analogous to the way train tracks shrink toward the horizon and make an angle with each other where they appear to meet it, even though they don't actually meet in the plane. These hyperbolic lines won't actually ever meet in the hyperbolic plane either but they approach the same point on the horizon.
That edge is basically an artifact of the model, you can equally model the hyperbolic plane space as a disk and then the boundary is a circle, or on an actual hyperboloid in 3D and it extends out forever.
The disk model of hyberolic geometry is made to map hyperbolic 2 space (which is infinite in area) into the finite interior of the disk. In order to capture this, the normal euclidean notion of distance is distorted by a function which allows "distances" to go to infinity as a curve approaches the boundary of the disk.
Let's go to to the normal infinite plane for a moment.
You can use a map that is inside a circle with r=1. The objects get deformed, but points have a 1 to 1 correspondence. Lines that pass though 0 look straight, but other lines are curved.
Measuring a distance is hard, you have to use some weird rules.
If you draw a segment of length 0.001 segment in the circular map, it has almost the same length in the real infinite map.
If you draw a segment of length 0.001 segment near the border of the circular map, it's a huge thing in the infinite map.
Moreover, a line that pass thorough 0 has apparent length 2 in the map, but represent an infinite length in the plane
Note that the border of the circle is outside the plane.
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The reverse happen if you have a map of the Earth. You can draw on the map with a pencil a long segment near the pole, but it represents a small curved segment in the Earth.
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Back to your question ,,,
It's on the hyperbolic plane, not in the usual euclidean plane. So the map is only the top half, and the horizontal line = axis x is outside, it's the border.
Length are weird, and a 0.001 segment draw with a pencil on the map far away from the x axis is small in the actual hyperbolic plane, but a 0.001 segment draw with a pencil on the map near the x axis is very long in the actual hyperbolic plane.
The circles "touch" the x axis. In spite they look short when you draw them with a pencil, they part that is close to the x axis has a huge length in the hyperbolic plane.
It must be that the figure with the half circles is just a representation of the hyperbolic space into 2D. Such projections are not faithful; you cannot take measurements in the projection and take them literally.
We can make an analogy to cartography: you can't trust areas and distances on distorted projections like Mercator.
Look, even the angles don't look to be zero in that diagram. We have to imagine that we zoom in on an infinitesimal zone around each corner to see the almost zero angle; i.e. the circle tangent lines actually go almost parallel. So to speak.
Thus the angles are locally correct, since they are measurable on arbitrarily small scales and can easily be imagined to be even when glancing at the entire figure. But distances between the points aren't localizable; they have to follow a measure which somehow correctly spans the abstract hyperbolic space that they represent.
How about this (almost certainly incorrect) imagining: pretend that the real line shown, on which the three points lie, is actually a horizon line, which lies in a vast distance (out at infinity). Just like the horizon when you do drawings with two-point perspective. Imagine the three points are vanishing points on the horizon. Vanishing points are not actually points; they just directions into infinity.
if, in a two-point perspective, you draw a curve whose endpoints are tangent to two vanishing point traces, that curve is infinitely long.
For instance if you draw an intersection between two infinite roads, where the curb has a round corner, you will get some kind of smiley curve joining two vanishing points. That curve is understood to be infinitely long.
Hyperbolic geometry! Note how lines on the chart don't appear straight. That's because this is just a projection of an infinite hyperbolic space. The rules of this projection move points at infinity to the real line, and straight lines to circles. That means each of the points on the mentioned triangle is infinitely far away in some direction.
A sequence of triangles, where the limit of the sum of angles is zero.
That's not a fucking triangle.
(It's Friday night people it's a joke and I have no idea what the article is talking about just looked at the picture)
I had to reason with my brain before it would accept it as a triangle. It has 3 sides and 3 corners so...
It's one of those things where it's technically correct but the headline is misleading. When you say "a triangle" without any qualification as the headline does, people are going to interpret that as a good old fashioned triangle. Using the term without clarification that you mean spherical geometry is kind of underhanded writing, imo.
The title attribute of the article is <title>A hyperbolic triangle with three cusps</title>
It's a mild form of clickbait.
I think it's just a normal ages-old pattern for writing headlines that pique people's curiosity. It's super common in popular math in particular, because math is always about generalizing. There's a fine line between that and actual clickbait meant to actively mislead.
Sides are half spheres but yeah it is not an euclidean triangle.