[HN Gopher] A triangle whose interior angles sum to zero
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A triangle whose interior angles sum to zero
Author : tzury
Score : 127 points
Date : 2025-11-29 00:26 UTC (22 hours ago)
(HTM) web link (www.johndcook.com)
(TXT) w3m dump (www.johndcook.com)
| dkdcio wrote:
| > infinite perimeter
|
| I don't follow, how/why?
| dadoum wrote:
| 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.
| ironSkillet wrote:
| 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.
| roywiggins wrote:
| 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.
| itishappy wrote:
| 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.
| jerf wrote:
| 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.
| volemo wrote:
| > 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.
| gus_massa wrote:
| 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.
| kazinator wrote:
| 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.
| ethmarks wrote:
| > 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.
| nhinck2 wrote:
| Gabriels Horn for another example.
|
| Doesnt seem that uncommon.
| JadeNB wrote:
| Gabriel's horn is the same phenomenon one dimension up:
| finite surface area but infinite volume.
| eru wrote:
| You mixed it up. The horn has infinite surface area but
| finite volume.
| JadeNB wrote:
| You're right. Thanks.
| IgorPartola wrote:
| 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.
| eru wrote:
| 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.
| almostgotcaught wrote:
| 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
| saithound wrote:
| > 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.
| almostgotcaught wrote:
| > 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.
| saithound wrote:
| > 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.
| genezeta wrote:
| 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:
| 1 ---------- | |n/2 |
| | 1 n| ---------- |
| |n/2 | |
| -------------------- 2
|
| 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_.
| ericol wrote:
| 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)
| anthonyIPH wrote:
| I had to reason with my brain before it would accept it as a
| triangle. It has 3 sides and 3 corners so...
| bigstrat2003 wrote:
| 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.
| eru wrote:
| It's a mild form of clickbait.
| Sharlin wrote:
| 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.
| cwillu wrote:
| The title attribute of the article is <title>A hyperbolic
| triangle with three cusps</title>
| nurettin wrote:
| Sides are half spheres but yeah it is not an euclidean
| triangle.
| abhashanand1501 wrote:
| >In spherical geometry, the interior angles of a triangle add up
| to more than p. And in fact you can determine the area of a
| spherical triangle by how much the angle sum exceeds p. 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.
| fluoridation wrote:
| >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.
| roywiggins wrote:
| >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.
| fluoridation wrote:
| 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.
| roywiggins wrote:
| 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
| fluoridation wrote:
| That doesn't help you if you're moving West to East,
| though.
|
| EDIT: Also, that's to measure distance, not direction.
| teo_zero wrote:
| > 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?
| fluoridation wrote:
| Flat-earthers don't accept that a flat plane implies
| infinite line of sight (especially at sea), so who knows.
| thaumasiotes wrote:
| > 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.
| lwansbrough wrote:
| 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.
| QuadrupleA wrote:
| 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.
| lordnacho wrote:
| 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.
| kakacik wrote:
| 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.
| immibis wrote:
| 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.
| dnemmers wrote:
| Hitler was enthralled by Henry Ford, and copied what he
| learned about anti-semitism.
|
| https://www.thehistoryreader.com/historical-
| figures/hitlers-...
| theoreticalmal wrote:
| The feedback response post is true, but specifically
| about Jews is not true. He hated Jews long before he rose
| anywhere near power
| fooker wrote:
| 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.
| teiferer wrote:
| 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.
| thomasahle wrote:
| 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
| zahlman wrote:
| 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?
| Sharlin wrote:
| As they say, you can't reason someone out of something they
| didn't reason themselves into in the first place.
| themafia wrote:
| > 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...
| taneq wrote:
| It also means that pi could be equal to 3 if you world is small
| enough.
| schobi wrote:
| 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.
| amelius wrote:
| No, you're talking about a hologram. Everything is flat.
|
| https://en.wikipedia.org/wiki/Holographic_principle
| anigbrowl wrote:
| Spherical geometers: the trolls of the math world
| sdeframond wrote:
| Ah! I just realized that there is an infinity of different
| triangles passing through those three points: two poles and any
| other point. Wild!
| vismit2000 wrote:
| 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)
| gorfian_robot wrote:
| Triangle Man _hates_ Person Man
| MathMonkeyMan wrote:
| A sequence of triangles, where the limit of the sum of angles is
| zero.
| gloftus wrote:
| 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.
| KuSpa wrote:
| 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).
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