A triangle whose interior angles sum to zero

Gabriels Horn for another example.

Doesnt seem that uncommon.

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.

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.

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.

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.