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Our paper "Random triangles and polygons in the plane"] – in which we give a novel answer to Lewis Carroll's question "What is the probability a random triangle is obtuse?" – was published recently in the American Mathematical Monthly: doi.org/10.1080/00029890.2019.

Here's an animated version of Figure 2 from the paper, showing a geodesic in triangle space. The geodesic starts at the equilateral triangle shown, and the three curved paths show the tracks of the three vertices.

@shonk I saw it in the Monthly! A very nice article.

@shonk Nice article; I see my co-author and one-time workshop-roommate Jason Cantarella is one of the authors. BTW, your metric tree in Fig.1 is the tight span of the three-point metric space; see en.wikipedia.org/wiki/Tight_sp

@11011110 At the origami workshop in Barbados? I heard that was very fun.

If I take the tight span of a (convex if you like) n-gon, is there any sensible way of extracting coordinates on n-gon space (as there evidently is for n=3)?

@shonk One of those workshops, yes. I'm heading to this year's in a couple weeks...

The tight span isn't as nice a space for more than three points (its dimension will typically be linear in the number of the points). It tells you all the distances between all pairs of points, which is all you need to know to find a congruent copy of the points (assuming they started out in the plane). But it doesn't tell you in what order they're connected to form a polygon.

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