I just posted this question on mathoverflow (I recommend looking into this new forum):

The mathedu mailing list has a recent longish thread which discusses among other things whether we should teach triangles as labeled or unlabeled to high school students (this is a vast oversimplification of the thread).  I have long been concerned with how we think (informally and formally) about mathematical objects, as for example my unfinished article here about the many ways of thinking about function.  So naturally, I started to consider how we think about triangles.

Consider circles.   Most informal and formal descriptions involve an embedding into R^2, but they *can* be characterized as manifolds (even as Riemannian manifolds) of dimension 1 with specific properties, independent of any embedding. This sort of thing has turned out to be a major way to think about all sorts of spaces.  So can we describe triangles in a similar way?

Unfortunately, manifolds are far removed from my usual mathematical work (category theory).  What I *think* I understand is that there can be *piecewise* linear manifolds, even Riemannian ones.  So perhaps we can say a triangle is a piecewise linear manifold of dimension 1 with certain properties.  Now, I want to define a triangle so that it comes complete with information about the lengths of its sides and what the three angles are.  Riemannian manifolds have a way to specify length and angles, and I can believe you can make the sides have specific lengths.  But the angles?  It seems to me that the tangent spaces (like those on a circle) result in all angles being 0 or pi, except at the corners where they don’t exist.  But I may not understand the situation correctly.

So my question is:  Is there a known methodology that allows triangles to be characterized independent of embeddings in such a way that incorporates information about side lengths and angles?

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