# Characterizing triangles unembeddedly

The mathedu mailing list has a recent longish thread at

http://www.nabble.com/Why-do-we-do-proofs--to25809591.html

which discussed 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, 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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I think the core question here is can one make sense of "side lengths" and "angles" without an embedding? – j.c. Oct 20 '09 at 1:12
I don't understand. The whole point of the manifold definition is that we want to ignore some information and preserve other information. If you want a notion of angle, you should be working in an inner product space. – Qiaochu Yuan Oct 20 '09 at 1:16