Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry

In the other replies there has been some mention of alternative methods for subdivision besides barycentric subdivision, but these are rarely encountered in algebraic topology. What are some of these other methods, in fact? Preferably they should be natural and canonical, not based on random choices. I dimly recall seeing somewhere (in a paper of Quillen or Segal?) a subdivision method generalizing the simple idea of subdividing a triangle into four triangles by adding new vertices at the midpoints of the three edges, but the generalization to higher dimensions isn't obvious. Does anyone know a reference for this? Another approach might be to use the canonical subdivision of an n-simplex into n+1 cubes, one at each vertex of the simplex, then subdivide each cube into small cubes in the obvious way, then subdivide the small cubes into simplices in some natural way. This seems a bit cumbersome, however.

A drawback of barycentric subdivision is that it takes some work to show that sufficiently many iterations of barycentric subdivision produce arbitrarily small simplices. It would be nice to have a subdivision method for which this was obvious.

I like to think of appendices (or supporting information, which is the same thing for a journal paper) in terms of the narrative structure of a text. The main text should contain everything that makes up the "story" of the work. In it, a reader who basically trusts that your methods are sound should find everything that they need to understand the work.

There are often, however, places where it is important to show your work, but that are not particularly interesting. If they are lengthy enough that they start feeling like a major detour in the flow of the narrative, then they are a good candidate for moving to an appendix.

Some examples from my own recent papers:

• Theorem and proof sketch in main text, boring exhaustive proof with lots of slightly different cases in appendix.
• Graph summarizing results plus an example of result detail in main text, all the rest of the results in appendix.
• Data from method presented in main text, data showing that plausible alternatives didn't work in appendix.
• Intuitive description of method and key mathematical concepts in main text, exhaustive mathematical details in appendix.

Exactly where to draw the line is somewhat subjective, but fortunately doesn't matter all that much unless you are dealing with format or length restrictions.

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Account Sharing Options,

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