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Let us consider a figure of the Euclidean plane comprised of finitely many non-degenerate non-overlapping triangles (i.e., no triangle has a zero area and no two distinct triangles have any inner point in common).

Two distinct triangles are said to be neighbors iff they have at least two points in common (i.e., they share a portion of side of non-zero length, so indeed infinitely many points).

Must there be at least one triangle that has at most three neighbors?

Is this a known problem?

Thanks in advance.

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    How could you cover the plane with finitely triangles? What do you think about this kind of pattern: http://www.metafysica.nl/turing/p31m_triangulatie.gif2012-06-29
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    I didn't say I was covering the plane. But on a torus, for instance, it's quite easy to come up with a counter-example in which each triangle has four neighbors. I suppose one could ask for the max of the minimal number of neighbors on more complex surfaces... There must be many other ways of generalizing the problem, but I want to keep it simple to begin with.2012-06-29
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    You can read about triangulation of surfaces. For example: http://rip94550.wordpress.com/2008/08/12/triangulations-of-surfaces-minimum-number-of-triangles/2012-06-29

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