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    $\begingroup$ For triangle tiles, there is this paper which I cannot access at the moment: "Classification of tilings of the 2-dimensional sphere by congruent triangles." Yoshio Agaoka and Yukako Ueno. Hiroshima Math. J.. Volume 32, Number 3 (2002), 463-540. $\endgroup$
    – Joseph O'Rourke
    Commented Nov 2, 2013 at 21:08
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    $\begingroup$ @JosephO'Rourke : Thank you. The open-access 78-page article is available @ projecteuclid.org/… $\endgroup$
    – Wlodek Kuperberg
    Commented Nov 2, 2013 at 22:12
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    $\begingroup$ @JosephO'Rourke : True even for non-convex tiles, only define a vertex as a point at which at least three tiles meet; then use the Euler characteristic for the sphere. $\endgroup$
    – Wlodek Kuperberg
    Commented Nov 3, 2013 at 1:59
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    $\begingroup$ By the way, the hyperbolic plane can be tiled by congruent copies of polygons with arbitrarily small diameters using "horobricks." $\endgroup$
    – Douglas Zare
    Commented Nov 3, 2013 at 18:32
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    $\begingroup$ The first questin is Problem 60 in the Scottish Cafe book of problems: Can one, for every $\varepsilon>0$, represent the surface of a sphere as a sum of a finite number of regions which are smaller in diameter than $\varepsilon$, closed, connected, congruent, and have no interior point in common? We assume that the boundaries of these sets are: (a) polygons, (b) curves of finite lengt, (c) sets of measure zero. (RUZIEWICZ) $\endgroup$
    – juan
    Commented Feb 25, 2014 at 20:28