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Is it possible to cover completely the surface of a Sphere with congruent, nonoverlapping arcs of Great Circles? Conway and Croft (1964) proved that it can be covered with half-open arcs, but not with open arcs. They also showed that the Plane can be covered with congruent closed and half-open segments, but not with open ones.
References
Conway, J. H. and Croft, H. T. ``Covering a Sphere with Great-Circle Arcs.'' Proc. Cambridge Phil. Soc.
60, 787-900, 1964.
Gardner, M. ``Point Sets on the Sphere.'' Ch. 12 in Knotted Doughnuts and Other Mathematical Entertainments.
New York: W. H. Freeman, pp. 145-154, 1986.