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.
Mikusiński's Problem
See also
Great Circle, SphereExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Mikusiński's ProblemCite this as:
Weisstein, Eric W. "Mikusiński's Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MikusinskisProblem.html