Roll a Sphere without Changing Orientation to a New Location in Two Straight Rolls

A sphere can be moved to any location with no net change in orientation by rolling without slipping in two straight-line rolls. Rolling a sphere in a straight line a distance of , where is any integer, returns the sphere to its initial orientation. By combining two such rolls, the sphere can be moved to any location.


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J. M. Hammersley proved that three straight-line rolls in the horizontal plane are sufficient to move a sphere to any desired position and orientation, but that three rolls can lead to arbitrarily long paths. In contrast, the total path length needed to move the sphere from a start position to a desired position at a distance away without changing orientation is , where is the ceiling function. The procedure is to first roll the sphere to a desired orientation with two straight-line rolls, and then move the sphere to the desired position with two straight-line moves.
By applying the Demonstration Re-Orient a Sphere with Two Straight Rolls, followed by this Demonstration, a ball can be rolled to any orientation and position in four rolls.
[1] J. M. Hammersley, "7. Oxford Commemoration Ball," in Probability, Statistics and Analysis: London Mathematical Society Lecture Note Series, No. 79 (J. F. C. Kingman and G. E. H. Reuter, eds.), Cambridge, UK: Cambridge University Press, 1983.
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