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

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

Contributed by: Aaron Becker (August 2012)
After work by: J. M. Hammersley
Open content licensed under CC BY-NC-SA



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.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.