Steinhaus' Billiard Ball Loop

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A billiard path is a polygon with vertices on the faces of a polyhedron such that if two segments meet at a vertex on a face , the plane through them is perpendicular to and the angle they form is bisected by the normal to at . A billiard ball loop is a closed billiard path.


This Demonstration shows a billiard ball loop in a cube discovered by Hugo Steinhaus. Each vertex lies at a distance one-third of the cube's edge length from two edges. The projection of the loop perpendicular to any face of the cube is a rectangle, and the projection along one of the main diagonals of the cube is a regular hexagon.


Contributed by: Izidor Hafner (October 2013)
Open content licensed under CC BY-NC-SA




[1] D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin Books, 1991, pp. 13–14.

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