page_title: Arithmetic billiards
section_title: Ideas of proof
hierarchical_section_title: Arithmetic billiards / Ideas of proof
caption_attribution_description: English: Mathematical image of an arithmetic billiard. Example explaining the reflection of the billiard, if the side ratio is 2/3.
context_page_description: In recreational mathematics, arithmetic billiards provide a geometrical method to determine the least common multiple and the greatest common divisor of two natural numbers by making use of reflections inside a rectangle whose sides are the two given numbers. This is an easy example of trajectory analysis of dynamical billiards.
  Arithmetic billiards have been discussed as mathematical puzzles by Hugo Steinhaus and Martin Gardner, and are known to mathematics teachers under the name 'Paper Pool'.
  They have been used as a source of questions in mathematical circles.
context_section_description: Reflecting the billiard: Consider a square with side . By displaying multiple copies of the original rectangle (with mirror symmetry) we can visualise the arithmetic billiard path as a diagonal of that square. In other words, we can think of reflecting the rectangle rather than the path segments.
  Reducing to the coprime case: It is convenient to rescale the rectangle dividing  and  by their greatest common divisor, operation which does not alter the geometry of the path (e.g. the number of bouncing points).
  Reversing the time: The motion of the path is “time reversible”, meaning that if the path is currently traversing one particular unit square
  (in a particular direction), then there is absolutely no doubt from which unit square and from which direction it just came.
  The proof can be found in a popularization article.
caption: By reflecting the billiard, we can visualize the path as a straight line. In this example the ratio of the two given numbers is 2/3.
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