page_title: Lissajous knot
hierarchical_section_title: Lissajous knot
caption_attribution_description: English: A Lissajous 821 knot.
context_page_description: In knot theory, a Lissajous knot is a knot defined by parametric equations of the form
  where, and are integers and the phase shifts, and may be any real numbers.
  The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves.
  Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajous
  knot isotopically into a billiard curve inside a cube, the simplest case of so-called billiard knots.
  Billiard knots can also be studied in other domains, for instance in a cylinder.
context_section_description: In knot theory, a Lissajous knot is a knot defined by parametric equations of the form
  where , , and  are integers and the phase shifts , , and  may be any real numbers.
  The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves.
  Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajous
  knot isotopically into a billiard curve inside a cube, the simplest case of so-called billiard knots.
  Billiard knots can also be studied in other domains, for instance in a cylinder.
caption: A Lissajous 821 knot
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