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In knot theory, a Lissajous knot is a knot defined by parametric equations of the form
where n x {\displaystyle n_{x}} , n y {\displaystyle n_{y}} , and n z {\displaystyle n_{z}} are integers and the phase shifts ϕ x {\displaystyle \phi _{x}} , ϕ y {\displaystyle \phi _{y}} , and ϕ z {\displaystyle \phi _{z}} 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 Lissajousknot 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 or in a solid torus.
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