  We classify 3-braids arising from collision-free choreographic motions of 3
bodies on Lissajous plane curves, and present a parametrization in terms of
levels and (Christoffel) slopes. Each of these Lissajous 3-braids represents a
pseudo-Anosov mapping class whose dilatation increases when the level ascends
in the natural numbers or when the slope descends in the Stern-Brocot tree. We
also discuss 4-symbol frieze patterns that encode cutting sequences of
geodesics along the Farey tessellation in relation to odd continued fractions
of quadratic surds for the Lissajous 3-braids.
