# Lissajous Knots

From Wikipedia: "In knot theory, a Lissajous_knot is a knot defined by parametric equations of the form:

x = cos(nxt + φx)
y = cos(nyt + φy)
z = cos(nzt + φz)
where nx, ny and nz are integers and the phase shifts φx, φy and φz 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."

## 'ABC' knots:

Some examples of Lissajous knots that result in the ABC's Lissajous curve when projected onto the plane z=0:

(nx, ny, nz) (φx, φy, φz)
Example 1 (1, 3, 1) (0, 0, 𝜋/2)
Example 2 (1, 3, 3) (0, 𝜋/2, 0)
Example 3 (1, 3, 2) (0, 𝜋/2, 𝜋/2)
Example 4 (1, 3, 4) (0, 𝜋/2, 𝜋/2)
Example 5 (1, 3, 5) (0, 𝜋/2, 𝜋/2)
Example 6 (1, 3, 5) (0, 𝜋/2, 0)
Example 7 (1, 3, 7) (0, 𝜋/2, 𝜋/2)
Example 8 (1, 3, 8) (0, 𝜋/2, 𝜋/2)
Example 9 (3, 9, 4) (0, 𝜋/2, 0)

To map onto the ABC Logo, ny/nx = 3. The z-component doesn't matter, so there is an infinite number of such knots. Example 3 is closest to the current 3D worm, but it would need some warping to match it exactly.

## Some other knots

Not Lissajous knots, and not related to the ABC logo.

x y y
Trefoil knot (2 + cos(3t))⋅cos(2t) (2 + cos(3t))⋅sin(2t) sin(3t)
Figure-eight knot (2 + cos(2t))⋅cos(3t) (2 + cos(2t))⋅sin(3t) sin(4t)
Cinquefoil knot (2 + cos(5t))⋅cos(2t) (2 + cos(5t))⋅sin(2t) sin(3t)
Septafoil knot (2 + cos(7t))⋅cos(2t) (2 + cos(7t))⋅sin(2t) sin(3t)
Torus knot (4 + cos(12t))⋅cos(t) sin(12t) (4 + cos(12t))⋅sin(t)
Spherical helix sin(t)⋅sin(2t) cos(t) sin(t)⋅cos(2t)