From Wikipedia: "In knot theory, a Lissajous_knot is a knot defined by parametric equations of the form:
x = cos(n_{x}t + φ_{x})where n_{x}, n_{y} and n_{z} are integers and the phase shifts φ_{x}, φ_{y} and φ_{z} may be any real numbers.
y = cos(n_{y}t + φ_{y})
z = cos(n_{z}t + φ_{z})
"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."
Some examples of Lissajous knots that result in the ABC's Lissajous curve when projected onto the plane z=0:
(n_{x}, n_{y}, n_{z}) | (φ_{x}, φ_{y}, φ_{z}) | |
---|---|---|
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, n_{y}/n_{x} = 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.
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) |