Torus Knot

What it draws

Read 2.5 + cos(3u + t) as a distance from the vertical axis: it runs between 1.5 and 3.5. The cos(2u) and sin(2u) it multiplies place the point at that distance and at the angle 2u around the axis. The second expression, sin(3u + t), is the height, and it stays between −1 and 1.

Height and distance are driven by the same 3u + t, so the point never leaves a torus: central circle of radius 2.5 lying flat, tube of radius 1, axis vertical. The angle around the axis is 2u, and the angle around the tube is 3u + t. Over one turn of u the strand goes twice around the axis, three times around the tube, and closes.

Why it is a trefoil

A curve winding p times around the axis of a torus and q times around its tube is the (p, q) torus knot. It is one closed strand rather than several exactly when p and q share no factor. Here p = 2 and q = 3, coprime, and the (2, 3) torus knot is the trefoil, the simplest knot there is. Looking straight down the axis from Top, two strands wind around and cross three times: the usual three-crossing picture of a trefoil.

The clock enters through the tube angle alone, and its effect is rigid. Since 3u + t = 3(u + t/3), the curve at time t is the curve at time zero turned by 2t/3 about the vertical axis. The knot does not deform as it plays. It rotates at two thirds of a radian per second, one full turn every 3π ≈ 9.42 seconds at Speed 1.

Try

• Top: down the axis, the three crossings are laid out plainly, and the tube's inner and outer edges become the two rings the strand weaves between.
• Turn on trace: the pen walks the strand from end to end, so the crossings can be followed in order and the knot seen to be a single closed piece.
• Change cos(2u) and sin(2u) to cos(4u) and sin(4u) for the (4, 3) torus knot: four strands from above, crossing nine times.
• Change all three 3u to 5u for the (2, 5) knot, the cinquefoil, with five crossings from above.

Read more

• Torus knot
• Trefoil knot
• Knot theory
• Torus