Knot Family
((2 + cos(q · u)) · cos(p · u), sin(q · u), (2 + cos(q · u)) · sin(p · u))
Open in the app The dials and keys named below are the app's.
What it draws
Read 2 + cos(q · u) as a distance from the vertical axis: the cosine runs between −1 and 1, so the distance runs between 1 and 3. The cos(p · u) and sin(p · u) it multiplies put the point at that distance and at the angle p · u around the axis. The second expression, sin(q · u), is the height, and it stays between −1 and 1.
Height and distance are driven by the same q · u, so the point never leaves a torus: centre circle of radius 2 lying flat, tube of radius 1, axis vertical. The angle around the axis is p · u and the angle around the tube is q · u, so over one turn of u the strand goes p times around the axis and q times around the tube. Nothing here depends on t, and the figure sits still.
Knot, retrace, or open arc
A strand winding that way is the (p, q) torus knot, and the two sliders choose which one. Three things can happen. When p and q are whole numbers sharing no factor, the strand closes into one knot. From Top it then crosses itself q · (p − 1) times: 8 at the opening pair, 3 at (2, 3), 5 at (2, 5), 12 at (7, 2). Swapping the two gives the same knot back, so that last picture is a more tangled drawing of a knot that needs only seven crossings.
When the two share a factor the strand closes early and then goes round again over its own path. At (3, 3) one loop is drawn over and over, and nothing is added after the first thirtieth of u.
When either slider leaves the whole numbers the strand needs more of u before its ends can meet. Turns of u opens at 10 to give it that: every stop of the two sliders is a whole number of tenths, so ten turns close every pair the sliders can reach. A whole coprime pair is simply drawn ten times over.
Try
- Set p to 2 and q to 3: the trefoil, the simplest knot there is, and the same knot the preset Torus Knot ties on a slightly wider ring.
- Drag q to 5, leaving p at 3: the pair is coprime again and the count from above rises to 10.
- Drag q to 3 instead: now both are 3, and the strand is one loop traced over and over in place.
- Drag p to 3.5: ten turns still close it. Lower Turns of u to 1 and the same strand stops 6 short of its own start, the full width of the figure.
- Press Top: looking down the axis, the crossings sit in a ring between the tube's inner and outer edges.