This simplest of all knots, known as the "trefoil", readily lends itself to iteration due to its simplicity and three-fold symmetry.

One possible starting self-similar knot:

After one iteration:

After three iterations:

Flash animation illustrating that the second iteration of this version of an iterated trefoil knot is unicursal.

An alternative approach for this knot is to first rearrange it into a figure-8 configuration. The iterative procedure now doubles the number of copies with each generation, as opposed to the tripling in the triangular version above.

The starting self-similar knot:

After the first iteration:

Flash animation illustrating that the third iteration of this version of an iterated trefoil knot is unicursal.

To see a digital artwork based on this iterated knot, click on the thumbnail below.

A 3-dimensional iterated trefoil knot is shown below. These figures were created using KnotPlot, which allows manipulation of the knots in 3-D. The KnotPlot files are available by clicking on the links below.

The starting knot:

The KnotPlot file for the starting knot

The knot after one iteration:

The KnotPlot file for the knot after one iteration

The starting knot:

The KnotPlot file for the knot after two iterations

All images copyright Robert Fathauer

Webmaster: rob@tessellations.com