The starting knot has three crossings, and since there is only one knot with three crossings, this knot must be a projection of the trefoil knot.
The starting curve for designing this knot is a rectangle. Two smaller rectangles are arranged within it, with a rectangular space of the same size and shape between them. The smaller rectangles are rotated 90 degrees with respect to the starting rectangle, and the scaling factor for this arrangement is easily calculated to be the inverse of the square root of 3. The number of crossings for the starting knot and first five iterations are 3, 7, 15, 31, 63, and 127, and the increase in the number of crossings with each successive iteration is 4, 8, 16, 32, 64, ...
The starting knot and the first two iterations, with the rectangles shown in dotted lines:
After five iterations:
After five iterations, with half the knot given a second color, to better show the path of the strand:
To see a digital artwork based on this iterated knot, click on the thumbnail below.
A 3-dimensional version of this knot after two iterations is shown below. This figure was created using KnotPlot, which allows manipulation of the knot in 3-D. The KnotPlot files are available by clicking on the links below.
The KnotPlot file for the starting knot
The KnotPlot file for the knot after two iterations
The KnotPlot file for the knot after two iterations, modified version (the one shown above)
All images copyright Robert Fathauer
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