**Artwork name: **Opus 325846

**Artist: **Opus 325846

**Description: **

There are (infinitely) many different knots, the simplest non-trivial knot being the trefoil knot. Opus 325846 (nicknamed Koos’ Knoopje) is a figure-eight knot, which you could call the second simplest knot. While the left-handed and right-handed trefoil knot are truly different knots, this is not the case for the figure-eight knot. Surprisingly, the figure-eight knot does *not *have a mirror symmetry: you cannot find a mirror plane. But if you turn the figure-eight knot upside-down and give it a quarter turn, you will see that you have obtained a mirror image. (It has a so-called *roto-reflective symmetry*.) A phenomenon like this is impossible in 2D.

Knots are familiar shapes yet can be dauntingly mysterious (especially when trying to untangle a messy one). Knowledge of some knot formations is a necessity for sailors, yet there is much we don’t know about knots – so mathematicians study “knot theory.”

Mathematical knots are closed. They do not have two loose ends, like shoelaces that can be untied. To make the simplest mathematical knot, take a length of string or flexible wire and bend it so the two ends cross each other. Now take the end that is “on top” and twist it to go under, then over the other end. Finally, glue the two ends together. This is called a *trefoil knot*. In its most symmetric presentation, it looks like three identical rings woven together. It is impossible to undo this or any mathematical knot without cutting it.

When Bakker gives instructions to his computer program to connect copies of a polyline generator in order to form closed circuits in space, some of the circuits among the thousands produced may be mathematical knots. The program contains a filter that can identify which of the circuits are knots, and from these the artist can select what becomes the basis for a knotted sculpture.

Polylines have abrupt, often sharp corners as they trace out a circuit. These paths do not flow, they jerk. To smooth a polyline path into a flowing curve, Bakker uses what mathematicians call *spline interpolation*. This is a bit like fitting a thin springy strip of steel around a set of pegs to form a curved path that touches each peg.

Cubic functions (the simplest is *y* = *x*^{3}) have curvy, S-shaped graphs. They have the remarkable property that, given four points (not all on a line), there is a cubic function whose graph goes through those four points. If the three points are fairly close to each other, the piece of the cubic curve running through them (called a spline) closely approximates line segments that connect the points. Using splines, Bakker can replace each sharp **V**-shaped corner of a polyline path with a **U**-shaped curve. The result is a smoothly curvaceous circuit that travels through all the corners of the polyline path.

The curved loop that results from smoothing a polyline circuit in space is merely a skeleton doodle with no thickness and no body. These must be provided by the artist. A simple thickening coats the curve so it has a uniformly shaped cross-section such as a circle (which produces a tube covering), a square, or triangle. The width and thickness of the curve’s covering can be varied for aesthetic reasons. This can suggest a change of speed and spread as the curve flows, much like water flowing in a creek that meanders through changing terrain.