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Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil

A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles stationary and rotate the other we create the twisting motion that creates cones and hyperboloids and if we rotate one of the circles 360° we get back to our original cylinder. These rotation of cylinder to hyperboloid to cone can be found below.

A cylinder is really a rather basic ruled surface; other more "interesting" surfaces do exist and can often be created relatively simply. For example, most planes are ruled surfaces as are many solids, even more unique surfaces like the Möbius strip (shown below) are ruled surfaces.

The surface we are going to be looking at is very similar to the Möbius strip which is the Trefoil knot, which is shown below.

This surface is parametrically defined by the following \[ x = sin(t) + 2sin(2t) \\ y = cos(t) -2cos(2t) \\ z = -sin(3t). \] In order to actually create the solid model of the trefoil knot ruled surface I created the outer edge with the above formulas and the inner edge by halving its size. This allowed the actual ruled surface of the trefoil knot to be easily visible. Furthermore, the rulings are also easily visible in the image below which is our 3D model of the trefoil knot ruled surface. The actual physical model is approximately 2"x2"x1".

When deciding a ruled surface to model I was immediately drawn to the Möbius strip which is a somewhat of a standard surface. As such I went on to explore similar surfaces, and landed on the trefoil knot. Interestingly, the trefoil knot can be considered a type of Möbius strip as it only has one side and one boundary strip. Additionally, later in the semester we will (I believe) have a discussion on knots and knot theory, so consider this a bit of a foretaste—or if we do not cover knots and not theory an exciting foray into knots, and it might also pique your interest in the Christian symbolism of the trefoil (or Trinity) knot.

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