Student Post: Rolling Trefoil Knots

Introduction:

Dr. Laura Taalman and Dr. Stephen Lucas have been researching rolling trefoil knots since before we (Abby Eget and Harley Meade) became involved with the project. When we joined the research team, we were fortunate enough to come into the project with a lot of previously coded information in OpenSCAD including the general shape of the Morton trefoil, which is tritangentless, which is what allows it to properly roll. Then we helped the team optimize the knot using a parameter a, in accordance to the formula in the section below, so that there was minimal variation in the height of the center of mass.

Optimizing the z value using the a value:

Originally the plan was to optimize the a value by scaling by 1 in the z direction. This was done surprisingly quickly, so we shifted directions. Instead of optimizing the a value for scaling by 1 in the z direction, we decided to try to optimize the scaling in the z direction based on varying a values. We used a values of 0.3, 0.5, 0.7, and 0.9.

Coded knot with a = .5

We printed the four trefoil knots with the four different a values from Shapeways. Then we printed four corresponding convex hulls in our own Maker Lab. These are pictured below.

This image has an empty alt attribute; its file name is img-6355-e1553031369202.jpg
a = .3 scaling = .4629
This image has an empty alt attribute; its file name is img-6356-1-e1553031560633.jpg
a = .5 scaling = .8210
This image has an empty alt attribute; its file name is img-0232-1-e1553031667294.jpg
a = .7 scaling = 1.3162
This image has an empty alt attribute; its file name is img-0231-1-e1553031817729.jpg
a = .9 scaling = 2.4954
This image has an empty alt attribute; its file name is graphcomparingknotssnip.png
Varying heights of the center of mass for knots of varying a values with z-scale 1.

Knot paths:

After considering how the knots rolled in relation to their convex hulls, we decided to use the triangles from the convex hulls’ code to see what kind of paths the knots make when rolling on the table. Two of these are pictured below, with the black line showing the horizontal shift in center of mass.

This image has an empty alt attribute; its file name is knotpathssnip.png
Path of Morton trefoil knots with a = 0.3 and a = 0.5

Two other parameters that we changed are the p and q values of the knots. These are related to the fact that the trefoil is a torus knot, where the p represents how many times the knot wraps around a torus meridianally and the q represents how many times the knot wraps around the torus longitudinally. We also decided to change these values, while keeping p and q relatively prime so as to not go from a knot to a link, to see if these knots would at least be externally tritangentless, and how their paths on a table would be affected.

This image has an empty alt attribute; its file name is img-0233-1-e1553033443463.jpg
p = 7 q =2

After exploring some of the variables and parameters surrounding rolling trefoil knots, we are excited to continue optimizing these knots by looking at other factors. In the future, we plan to optimize the materials, infill, and consistency in sizing of the knots to minimize differences not currently accounted for.

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