Trig Trials

Written by Claire Gilreath. Joanne Wang, and Selihom Gobeze (students in Math 383D Knot Theory Spring 2023).

Finding the Coordinates of Petal Knots

We decided we wanted to try to 3D print the petal projections of knots for our project because they looked pretty and seemed like a challenge.

Image showing the trefoil knot in petal form.

A petal projection of the trefoil knot.

Petal projections of knots are projections where all of the crossings are aligned through a single line in 3d space, giving the knot a flower-like appearance. A two-dimensional petal projection of the trefoil (3_1) is pictured to the right (via this image from Wikipedia). We decided to work on creating petal versions of 3_1, 5_1, 6_2, and 6_3 knots. We relied heavily on the work described in Colin Adams’ “Knot Projections with a Single Multi-Crossing” paper. In the appendix of this paper, Adams and his coauthors described the order of each petal for the knots we hoped to construct. Also, we utilized the Wikipedia page  about petal projections of knots to better understand their structure.

We were a bit uncertain of the best method to create these knots, since as far as we knew, there were no equations we could plug into Cinema 4D, nor were there coordinates constructed by someone else. After talking to Professor Denne, we felt that our best bet would be creating our own sets of coordinates using trig and polar coordinates. We decided to find four points per petal: the center, the top of the arc of each petal, and the end of both edges. To do this we constructed a circle through the ends of the edges, shown in red in the diagram of the trefoil below.

This figure shows the details behind the computations needed to find the coordinates of points along the petal trefoil knot.

This figure shows the details behind the computations needed to find the coordinates of points along the petal trefoil knot.

We then found the angle between each of the edges using the formula Pi/(number of petals), so for the trefoil, we had Pi/5. The length of each red edge was the radius of the circle, r, which we determined in a later step after finding the heights. Next, we added lines from the top of each petal to the center of our circle, shown in green, and found the angle between these lines using the formula 2Pi/(the number of petals), this was 2Pi/5 for the trefoil. To find the length of the green lines, we treated the curved part of the petal as a semicircle and found its radius by finding the length of the chord between the two red edges and dividing by 2. We then found the length of the other piece by using the Pythagorean theorem and added both measurements together. These calculations are shown in purple in the diagram above, where the length of the green line is given by the following formula: the square root of (r2.-(r*sin(Pi/5))2) plus r*sin(Pi/5). Using trig, this is r(cos(Pi/5)+sin(Pi/5)). We called this quantity x as we labeled the points of the trefoil in the diagram above.

The next step was to find the heights of each point and then convert everything into Cartesian coordinates in order to have points that Cinema4D could understand.

Image of the petal trefoil knot with details of the heights of the points.

This figure shows the heights given to the points on the petal trefoil knot.

We used the orders from the last page of Adams’ paper which gave us the heights for the edges at (0,0). We also assigned heights to the points we found by averaging the heights of the edges to find the height for the top of the petal. We then averaged the height of the top of the petal and the edges at (0,0) to find the height of the end of each edge. For the trefoil, we came up with the heights to the right and then scaled them up by 1.5, so our model would be a bit larger after printing. This also guided our decision to make the radius of our circle (r) be 3 for 3_1 and 5_1. We used r = 3.5 for 6_2 and 6_3.

 At this point, we had found the polar coordinates for all of the points we wished to plot in Cinema4D, so we converted them to decimal approximations with the help of WolframAlpha. The Cartesian coordinates of 3_1 are shown in the table below. We followed a similar process for 5_1, 6_2, and 6_3 knots. Points completed!

Vertex x y z
1 0 0 6
2 -0.927050983 2.853169549 4.875
3 0 4.19040674 3.75
4 0.927050983 2.853169549 2.625
5 0 0 1.5
6 -0.927050983 -2.853169549 2.25
7 -2.463059283 -3.390110266 3
8 -2.427050983 -1.763355757 3.75
9 0 0 4.5
10 2.427050983 1.763355757 3.375
11 3.985313636 1.294906896 2.25
12 3 0 1.125
13 0 0 0
14 -3 0 0.75
15 -3.985313636 1.294906896 1.5
16 -2.427050983 1.763355757 2.25
17 0 0 3
18 2.427050983 -1.763355757 3.75
19 2.463059283 -3.390110266 4.5
20 0.927050983 -2.853169549 5.25
21 0 0 6