Written by Claire Gilreath. Joanne Wang, and Selihom Gobeze (students in Math 383D Knot Theory Spring 2023).
Building Petal Knots in Cinema 4D
Once we had found all of our coordinates for our petal knots, we were excited to actually build the knots in Cinema4D! Of course, this was not without its challenges. We started with 3_1. Our first step was to to import our points to a spline. Naturally, we ran into issues here because we were not aware that the points from the first row are read as the x, y, and z labels, so the data is not actually imported. We fixed this by editing our .txt file to include a row of 0s at the top. Also, one of the points was incorrect and we had to go back to WolframAlpha for a quick fix (this happened a couple of times throughout the process).
At this point, we were not aware that we needed to connect the last point to the first point, so we skipped that step and decided to try the Spline Smooth tool to round out our polygonal edges. We found that the result was not as uniform as we had expected but we decided to keep going to see what would happen after the sweep. We made a circle and decided to set the radius at 0.25cm in order to avoid self-intersections but give the model enough structure to support itself. Then, we made the sweep and looked at our slightly wonky 3_1 knot. We realized that the ends were not connected so we used stitch and sew to fix that (this was a result of failing to connect the last two points of the spline). To the right is our final improved 3_1 knot.
We moved on to the 5_1 knot and followed the same procedure. Once we had built it, Professor Denne looked at our slightly wonky 5_1, and suggested we try to use Chamfer tool instead of the Spline Smooth tool to make the model look smoother and more uniform, with less harsh edges of the petals. In doing this, we discovered that our last point vanished when we tried to Chamfer. We realized that we needed to close the spline by deleting our last point at (0,0,0) and using the Spline Pen to connect the first point to the last point to close the gap. We set the radius of the Chamfer to 3cm and compared our new 5_1 to the wonky one. We decided we liked the Chamfer version better and fixed our 3_1 the same way. This time we set the radius to 5cm, which we found looked much smoother and decided to stick with that for all knots. Our completed 5_1 knot is pictured above.
We repeated this same process with 6_2, expecting everything to be a lot easier now that we had solidified our methods. However, this was not the case. As we rotated around the knot after sweeping, we noticed that we had a self-intersection at one point and two strands that were concerningly close together. Though we’re not entirely sure why this happened, we think the Chamfer forced the two strands too close together. We first tried to make the radius of the circle inside the sweep smaller, but the strands were still touching at 0.17cm, so we decided to manually pull the spline points apart after the Chamfer but before the sweep. This fixed our problem and everything looked good after the sweep this time. Our completed 6_2 knot is pictured above.
As we built 6_3 we were worried about self-intersections, since the (x,y) coordinates were the same as in 6_2. We did not run into any problems this time, perhaps because of the different heights. Below is a picture of 6_3 from four angles, showing the “nice” petal view as well as the less attractive side views.