Two versions of a Torus Sliceform
On the left is a twenty four slice version and on the right is a sixteen slice version
The torus is a beautiful design. I am fascinated by the dynamics of its construction. Toruses are made with two types of Villarceau circles which are half moon slices. The two types of Villarceau circles have opposite side slits that are slid into one another to form the torus. Toruses must be made with a flexible material because the Villarceau circles needs to be bent and manipulated to be slid into one another. In the past, I have altered the edges and angles of Villarceau circles and I created two beautiful flowers, a chrysanthemum https://papercraftetc.blogspot.com/2013/10/pom-pom-chrysanthemum-season.html and an amaryllis flower https://papercraftetc.blogspot.com/2014/03/dreaming-of-flowers-amaryllis-torus.html.
Recently, I discovered this paper, Building a Torus with Villarceau Sections by Marıa Garcıa Monera and Juan Monteabout from the University of Valencia, published in the Journal for Geometry and Graphics Volume 15 (2011), No. 1, 93–99. https://www.heldermann-verlag.de/jgg/jgg15/j15h1mone.pdf and I wanted to perfect the toruses that I made in the past. The paper explains the mathematics behind the creation of the torus. Using their formulas, I was able to create an accurate sixteen slice version of the torus. I have included the angle measurements in my files below. In the past, I felt that my sixteen slice torus looked good but I felt that the angles in the Villarceau circle were not accurate. I didn't know the formula to correct the inaccuracy. After discovering this paper, my intuitions were correct. I discovered using their calculations that two of the angles of the sixteen slice torus were off by a tenth of a degree. This goes to show that a slight difference in a complex system can make a big difference.
I have included two versions of the toruses in my files. The twenty four slice version is a more complex design because of the number of Villarceau circles. The sixteen slice version is easier to put together and I recommend completing this design before attempting the twenty four slice version. The cutting and assembly of the sixteen slice version will take about thirty minutes to complete. The twenty four slice will take approximately an hour to put together. Please note, it will require a lot of patience when the final slices are stretched and maneuvered to be put into place. Here is a basic tutorial of the weaving of the Villarceau circles, https://papercraftetc.blogspot.com/2013/10/a-honeycomb-pumpkin-decoration-for-fall.html Once the stack of slices are woven together, the torus must be made into a donut shape and the corresponding slices need to be slid into one another. It is difficult to photograph and I recommend googling a YouTube video on how to make a torus sliceform for further instruction.
Here is the PDF. I used 65 lb cardstock in two different colors, one for each type of Villarceau circle. I recommend double cutting the pattern as the slits do not cut precisely. It was frustrating and time consuming for me to cut the hanging chads after the Villarceau slices had been cut with the Silhouette after just one pass.
Here is the .Studio file.
Here is the SVG.
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