# How is this "slide-together" built of pseudo-lunes supposed to yield a complete and proper ring torus on assembly?

I was working through the math and graphing of the equations in:

Building a Torus with Villarceau Sections

and made 20 moon-shaped pieces that I attempted to build a torus with.

The problem I see is that there is a major incongruity between what's going on in Figure 3, and what is shown as the fully assembled torus in Figures 7, 8, and 9.

If we extrapolate Figure 3, we would get a shape that is concave on the outside, but convex on the inside. The "slices" are moon-shaped after all.

However, through some major misunderstanding on my part, perhaps due to a failure to illustrate or describe how the congruent slices are made to form a circular cross-section.

I was hoping that someone who reads the paper, or who has previously explored this sort of modeling technique could either point out what I'm doing wrong, or confirm that something is amiss, or outright missing between the beginning of the paper, up to page 97, and the rest of the paper that shows a ring torus - purportedly assembled from the components in the first half of the paper.

Also, since it is not mentioned explicitly in the paper, but is critical to a properly built model, I'd like to know how many pieces are needed to assemble a complete torus - based on the number of slots in a single half-moon piece.

I have 9 slots, and 20 pieces - which seems inadequate to the task. However, this is what seems to be implied by Figures 8 and 9, and the first paragraph of Section 5.

So what have I independently done up to this point? I worked most of the math out in a Desmos calculator to graph the boundaries of the moons, the central curve, and the endpoints of the slots. Villarceau Moons Then I made 2 heavy cardstock templates, cut out 20 light card moons, pierced the 180 holes along the central curves, drew the 180 slot lines, cut the 180 slots, and assembled all of the pieces into a partial model.

I considered offsetting the slots on each piece to give the pieces in the ring a "twist"; however, it's immediately obvious that such a course of action won't work, because the intersecting slots are different sizes, which would result in a mismatch. It's also strongly implied by Figure 3 that rotating the pieces in such a way is NOT how to assemble the model.