Is there a simpler single polygon toroid?

Question

In B.M. Stewart's book Adventures Among the Toroids, toroidal shapes of many sorts are made. One of them is the ring of 8 octahedra, with 48 faces. The toroid is made with a single polygon -- the equilateral triangle.

Are there single non-regular polygons that can make a toroidal shape with fewer than 48 faces? One restriction -- all neighboring polygons must be in different planes, to prevent things like the ring of 8 cubes.

The faces should be non-intersecting. The underlying graph of edges might be one of these, maybe.

Answer 1

Let $y=2, x=\sqrt{5+2 \sqrt{2}}\approx 2.79793 $, Then the following toroid with green $y$ and blue $x$ lengths is made from 24 identical triangles. But there may be something smaller.

Took less than 12 hours for someone to build it.

The net, with green points the 6 outer vertices:

Is there anything smaller than 24 faces? Here's something larger.

Let $y=2, -127 + 124 x^2 - 26 x^4 - 4 x^6 + x^8=0, x\approx 2.31498614558$. Then the following toroid with green $y$ and blue $x$ lengths is made from 32 identical triangles.