There are chains of octahedra and chains of icosahedra that meet end-to-end to form a ring, amounting to a toroidal (or genus-one) deltahedron. Here is Figure 1 in a paper by Michael Elgersma and Stan Wagon, The Quadrahelix: A Nearly Perfect Loop of Tetrahedra [new window; arxiv.org], showing deltahedral torids with 48 (the 8 octahedra) and 144 (the 8 icosahedra) triangular faces.

John Horton Conway created a toroidal deltahedron with fewer triangular faces than required by those above. He said so on the thread here [Google Groups; opens in new window]. In that same thread, Heidi Burgiel gave a simple description of the result (thank you Heidi!) that I have used to construct the beast. The result is below. Click the play button and you'll be taken through the steps that Heidi listed (which I have quoted after the video).

JHC claimed, without proof (but surely without refutation), that the number of faces he used —36— was the minimum possible:

"Martin Gardner asked for the smallest number of equal equilateral triangles that can be used to make an embedded torus, giving some answer that I improved to 36 (if I've just counted them correctly from the model I have here). I'm sure 36 is smallest, and uniquely so.

"Start with an octahedron with one face up and one face down. It will have six "side" faces. Fit three more octahedra face to face with three of these side faces (equally spaced, of course). Looking down on the object, you should see three square-pyramidal shapes pointing up and away from the central octahedron.

"Place tetrahedra face-to-face with the other three side faces of the central octaedron, and pack the spaces between the outer octahedra and the tetrahedra with six more tetrahedra. Remove the central octahedron (I guess you have to drill it out, or partially dissasemble your model at this point) and you have a torus! It rests nicely on the exposed faces of the first three tetrahedra, with the three pyramidal octahedron peaks pointing up and out."