©

In 2018 E. Pegg provided this toroid, which uses just a single shape for faces. These are triangles {aab}, where a = sqrt[5+2 sqrt(2)]/2 = 1.398966 (the blue ones in the pic) and b = 1 (the green ones). The central green triangles of the pic mark the doorways of the corridor, while the outer blue hexagon marks the equator.

Incidence matrix

6 * | 2 2  2 1 0 | 3 2  2  at corridor
* 6 | 0 0  2 1 2 | 0 1  4  at equator
----+------------+-------
2 0 | 6 *  * * * | 2 0  0  a corridor lacings
2 0 | * 6  * * * | 1 1  0  b upper/lower doorway
1 1 | * * 12 * * | 0 1  1  a
1 1 | * *  * 6 * | 0 0  2  b
0 2 | * *  * * 6 | 0 0  2  a along equator
----+------------+-------
3 0 | 2 1  0 0 0 | 6 *  *  {aab}
2 1 | 0 1  2 0 0 | * 6  *  {aab}
1 2 | 0 0  1 1 1 | * * 12  {aab}