As the Bonnet rotation angle increases, a continuous family of minimal surfaces is generated. Minimal surfaces that are ninety degree Bonnet rotations of each other are called adjoint surfaces. For example, the catenoid and helicoid are adjoints. 45 o rotation. A multiply connected surface must be instead treated as its universal covering surface, as the catenoid-helicoid shows.
Some pairs are adjoints, though, such as the P and D surfaces.
An important property of the adjoint transformation is that mirror planes transform to embedded two-fold rotation axes, and vice versa. For example, the equator of the catenoid turns into the midline of the helicoid. A fact useful for generating TPMS shown in these pages is that the adjoint of an unstable fundamental region may be stable. All the free contact lines on the faces of the kaleidoscopic cells transform to fixed straight line boundaries, which are much more stable. This is the notorious period killing problem of proving a conjectured TPMS exists.