In 1990, Sven Lidin and Stefan Larsson found in the associate family of a particular member of the Schwarz H-surfaces a new embedded triply periodic minimal surface, similar in appearance to Alan Schoen‘s Gyroid. Below are corresponding pieces of the H surface, an intermediate member of the associate family, and the Lidinoid.
A much larger piece is shown in the stereo pair below:
There is strong numerical evidence (see Adam Weyhaupt’s paper) that the Gyroid can be deformed into the Lidinoid through embedded minimal surfaces that all have an order 3 rotational symmetry. It is not known whether either surface can be deformed (through embedded minimal surfaces) into any other known triply periodic minimal surface.
A. Weyhaupt: Deformations of the Gyroid and Lidinoid Minimal Surfaces, Pacific Journal of Mathematics 235 (2008), 137–171.