Triply periodic minimized structures

Minimal surface areas have been researched for more than 200 years. A minimal surface area is a surface that has the smallest area bounded by a given contour. Triply Periodic Minimal Surface (TPMS) are minimal surfaces in all three x,y, and z directions. They are defined by implicit functions, and many of them include trigonometric functions. For example, the Gyroid structure is obtained by evaluating the function of over a finite space: $$f = { sin(x)\cdot cos(y)+ sin(y)\cdot cos(z)+ sin(z) \cdot cos(x) }$$ The marching cubes algorithm is often used to go from the implicit function to the mesh state. There are many types of TPMS, some of which can be seen below:

To make some of those structures manufacturable, it is important to give them thickness. This can easily be done by changing the iso-value of the marching cube function and constrain it in any form of solid. TPMS can then be inscribed inside a primitive.