I made a study on how to make a Gyroid which is a minimal surface used a lot when additive manufacturing is OK.

there is a built in function int CATIA for additive manufacturing preparation 22mn40s + use case

https://www.youtube.com/watch?v=oKylME0JhYc

Beyond value for industry, it is a nice surface and it is cool to understand it (I think).

A minimal surface has its mean curvature = zero. So, in any vicinity on the gyroid surface, it is like a saddle! (everywhere like a saddle shape)

A Gyroid has an implicit equation sin x cos y + sin y cos z + sin z cos x = 0 => many many symmetries.

Here is the logic of the modeling

1. create a curve following the implicit equation z=0,  x=[0 ;Pi/4]
2. rotation the curve and join (you have 2 curves)
3. symmetry of the join => to get the base patch 1/48 unit patch
4. symmetry and rotation of the patch to get 1/8 unit patch
5. rotation 180deg against the red curves to get 4/8 unit surface
6. central symmetry to get full unit surface

1. create a curve following the implicit equation z=0,  x=[0 ;Pi/4] -> the yellow curve

2. rotation/180 symmetry the curve and join by swapping the reference axis -> the orange curve

3. symmetry of the join against the green line => to get the base patch 1/48 unit patch -> you get the red curves

and just fill -> check that gaussian map is zero...

4. 180 rotation (red line) and adding red plane symmetry for the yellow patches to get the 1/8 unit

5 rotation 180deg against the red curves to get 4/8 unit surface

6.central symmetry to get full unit surface

What is nice is the discrete gyroid as described in this paper

this 3/4 of 8 unit gyroid pattern

and this is for fun -> meshed - dual mesh - thickened mesh - subdivided mesh