In fact, we have already created these two surfaces at the intermediate stages in weaving torus and nanospring. One can obtain these two kinds of minimal surfaces simply by throwing away all pentagons.
I don't have good examples to illustrate this point, hopefully, we can create some beaded models to demonstrate this. Here is a picture that I tried to show how to make T120. Interestingly, in the intermediate stage, I have a catenoid.