It turns out there are ten topologically distinct spherical weaves with 24 strands and cubic symmetry. This means that each strand maps to each other strand by a symmetry of the model. There are also ten other arrangements of 24 great circles that do not admit a proper weave. I could not find any literature on spherical weaves so I had to do my own analysis (and I could be wrong!). Please let me know if you know of any existing research in this area. Next, I will look into the weaves with dodecahedral symmetry.
This is the first model I have made with 24 strands. All the symmetry points of the cube can be seen in the photos - the square faces correspond to the six large almost-square octagonal holes, the corners correspond to the eight almost-triangular hexagonal holes, and the edges correspond to twelve squished octagonal holes.
This was not an easy project. I first tried with strips of card, but these were too flimsy to make a self-supporting model. I then tried using a double thickness of card, but that was too difficult to work with. Finally I used chair-cane. This was exactly the right stiffness but it was uneven and produced a model that is a bit lopsided. Oh well, it will have to do for now - I still think it looks cool; it is springy and nice to handle too.
